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

TL;DR

This paper shows that a specific SUSY SO(10) GUT model predicts an enhanced dimension-6 proton decay rate, making it detectable at Hyper-Kamiokande, unlike minimal SUSY SU(5) models.

Contribution

It identifies a SUSY SO(10) GUT framework with specific Higgs fields that leads to a larger, observable proton decay rate due to Planck-suppressed SU(5)-breaking VEVs.

Findings

Proton decay width is within Hyper-Kamiokande's detection range.

SU(5)-breaking VEVs are enhanced by Planck-suppressed terms.

The gauge boson masses are smaller, increasing decay rate.

Abstract

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.

Tables

Table 1. Table 1: The p → π 0 ​ e + → 𝑝 superscript 𝜋 0 superscript 𝑒 p\to\pi^{0}e^{+} partial lifetime and the mass spectrum for various input values of λ 1 , λ 2 , κ subscript 𝜆 1 subscript 𝜆 2 𝜅 \lambda_{1},\lambda_{2},\kappa and m χ subscript 𝑚 𝜒 m_{\chi} . The masses and VEV are in units of 10 16 superscript 10 16 10^{16} GeV. From (i-A) to (ii-B), m A < 0 subscript 𝑚 𝐴 0 m_{A}<0 and thus a 24 subscript 𝑎 24 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 subscript 𝑎 24 0 a_{24}>0 , and (i-B) and (ii-B) to the cases with a 24 < 0 subscript 𝑎 24 0 a_{24}<0 . In (iii), m A > 0 subscript 𝑚 𝐴 0 m_{A}>0 and thus a 24 subscript 𝑎 24 a_{24} is complex. We change one of λ 1 , λ 2 , κ subscript 𝜆 1 subscript 𝜆 2 𝜅 \lambda_{1},\lambda_{2},\kappa in (iii ′ ), (iii ′ ′ {}^{\prime}{}^{\prime} ), (iii ′ ′ ′ {}^{\prime}{}^{\prime}{}^{\prime} ).

	$({\lambda }_1,{\lambda }_2,\kappa )$	$m_{\chi }$	$m_A$	$M_X$	$M_Q$	$M_3$	$M_8$	$v_R$	${\tau }_p\left({10}^{34}\mathrm{years}\right)$
(i-A)	$(1,1,1)$	0.1	$-0.47$	$2.8$	$0.55$	$0.25$	0.068	0.24	3.4
(i-B)	$(1,1,1)$	0.1	$-0.46$	$2.8$	$0.61$	$0.25$	0.057	0.25	4.7
(ii-A)	$(1,1,1)$	0.01	$-0.48$	$2.8$	$0.59$	$0.26$	0.069	0.25	4.3
(ii-B)	$(1,1,1)$	0.01	$-0.44$	$2.7$	$0.57$	$0.24$	0.059	0.24	3.7
(iii)	$(1,1,1)$	0.1	$0.46$	$2.8$	$0.58$	$0.25$	0.063	0.24	4.0
(iii′)	$(1,0.5,1)$	0.1	$0.58$	$3.2$	$0.67$	$0.17$	0.043	0.22	7.2
(iii${}_{}{}^{\prime }{}^{\prime }$)	$(0.1,1,1)$	0.1	$0.12$	$2.8$	$0.58$	$0.25$	0.063	0.24	4.0
(iii${}_{}{}^{\prime }{}_{}{}^{\prime }{}^{\prime }$)	$(1,1,0.5)$	0.1	$0.48$	$2.8$	$0.62$	$0.26$	0.066	0.36	5.1

Equations

$$M_X^2/g_5^2$$

$$M_Q^2/g_5^2$$

$$M_U^2/g_5^2$$

$$M_E^2/g_5^2$$

$$M_X:M_Q:M_U:M_E\simeq 5:1:4:6.$$

$$\Gamma \propto A_R^2{\left(\frac{1}{M_X^2}+\frac{1}{M_Q^2}\right)}^2+\frac{4A_L^2}{M_X^4},$$

$${\Gamma }^{SU(5)}:{\Gamma }^{\mathrm{now}}\simeq 1:136,$$

$$W=\frac{1}{2}{m}_A{A}^2+\frac{{\lambda }_1}{M_{\ast }}(A^2{)}^2+\frac{{\lambda }_2}{M_{\ast }}{A}^4+m_{\chi }\chi \bar{\chi }+\kappa A\chi \bar{\chi },$$

$$W{|}_{\mathrm{SM}\mathrm{singlets}}$$

$$a_1=-\frac{m_{\chi }}{\sqrt{5}\kappa },$$

$$a_{24}=\frac{m_{\chi }}{\sqrt{5}\kappa }a,$$

$$v_R{\bar{v}}_R=\frac{4m_{\chi }^3}{25{\kappa }^4{M}_{\ast }}\left({\lambda }_1(1+a^2)+\frac{{\lambda }_2}{5}(1+3a^2-\frac{1}{\sqrt{6}}{a}^3)\right)+\frac{m_{\chi }{m}_A}{5{\kappa }^2},$$

$${\lambda }_1+\frac{3}{5}{\lambda }_2+\frac{5{\kappa }^2{m}_A{M}_{\ast }}{4m_{\chi }^2}-\sqrt{\frac{3}{2}}\frac{{\lambda }_2}{5}a+({\lambda }_1+\frac{7}{30}{\lambda }_2)a^2=0.$$

$$a_{24}^2\sim -\frac{m_A{M}_{\ast }}{4({\lambda }_1+\frac{7}{30}{\lambda }_2)},v_R{\bar{v}}_R\sim \frac{{\lambda }_2}{5\sqrt{30}\kappa }\frac{m_A{a}_{24}}{{\lambda }_1+\frac{7}{30}{\lambda }_2}.$$

$$-2\ln \frac{m_{SUSY1}}{m_Z}$$

$$8\ln \frac{m_{SUSY2}}{m_Z}$$

$$-\frac{2}{{\alpha }_3(m_Z)}+\frac{3}{{\alpha }_2(m_Z)}-\frac{1}{{\alpha }_1(m_Z)}=\frac{1}{2\pi }\left(\frac{12}{5}\ln \frac{M_H}{m_Z}-2\ln \frac{m_{SUSY1}}{m_Z}\right),$$

$$-\frac{2}{{\alpha }_3(m_Z)}-\frac{3}{{\alpha }_2(m_Z)}+\frac{5}{{\alpha }_1(m_Z)}=\frac{1}{2\pi }\left(36\ln \frac{M_G}{m_Z}+8\ln \frac{m_{SUSY2}}{m_Z}\right),$$

$$M_H$$

$$M_G^6$$

$$\sum _i{l}_A^i=0,\sum _i{l}_B^i=2.$$

$$M_H\sim 10^{15}-10^{16}\mathrm{GeV},M_G\sim 2\times 10^{16}\mathrm{GeV},$$

$$M_{{\chi }_Q}:M_{{\chi }_U}:M_{{\chi }_E}\simeq 1:4:6.$$

$$M_8$$

$$M_3$$

$$M_8:M_3\simeq 1:4.$$

$$M_H$$

$$M_G^6$$

$$\frac{M_Q^4}{M_E{M}_U^3}\frac{(M_{{\chi }_E}{M}_{{\chi }_U}^3{)}^{1/2}}{M_{{\chi }_Q}^2}{\left(\frac{M_8}{M_3}\right)}^{5/2}\simeq \frac{1}{576},$$

$$\frac{M_E^2{M}_U^2}{M_Q^4}\frac{M_{{\chi }_Q}^2}{M_{{\chi }_E}{M}_{{\chi }_U}}\simeq 24.$$

$$M_X\simeq M_G\frac{1}{(6{\rho }^2{)}^{1/6}},$$

$$\rho =\frac{M_3}{M_X}\simeq \frac{16}{5}{\lambda }_2\frac{1}{g_5^2}\frac{M_X}{M_{\ast }}.$$

$$\rho \simeq 2{\left(\frac{{\lambda }_2}{g_5^2}\right)}^{3/4}{\left(\frac{M_G}{M_{\ast }}\right)}^{3/4},$$

$$\Gamma =\frac{\pi }{4}\frac{m_p{\alpha }_H^2}{f_{\pi }^2}(1+F+D{)}^2{\alpha }_U^2\left[A_R^2{\left(\frac{1}{M_X^2}+\frac{1}{M_Q^2}\right)}^2+\frac{4A_L^2}{M_X^4}\right],$$

$$M_X:M_3:M_8=1:\rho :\rho /4,$$

$$M_{{\chi }_Q}:M_{{\chi }_U}:M_{{\chi }_E}=1:4:6,$$

Full text

**Detectable dimension-6 proton decay

in SUSY SO(10) GUT at Hyper-Kamiokande **

Naoyuki Haba,  Yukihiro Mimura  and  Toshifumi Yamada

*Institute of Science and Engineering, Shimane University, Matsue 690-8504, Japan *

Abstract

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.

1 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 $\tau(p\to\pi^{0}e^{+})>1.6\times 10^{34}$ years (90% confidence level) [2], and it will be searched up to $6.3\times 10^{34}$ years at $3\sigma$ 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 24H cannot be determined theoretically. The GUT gauge boson mass becomes heavier (the proton lifetime is longer) if the self-coupling of 24H is smaller. 111 In non-SUSY SU(5) GUTs, some choices of GUT Higgses yield models that survive the SK bound and will be explored at HK [16].

The SUSY GUTs also predict the dimension-5 proton decay via colored Higgs exchange [7], such as $p\to K^{+}\bar{\nu}$, whose current bound reads $\tau(p\to K^{+}\bar{\nu})>5.9\times 10^{33}$ 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 $p\to\pi^{0}e^{+}$ 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 $\sim$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 ${\bf 45}_{\rm H}+{\bf 16}_{\rm H}+\overline{\bf 16}_{\rm H}$, 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 $({\bf 3},{\bf 2},1/6)$, which is absent in SU(5) GUT, is about $1/5$ lighter than the GUT gauge boson with SM charge $({\bf 3},{\bf 2},-5/6)$, 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 $({\bf 3},{\bf 2},1/6)$ 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 $p\to\pi^{0}e^{+}$ in the SO(10) model with ${\bf 45}_{\rm H}+{\bf 16}_{\rm H}+\overline{\bf 16}_{\rm H}$ 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. 222 Considering this enhancement and uncertainty of the Yukawa coupling unification, we omit an analysis of the dimension-5 decay in this paper.

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(10) gauge bosons that gain mass via symmetry breaking with ${\bf 45}_{\rm H}+{\bf 16}_{\rm H}+\overline{\bf 16}_{\rm H}$, and show that the dimension-6 proton decay width can be enlarged. In Section 3, we study how the characteristic SO(10) breaking vacua of the model are obtained. In Section 4, GUT-scale threshold corrections are evaluated for the calculation of the dimension-6 proton decay width. In Section 5, a detailed numerical result for the proton lifetime is presented. Section 6 is for the conclusion. In Appendix A, we show the mass spectrum of the multiplets which come from ${\bf 16}_{\rm H}+\overline{\bf 16}_{\rm H}$. In Appendix B, an alternative, renormalizable superpotential with ${\bf 54}_{\rm H}$ representation GUT Higgs field is shown.

2 Spectrum of the SO(10) gauge bosons and proton decay

There are many ways to break the SO(10) gauge symmetry to the SM gauge symmetry. The most economical choice of the Higgs representations to break SO(10) is ${\bf 45}_{\rm H}+{\bf 16}_{\rm H}$. We also introduce $\overline{\bf 16}_{\rm H}$ for $D$-flatness. ${\bf 45}_{\rm H}$ contains two SM singlets: a SU(5) singlet ($a_{1}$) and a SM singlet in SU(5) adjoint ($a_{24}$). By general vacuum expectation values (VEVs) of $a_{1}$ and $a_{24}$ (without particular relations between them), SO(10) is broken down to $SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}\times U(1)^{\prime}$. The SM singlet in ${\bf 16}_{\rm H}$ ($v_{R}$), which is a SU(5) singlet, develops a VEV to break $U(1)^{\prime}$ (the singlet in $\overline{{\bf 16}}_{\rm H}$ ($\bar{v}_{R}$) also gains a VEV with $|v_{R}|=|\bar{v}_{R}|$ to keep D-flatness). Due to the absence of cubic term of ${\bf 45}_{\rm H}$, $a_{24}$ cannot acquire a VEV from the renormalizable superpotential. However, by introducing non-renormalizable, quartic terms of ${\bf 45}_{\rm H}$, $a_{24}$ can acquire a VEV to break SU(5).

The GUT gauge boson masses generated by the VEVs of ${\bf 45}_{\rm H}$, ${\bf 16}_{\rm H}$ and $\overline{\bf 16}_{\rm H}$ are

[TABLE]

where $X$, $Q$, $U$ and $E$ denote SO(10) gauge bosons whose SM charges are $X:({\bf 3},{\bf 2},-5/6)$, $Q:({\bf 3},{\bf 2},1/6)$, $U:({\bf 3},{\bf 1},2/3)$ and $E:({\bf 1},{\bf 1},1)$, and $a_{1}$, $a_{24}$ and $v_{R}$ are the VEVs of canonically-normalized SM singlets. The extra U(1) gauge boson mass is $g_{5}\sqrt{5/2}v_{R}$.

When $a_{24}\gg a_{1},v_{R}$, we obtain

[TABLE]

This ratio is easily obtained by the rule $(3+2):(3-2):(2+2):(3+3)$.

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

[TABLE]

where $A_{L,R}$ are the renormalization factors for $q\ell(u^{c})^{\dagger}(d^{c})^{\dagger}$ and $qq(e^{c})^{\dagger}(u^{c})^{\dagger}$ operators. One finds that the $Q$ gauge boson exchange gives much larger contribution when $M_{X}:M_{Q}\simeq 5:1$. The ratio of the decay width in SU(5) GUT ($M_{Q}\to\infty$) and in the SO(10) GUT with $a_{24}\gg a_{1},v_{R}$ (now) is

[TABLE]

for $A_{L}\simeq A_{R}$, if the $X$ gauge boson masses are the same. Since the naive prediction of $p\to\pi^{0}e^{+}$ partial lifetime in SU(5) GUT is $\tau_{p}\sim 10^{36}$ years, the prediction in the SO(10) with $a_{24}\gg a_{1},v_{R}$ is $10^{34}$ years, which is on the current experimental bound at SK.

3 SO(10) breaking vacua in the model

We consider a superpotential for the GUT breaking Higgs fields ${\bf 45}_{\rm H}$ ($A$), ${\bf 16}_{\rm H}$ ($\chi$) and $\overline{{\bf 16}}_{\rm H}$ ($\bar{\chi}$),

[TABLE]

where we define $A^{2}\equiv A_{ab}A_{ab}/2$, and $A^{4}\equiv A_{ac}A_{ad}A_{bc}A_{bd}/2$ 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 $\chi$ (Clebsch-Gordan coefficients for $\bf 16$ representation can be found in [17]) is given by

[TABLE]

The $F$-flat conditions read

[TABLE]

where $a$ is a solution to

[TABLE]

In Eq. (10), the condition $\partial W/\partial v_{R}=(m_{\chi}+\sqrt{5}\kappa a_{1})\bar{v}_{R}$ fixes the VEV of $a_{1}$ to be around $m_{\chi}$. 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 $|m_{A}M_{*}/m_{\chi}^{2}|\gg 1$, in which case we obtain

[TABLE]

Thus, vacua with $|v_{R}|,|a_{1}|\ll|a_{24}|$ are obtained333 If one adds a non-renormalizable term $(\chi\bar{\chi})^{2}/M_{*}$ to the superpotential, new vacua with $v_{R}\sim\sqrt{m_{\chi}M_{*}}$ appear. However, the vacua we obtain in the main text remain stable with a correction of $O(v_{R}^{2}/M_{*}^{2})$.

with a feasible assumption $m_{\chi},m_{A}\ll M_{*}$.

4 GUT-scale threshold corrections for the gauge coupling unification

The gauge coupling unification conditions [24] in SUSY SO(10) GUT are written as444 The SUSY particle threshold contributions in the respective equations are more precisely written as

$\displaystyle-2\ln\frac{m_{\rm SUSY1}}{m_{Z}}$ $\displaystyle=$ $\displaystyle-\frac{8}{5}\ln\frac{\mu_{H}}{m_{Z}}-\frac{2}{5}\ln\frac{m_{H}}{m_{Z}}+4\ln\frac{M_{\tilde{g}}}{M_{\tilde{w}}}+\frac{3}{5}\ln\frac{m_{\tilde{q}^{c}}^{3}m_{\tilde{d}^{c}}^{2}m_{\tilde{e}^{c}}}{m_{\tilde{q}}^{4}m_{\tilde{\ell}}^{2}},$

(15)

$\displaystyle 8\ln\frac{m_{\rm SUSY2}}{m_{Z}}$ $\displaystyle=$ $\displaystyle 4\ln\frac{M_{\tilde{g}}M_{\tilde{w}}}{m_{Z}^{2}}+3\ln\frac{m_{\tilde{q}}^{2}}{m_{\tilde{u}}m_{\tilde{e}^{c}}},$

(16)

where $\mu_{H}$, $m_{H}$ are higgsino and heavier Higgs masses, $M_{\tilde{g}}$ and $M_{\tilde{w}}$ are gluino and wino masses. From these equations, one finds that the colored Higgs mass is larger for a smaller ratio of $M_{\tilde{g}}/M_{\tilde{w}}$, and the unification scale $M_{G}$ becomes smaller for heavier wino and gluino masses.

[TABLE]

[TABLE]

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 $l_{A}=\frac{5}{12}(2l_{3}-3l_{2}+l_{1})$ and $l_{B}=\frac{1}{6}(2l_{3}+3l_{2}-5l_{1})$ where $l_{i}$ gives the beta coefficient contribution of the respective multiplet, $l_{i}=\Delta b_{i}^{\rm SUSY}$. Because the would-be-Goldstone modes which are eaten by the gauge bosons lack from the multiplets, we obtain

[TABLE]

The RGEs give

[TABLE]

and the GUT gauge boson and colored Higgs masses depend on threshold corrections of GUT-scale particles.

The $\bf 45_{\rm H}$ and $\bf 16_{\rm H}$ representations are decomposed under SU(5) as ${\bf 45}_{\rm H}={\bf 24}+{\bf 10}+\overline{\bf 10}+{\bf 1}$. and ${\bf 16}_{\rm H}={\bf 10}+\overline{\bf 5}+{\bf 1}$. One linear combination of the ${\bf 10}$’s (and $\overline{\bf 10}$’s) from $\bf 45_{\rm H}$ and $\bf 16_{\rm H}$ ($\overline{\bf 16}_{\rm H}$) is absorbed by the gauge bosons $Q,U,E$. For $|v_{R}|\ll|a_{24}|$, the linear combination to be absorbed mainly comes from $\bf 45_{\rm H}$. The other linear combination is a physical mode and we denote its components by $\chi_{Q},\chi_{U},\chi_{E}$ (which respectively have the same SM charge as $Q,U,E$). For $|m_{A}|,|m_{\chi}|\ll M_{*}$, their masses satisfy the ratio (see Appendix A for the derivation)

[TABLE]

The $\bf 24$ representation in $\bf 45_{\rm H}$ contains a SU(3)c adjoint $({\bf 8},{\bf 1},0)$ and a SU(2)L adjoint $({\bf 1},{\bf 3},0)$ as physical modes. Their masses can be calculated (using the minimization conditions) as

[TABLE]

and when $|a_{1}|\ll|a_{24}|$, we find

[TABLE]

The other physical Higgs modes are the ${\bf 5}+\bar{\bf 5}$ in ${\bf 16}_{\rm H}+\overline{\bf 16}_{\rm H}$. We additionally introduce two $\bf 10_{\rm H}$ representations to generate renormalizable Yukawa couplings that give fermion masses after electroweak symmetry breaking. Then, there are three heavy colored Higgs fields $H_{C}$, $H_{T_{1,2}}$, and two heavy Higgs doublets $H_{D_{1,2}}$ around the GUT scale.

Now the gauge coupling unification conditions are specified as

[TABLE]

In the vacua with $|v_{R}|,|a_{1}|\ll|a_{24}|$, we obtain from Eqs. (5),(23),(26),

[TABLE]

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 $M_{G}^{6}=M_{X}^{4}M_{8}M_{3}$ and $M_{8}=M_{3}=\lambda M_{X}$ where $\lambda$ is proportional to the self-coupling of the SU(5) adjoint representation. $\lambda$ is arbitrary unless it far exceeds $O(1)$, and people often assume $\lambda\sim 1$, which gives $M_{X}\sim M_{G}$. In the current SO(10)-breaking vacua $|v_{R}|,|a_{1}|\ll|a_{24}|$, if we write $4M_{8}\simeq M_{3}=\rho M_{X}$, $\rho$ 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),

[TABLE]

and from Eqs. (1),(25),

[TABLE]

Therefore we find

[TABLE]

which equals 0.1 for $\lambda_{2}=1$ and $M_{*}=2\times 10^{18}$ GeV. It follows that $M_{X}$ is a little larger than $M_{G}$. Nevertheless, the $Q$ gauge boson satisfying $M_{X}:M_{Q}\simeq 5:1$ enhances the dimension-6 proton decay width compared to the minimal SU(5) model.

To summarize, in the SO(10)-breaking vacua with $|a_{24}|\gg|a_{1}|,|v_{R}|$, 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 $\tau_{p}\sim 10^{34}$ 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 $M_{\tilde{g}}/M_{\tilde{w}}$ smaller, and enhancement of the dimension-6 proton decay is achieved by increasing their product $M_{\tilde{g}}M_{\tilde{w}}$, 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 $|a_{24}|\gg|a_{1}|,|v_{R}|$, 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 ${\bf 16}_{\rm H}$ is replaced by $\bf 126_{\rm H}$ representation. In this case, when a vacuum with $|a_{1}|\ll|a_{24}|$ is chosen, $({\bf 6},{\bf 3},1/3)$ 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.

5 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, $\alpha_{s}^{(5)}(M_{Z})^{\overline{\rm MS}}=0.1181\pm 0.0011$ [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 $+3\sigma$. 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 $p\to\pi^{0}e^{+}$ is [21]

[TABLE]

where we use proton mass $m_{p}=0.9383$ GeV, chiral Lagrangian parameters $F=0.46$, $D=0.80$, hadron matrix element for proton decay $\alpha_{H}=-0.014\ {\rm GeV}^{3}$ at 2 GeV [22], decay constant $f_{\pi}=0.1307$ GeV, renormalization factors down to 2 GeV, $A_{L}=2.91$, $A_{R}=2.78$ (The two-loop renormalization factors are calculated in [23]). From the SO(10) gauge coupling unification, we obtain $1/\alpha_{U}(M_{X})=4\pi/g_{5}^{2}\simeq 25.2$.

Before presenting the main result, we show an estimate on the partial proton lifetime. Under the approximations with

[TABLE]

and

[TABLE]

the partial proton lifetime is found to be

[TABLE]

As discussed in the previous section, $\rho\ll 1$ 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 $\rho\simeq 0.1$ for $\lambda_{2}=1$ and $M_{*}=2\times 10^{18}$ 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 $M_{8}=M_{3}=\lambda M_{X}$ where $\lambda$ is proportional to the self-coupling of the adjoint field that breaks SU(5). Then, the partial proton lifetime is found to be

[TABLE]

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 $\rho=0.1$ and $\lambda=1$.

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 $M_{*}=2\times 10^{18}$ GeV, and take benchmark values for $\lambda_{1},\lambda_{2},\kappa$ and $m_{\chi}$. We solve the $F$-flat conditions Eqs. (10)-(13) and the unification conditions by varying $m_{A}$ and the colored Higgs mass. Since Eq. (13) is a quadratic equation, there are two solutions. If $m_{A}<0$, $a_{24}$ is real and the mass spectrum splits into two, both of which are tabulated. If $m_{A}>0$, $a_{24}$ is complex and the two solutions yield the same mass spectrum in terms of the absolute values.

From (iii) and (iii${}^{\prime}{}^{\prime}$) of Table 1, we find that the mass spectrum is not sensitive to $\lambda_{1}$. This is because the relation $|a_{1}|,|v_{R}|\ll|a_{24}|$ gives $M_{3}/a_{24}\propto\lambda_{2}a_{24}/{M_{*}}$. Although $a_{24}$ depends on $\lambda_{1}$, the ratio $M_{3}/M_{X}$ does not depend on $\lambda_{1}$ for $|a_{1}|,|v_{R}|\ll|a_{24}|$. As a result, once $M_{X}$ is chosen to realize the gauge coupling unification, the mass spectrum is almost independent of $\lambda_{1}$. On the other hand, when $\lambda_{2}$ is smaller, the SU(3)c adjoint and SU(2)L adjoint particles become lighter ($\rho=M_{3}/M_{X}$ 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 $\lambda_{2}\sim 1$, the dimension-6 proton decay is detectable at HK.

In the benchmarks of Table 1, the effective colored Higgs mass, $M_{H_{T}}=\frac{M_{T_{1}}M_{T_{2}}M_{H_{C}}}{M_{D_{1}}M_{D_{2}}}$, is $2\times 10^{17}$ 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.

6 Conclusion

We have studied the dimension-6 proton decay in a SUSY SO(10) GUT with only ${\bf 45}_{\rm H}+{\bf 16}_{\rm H}+{\bf\overline{16}}_{\rm H}$ GUT-breaking Higgs fields. Since the SU(5)-breaking VEV of ${\bf 45}_{\rm H}$ is induced by the Planck-suppressed, quartic superpotential for ${\bf 45}_{\rm H}$, this VEV is larger than the SU(5)-conserving VEVs. This results in a $1/5$ suppression of the $Q({\bf 3},{\bf 2},1/6)$ gauge boson mass compared to the $X({\bf 3},{\bf 2},-5/6)$ gauge boson mass. On the other hand, the masses of the SU(3)c adjoint and SU(2)L adjoint particles from 45H are much smaller than the $X$ gauge boson mass and this enhances the latter when the unification condition is fulfilled. Still, the mass of the $Q$ gauge boson can be below $0.6\times 10^{16}$ GeV (for $\lambda_{2}=1$ and $M_{*}=2\times 10^{18}$ GeV) and can thus give rise to a fast dimension-6 proton decay detectable at Hyper-Kamiokande.

Appendix A Mass spectrum

${\bf 45}_{\rm H}(A)$ and ${\bf 16}_{\rm H}(\chi)$ representations are decomposed under SU(5) as ${\bf 45}_{\rm H}={\bf 24}+{\bf 10}+\overline{\bf 10}+{\bf 1}$ and ${\bf 16}_{\rm H}={\bf 10}+\overline{\bf 5}+{\bf 1}$. One linear combination of the ${\bf 10}$’s (and $\overline{\bf 10}$’s) from $\bf 45_{\rm H}$ and $\bf 16_{\rm H}$ ($\overline{\bf 16}_{\rm H}$) is absorbed by GUT gauge bosons, $Q:({\bf 3},{\bf 2},1/6)$, $U:(\bar{\bf 3},{\bf 1},-2/3)$, and $E:({\bf 1},{\bf 1},1)$. The other linear combination yields physical modes $\chi_{Q}:({\bf 3},{\bf 2},1/6)$, $\chi_{U}:(\bar{\bf 3},{\bf 1},-2/3)$, and $\chi_{E}:({\bf 1},{\bf 1},1)$. The mass matrix of each component of the ${\bf 10}+\overline{\bf 10}$’s can be written as

[TABLE]

where

[TABLE]

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 $M_{AA}+M_{\chi\bar{\chi}}$. In the limit with $m_{A},m_{\chi}\ll M_{*}$, $M_{\chi\bar{\chi}}$ dominates, but $M_{AA}$ can be non-negligible for $\chi_{Q}$ due to the large factor $D_{Q}/C_{Q}$. Using the minimization condition, we obtain $M_{AA}\simeq m_{A}(7-D_{Q,U,E})\lambda_{2}/(30\lambda_{1}+7\lambda_{2})$ for $m_{A},m_{\chi}\ll M_{*}$.

The masses of isospin doublet and color triplet Higgses are obtained from the superpotential

[TABLE]

and the mass term is

[TABLE]

[TABLE]

where

[TABLE]

and $(c_{D},c_{T})=(3,-2)$. The doublet-triplet splitting needs fine-tuning. Without loss of generality, $\lambda_{\chi}^{1}$ is set to zero by a rotation of $(H_{1},H_{2})$. In this basis, by the fine-tuning $M_{11}=M_{12}+\lambda_{H}A_{D}=0$, 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 $\sim 5/3\lambda_{H}A_{D}$ for $|a_{1}|,|v_{R}|\ll|a_{24}|$.

Appendix B Renormalizable model obtained by employing ${\bf 54}_{\rm H}$

In the main text, we have considered the model with ${\bf 45}_{\rm H}+{\bf 16}_{\rm H}+\overline{\bf 16}_{\rm H}$ and with non-renormalizable quartic terms of $\bf 45_{\rm H}$. In this appendix, for readers who prefer renormalizable models, we show that a renormalizable superpotential with $\bf 54_{\rm H}$ (whose SM singlet component is denoted by $E$) can also provide the wanted vacua where $|a_{24}|\gg|a_{1}|,|v_{R}|$ (and $|a_{24}|\gg|E|$).

The superpotential for the SM singlets is

[TABLE]

From the $F$-flat conditions, we obtain

[TABLE]

where $a$ is a solution of the following equation:

[TABLE]

Vacua with $|E|,|a_{1}|,|v_{R}|\ll|a_{24}|$ are obtained by assuming $m_{\chi},m_{A}\ll m_{E}$, which gives

[TABLE]

The $\bf 54_{\rm H}$ is decomposed as ${\bf 54}_{\rm H}={\bf 24}+{\bf 15}+\overline{\bf 15}$ under SU(5). The mass matrices of the adjoint representations after SU(5) breaking are

[TABLE]

for $({\bf 8},{\bf 1},0)$ and

[TABLE]

for $({\bf 1},{\bf 3},0)$. For $a\gg 1$, $\kappa_{2}E\simeq-m_{A}$, and we obtain the masses of the lighter adjoint fields (using $2m_{A}m_{E}\simeq(\kappa_{2}a_{24})^{2}$) as $M_{8}\simeq(3-4\times 2)m_{A}=-5m_{A}$ and $M_{3}\simeq(-2-9\times 2)m_{A}=-20m_{A}$, and hence $M_{8}:M_{3}\simeq 1:4$. We have thus verified that the mass ratio $M_{8}:M_{3}$ is the same as the model with the non-renormalizable terms, which is obtained by integrating out $\bf 54_{\rm H}$.

Acknowledgement

This work is partially supported by Scientific Grants by the Ministry of Education, Culture, Sports, Science and Technology of Japan, Nos. 17K05415, 18H04590 and 19H051061 (NH), and No. 19K147101 (TY).