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JHEP02(2021)181 . . + ν K SU(5) SU(5) SU(5) → Springer p by nonrenor- October 8, 2020 January 7, 2021 : December 2, 2020 R : February 22, 2021 : , and : + U(1) ν π Received Revised → Accepted , p Published , 0 b K ) + , µ + Published for SISSA by e https://doi.org/10.1007/JHEP02(2021)181 ( → p , and Qaisar Shaﬁ 0 [email protected] π a , ) + , µ + e symmetry which plays an essential role in implementing ( R → p U(1) . 3 Mansoor Ur Rehman a . 2010.01665 The Authors. c Beyond Standard Model, GUT, Supersymmetric Eﬀective Theories, Neutrino

We explore proton decay in a class of realistic supersymmetric flipped , SU(5) [email protected] symmetry is utilized to show how such intermediate scales can arise in flipped University of Delaware, Newark, DEE-mail: 19716, U.S.A. [email protected] Department of Physics, Quaid-i-AzamIslamabad University, 45320, Pakistan Bartol Research Institute, Department of Physics and Astronomy, 4 b a Open Access Article funded by SCOAP ArXiv ePrint: and flipped Keywords: Physics decay channels that include We identify regions of thetestable parameter at space Hyper-Kamiokande that and yieldgauge proton coupling other unification lifetime next in estimates generation the whichZ experiments. presence are of intermediate We scale discussFinally, particles we how is compare realized, our and predictions a for proton decay with previous work based on models supplemented by a hybrid inflation. TwoI distinct seesaw, neutrino are mass identified, models,malizable with superpotential based the terms. on latter inverseset arising Depending of seesaw from intermediate on and the scale the type breaking color neutrino triplets of from mass the model Higgs an superfields appropriate play a key role in proton Abstract: Maria Mehmood, Observable proton decay in flipped SU(5) JHEP02(2021)181 ]. 32 and SU(5) ], and Hyper- 9 , gauge symmetry, 8 has been under in- SU(5) ]. This observation leads SU(5) 24 ], DUNE [ – 7 parity. 21 R 6 ], and the anticipated experimental 6 – 1 matter parity. In order to study the con- 2 Z 15 2 symmetry ] based on the flipped R ] is based on a no-scale supergravity model of – 1 – 15 37 32 – 35 (4-2-2) model mediated by color triplets of intermedi- R 18 symmetry and modified , in minimal supersymmetric ν 2 symmetry and + Z R SU(2) K × ] should provide valuable information for comparing the proton → L U(1) ] an exciting possibility of observable proton decay from a super- 10 p ] now also favored by LHC searches [ 33 4 ]. This decay mode can be suppressed by assuming suitably large 20 ] models via the dimension six operators as recently discussed in [ SU(2) – model considered in [ 12 31 , × 1 13 – c 20 11 ]. These studies have prompted the present paper where we consider a su- 25 [ 34 SU(5) symmetry and color triplet masses violation and proton decay SU(4) 4 R Z SU(5) In a recent paper [ 4.1 ate mass range was identified.as This shown 4-2-2 in model [ nicelypersymmetric implements shifted hybrid hybrid inflation inflation, modelsupplemented [ by a global tributions to proton decay from the color triplets of 5-plet and 10-plet Higgses, we employ to an interesting comparisonflipped of gauge boson mediatedThe proton flipped decay modes in inflation with an approximate symmetric decay predictions by GUT models.five operators In in this supersymmetric regard GUTsdominant proton has decay decay been mode, induced a by subjecttense the of dimension great scrutiny interest. [ Thesfermion expected masses [ Proton decay is rightlyels considered of an Grand important Unifieddecay observable channels Theories and by (GUTs). Super-Kamiokande discriminatorresults (Super-K) for The from [ mod- the current next lifetime generationKamiokande bounds (Hyper-K) experiments [ such on as various JUNO proton [ 1 Introduction 5 U(1) 6 Gauge coupling unification 7 Conclusion 3 Neutrino masses 4 Proton decay in FSU(5) with U(1) Contents 1 Introduction 2 Supersymmetric flipped SU(5) model JHEP02(2021)181 - R , the SU(5) charge, ]. The 31 X – SU(5) -symmetric R 25 [ U(1) ) belong in the . One is mostly c X gauge symmetry 3 . The conjugate N -symmetry violation U(1) R symmetry violation at × FSU(5) FSU(5) R ] where this mode is highly we discuss proton decay in 39 SU(5) , 4 ≡ we discuss proton decay arising 38 . Our conclusions are summarized , are presented. Especially, a unique 5 6 32 we briefly describe the flipped [ 2 SU(5) FSU(5) SU(5) are used to break the – 2 – . Here and later, if necessary, the 1 H − symmetry for naturally realizing intermediate mass -symmetric superpotential and some of its uniquely 4 from the renormalizable interactions. The estimates 10 R Z , FSU(5) 1 H ] where one of the color-triplets in Higgs 5-plet becomes 10 38 symmetry can naturally generate intermediate scale masses LLRR 4 Z model mediated via both color triplets and the superheavy gauge decay is found to serve as an additional discriminator between the ν + gauge group is defined as . The electroweak breaking is accomplished through the electroweak K ) SU(5) -symmetric model which leads to proton decay modes in the observable representations are labeled with superscripts. In contrast to → representations of R -symmetry breaking terms at nonrenormalizable level to generate the right GeV 5 p SU(5) R 1 16 . -symmetric interactions, and the second model assumes R 7 10 FSU(5) -symmetry breaking effects. This decay is especially relevant for the second model and R 3 ' ( − , of 5 The layout of this paper is as follows: in section ) , 1 X ( q right handed neutrinos arepair required of GUT by Higgs theto superfields, gauge the symmetry minimal in supersymmetricscale standard model (MSSM) gauge symmetry at the GUT 2 Supersymmetric flipped SU(5)The model Flipped MSSM matter superfields including the right10 handed neutrino superfield ( due to which allows handed neutrino masses. Theintermediate issue mass color of triplets gauge is couplingin discussed section unification in in section the presence of these for the proton partialrange lifetimes of for Hyper-K the alongcouplings. various with The channels role the of are lower anfor additional presented bounds the in on color the the tripletsfrom observable color is the triplet briefly mixing masses highlighted. of and color-triplets relevant In in section the 5-plet and 10-plet Higgs which becomes possible based on in the superpotential at nonrenormalizable level.symmetric flipped In section bosons. We mainly focusnonflipping on operators mediation of by the type color triplets which occurs via the chirality model a successful realizationintermediate of mass gauge color coupling triplets. unification is achieved in themodel presence including of its field content,attractive the features. Two models of neutrino masses are described in section prediction for present model and previous modelssuppressed. of An flipped additional for the color tripletsmodel from recently the considered Higgs in 5-plets.naturally [ light This and is contributes only in to contrast the to charged another lepton channels. Lastly, in the present nism with extra gaugenonrenormalizable singlets, while level the in second theWe model superpotential discuss assumes and the employs therange type-I mediated seesaw by mechanism. color tripletsvarious branching from fractions the and Higgs comparison with multiplets. The distinctive predictions of two models of light neutrino masses. The first model utilizes an inverse seesaw mecha- JHEP02(2021)181 . 2 Z (2.1) × in the R c with the N U(1) ↔ , gauge indices c multiplets into 1 1 1 2 E HN Z +1 +1 +1 +1 +1 − − − FSU(5) at the GUT scale W SU(5) and + ] c 2 FSU(5) − h D 37 5 SU(5) – 3 ) ↔ − j ⊂ 35 5 c R 1 1 1 0 0 0 0 0 5 i ( L U q 1 ) ]. e ( ij gauge symmetry breaking as the y 27 SU(2) + 2 h × playing the role of inflaton. The com- 5 c 3 2 h is given by [ S − j 5 model [ FSU(5) 1 1 5 ) is relevant for hybrid inflation with the 3) 6) 3) 2) 1 i 2) / / 3) − H 3) / 6) / 2) 3) 3) / 1 6) 1 Y / . It is clear from the table that we can obtain / 1 / 1 / / / SU(3) 2.1 , acquires a nonzero vacuum expectation val- 1 10 / 1 2 10 ) 1 1 − − − 1 − – 3 – − × − − H SU(5) − H u,ν , along their respective MSSM singlet directions. L ( , of and charge assignments under MSSM and ij 10 2 2 10 5 y , (1 1 0) g (1 1 0) 3 1 1 (1 1 0) (1 1 1) 3 1 1 mutliplets by flipping 3 1 1 8 λ × M (1 1 0) 1 H ( (3 2 1 c 3 1 + c (1 2 1 ( c ( 3 2 H c 2 (3 2 1 H ( c c (3 1 . The decomposition of various S ( h u = c (3 1 (1 2 H H 2 c E N 3 + 10 (1 2 N 2 h = Q d N h c H H D D − h H Q M D U 5 L 2 FSU(5) i 5 H 32 D Q D , − h 1 j g − c H 2 5 defines the gauge symmetry breaking scale and is related 1 − h 10 FSU(5) N 1 H − H 5 1 i c H M 10 10 10 N ) 1 H h 1 H d ( ij symmetry listed in table 10 y 10 = ) 2 8 1 8 λ X i ( Z 1 1 GeV. q 2 3 1 1 H 5 2 h + + are real and positive dimensionless couplings. The − H − H present in κS − h − S × 1 5 16 10 5 5 10 10 u κ 10 = R 10 1 H SU(5) ,H W ≈ 10 d and U(1) h H λ M 5 . The superfield content of g λ, = The superpotential suitable for supersymmetric hybrid inflation in G pletion of hybridconjugate inflation pair is of followed 10-pletues by Higgses, (vev), the Here the massto parameter the unified gaugeM coupling, where will be suppressed.scalar The component of first the term gauge of singlet eq. superfield ( additional doublets, their MSSM components are shownthe in table MSSM decomposition of corresponding multiplets of the standard Table 1 JHEP02(2021)181 . R R ), d , is , in b (3.2) (3.1) S ,H ) = 1 U(1) U(1) a matter HN u a S S 2 H W ( ] with the ab Z R µ respectively 45 – basis can now ) 43 λ M S term, however, is ]. The significance , c µ ) provide the Dirac 29 and 2.1 term to all orders while ]. Note that the N,N λ M ( 2 h 40 5 , 2 − h    5 for successful realization of susy 0 γM S , assuming even matter parity for , ]. The MSSM ) 1 H 35 h.c u,ν 0 10 ( + 1 i γM µ ) residing in the Higgs superfield pairs, m b h 10 S ) a D a S , S u,ν 0 0 matter parity. The key feature of ( c H ai † P , in third line of eq. ( 2 . 2 ) γ Σ – 4 – which have odd matter-parity with m m 2 e D . However, their contribution to proton decay . However, an explicit mass term, ) provide heavy masses Z ( / ij mass term [ 5 a    a 3 ab = 1 S S , y 2 h y 3 2.1 m ) = 5 − d ) ( ij , appear at the renormalizable level. Although 2 ab I ¯ 5 HN 1 i S y 3 − h , ) and ( , y 5 c − W ) h 10 ∼ ¯ 5 2 u,ν 1 H h − ab ( ij ,D N,N 5 ( y µ 10 c H 1 H a ], we can employ a inverse seesaw mechanism [ M D S 10 . It is important to note that the electroweak doublets, ( ) 42 , 2 h 5 and 41 and -symmetry. We, therefore, include a spurion gauge singlet superfield , 1 3 3 R − H . Other terms at the nonrenormalizable level relevant for proton decay i − − . An intermediate scale supersymmetry breaking by the spurion field, 3 5 ¯ 5 10 , 1 h 2 , can lead to ( 2 5 P , 10 1 H 1 m do not acquire mass from these terms. This ultimately provides the solution of 2 and 10 / 10 = 1 ) ) (Σ) = 2 3 symmetry is quite evident here as it forbids the , 2 h 1 H 2 1 i R 5 m ) is responsible for generating the heavy Majorana neutrino masses necessary for − h R , i, a 10 5 10 2 a 1 , 2.1 − h A mass matrix for neutrinos and gauge singlet fields in the The Yukawa couplings, The terms in the second line of eq. ( S − H i ∼ 1 H U(1) with (5 through the Kähler potential term, Σ 10 F h be written as rates is highly suppressed.term To implement for a the double seesaw gaugenot mechanism allowed singlet we due also superfields to need aΣ mass Σ izable level, where are In order to accommodateneutrino the oscillations light [ neutrino masseshelp responsible of for extra solar gauge andThis singlet atmospheric allows superfields us to include the following additional term in the superpotential at renormal- forbidding the dangerous dimensioneq. ( four proton decaythe implementation terms. of seesaw The mechanism last as described term, in the next section. 3 Neutrino masses masses for all fermions.tion with The proton discussion decay ofS is tiny included neutrino in massessymmetric the and these next its terms sections. possible areparity connec- Some forbidden lies additional by in terms making such the as, lightest supersymmetric particle a dark matter candidate while keeping the electroweak Higgs doublets massless, anddecay by mediated also avoiding via dimension five the proton assumed expected to be generatedsymmetry also by forbids the the Giudice-Masiero quadratichybrid and mechanism inflation. cubic [ terms of to the color triplet(10 pairs ( in doublet-triplet splitting problem via theof missing partner mechanism [ JHEP02(2021)181 . ] ) ' ' 7 ) 47 and − , (3.5) (3.4) (3.3) M c u,ν 10 ( 46 N ! × m ) 1 8 . − H 1 . This is in , 10 matter parity 9 † N 1 − 2 U Z ν 10 (10 P · m × GeV. ) m ∗ N ) 1 9 . 9 U symmetry these values − H 2 , 4 10 10 9 = Z 1 , − × ! is assumed. See refs. [ 5 10 , . (10 5 | . The mixed states of ) diag ν are real and diagonal, 2 1 4 ×

, can be determined as a function , 3 m µ M 7 u,ν 6 2 10 ( . − 4 ! N γ ja ) 5 10 U m (1 γ c ∼ 1 1 P and ν + u,ν × where rapid proton decay mediated 2 − H ( − u ) m ) 1 M ! | 5 . υ diag m γ 10 2 ) 4 a ( ) 1 ) , /M 2 1 µ , namely − H 7 µ u,ν ' P − H ( u,ν N 1 0 P M ( y 10 γ 10 m − U 10 basis can be written as ( m )

m 1 = × . Applying the inverse seesaw mechanism with ) T – 5 – ) 4

c ) 3 8 γ 3 γ . (10 ( ) = u,ν (2 , µ (

+ ) eV, for 2 c u,ν 1 m T ( N,N 4 ! symmetry breaking terms at the nonrenormalizable γ ( , µ , the mixing matrix 3 diag 05) 1 m . ia − N,N R + µ ( 0 γ 5 ( , = ' 1 , are the dimensionless matrices with family indices sup- ! M 4 ν 2 10 , in term of the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) ) P U(1) m diag 1 3 TeV and obtain normal mass hierarchy for the light neutrinos, 1 H 0086 γM , m . − H 2 . This enables us to estimate all relevant proton decay rates 0 P 10 10 = , , 10 † N 1 1 H 7 m ∗ µ ∼ PMNS 1 H , U − is a global symmetry it could be broken in the hidden sector while ∗ 10 U L a 10 µ U (10

R = 0 and ' 3 ×

equal to a unit matrix. This also allows us to write the mixing matrix 8 -symmetry breaking occurs in such a way that it only allows terms with ) = γ k 1 0 t ∗ . N 4 R γ N U(1) + U (1 U , m , we obtain the light neutrino mass matrix, . | c ⊃ 5 gauge symmetry. To provide an estimate for the relevant couplings, we set − GeV and PMNS M diag , with , m ∗ PMNS U ]. As II k 10 16 u ja HN U γ charge in the superpotential at the nonrenormalizable level. With γ ' 48 m 10 W . ( . However, for numerical estimates we will assume normal-ordered (NO) light neutrino couplings can be boosted by the factor A neutrino mass matrix in the An alternative interesting possibility for generating light neutrino masses can be re- In general for a given matrix = 4 a , × obtain masses of order, | FSU(5) γ 3 µ = 0 , diag ν 4 L a a 2 . diag by the colorγ triplets in the Higgs 5-plets and 10-plets strongly restricts the couplings, where pressed. This model is briefly discussed in section present only an even numberleading of matter order, superfields the appear following with terms the are 10-plet allowed Higgs in fields. the To superpotential, alized by allowinglevel explicit [ mediating breaking effects tosume the that visible the sector viaR gravitational interactions. We will as- of masses with U matrix, mediated by the color triplets in the Higgs 5-plets as discussed in the next section. m As we discuss in theof next section, with the helpS of an additional which is diagonalizedcontrast by to a the double unitary seesawfor matrix mechanism a where recent analogous treatmenton of double seesaw mechanism1 in an inflation model based µ where we adopt a basis in which both JHEP02(2021)181 , 2 P M and 1 m (4.1) (4.2) (3.6) γ ]. ) field is h 2 ]. Note 55 5 GeV for = , S ) 63 2 c -symmetry ν is required h − 14 model. In a R M (5 S , are also not 10 eV, and with H inflation model × 05) 10 . 0 . 0 H SU(5) 6 is real and diagonal, , , 10 ij , . For a normal mass 11 ) 1 FSU(5) P 0086 c 10 γ k . . For numerical estimates ( 0 U † N /m × , c j 2 7 U 7 D − . ν c i M 1 i m , , 10 D ) ) ∗ 1 N 10 γ + × . U u,ν ) ( 10 k 1 c k . = L parity. In this section we will explore ], m . The light neutrino mass matrix is = ( c × j E 1 (1 , can appear at nonrenormalizable level 1 j provides the mass matrix, 54 1 γ R i symmetry 1 D − , as discussed in a recent paper on 4-2-2 . L i – diag ν ) i matter parity even if we allow 10 M c R Q (5 diag , and Higgs 10-plets, L m ν 2 49 II HN ( h ⊂ Z 5 M 2 as W c ( h γ ' diag and M M LLRR ) – 6 – 2 P 2 P 5 i i D N ) m m model suitable for susy hybrid inflation model. As symmetry these operators can lead to fast proton S S u,ν ij U T ( h h ' δ i and R m diag ν c ⊃ ⊃ . These values of the right handed neutrino masses are ν M m = ( 4) k k symmetry plays an important role in suppressing various M . U(1) ν 5 1 = 7 2 FSU(5) j j , m 5 Z 3 i ij RRRR 10 2 P ) i 5 − × , 2 P c m H ν R 10 10 m ] mostly in comparison with the unflipped mediated by the superheavy gauge bosons has been extensively M 10 H symmetry and modified × ( 62 S 0 2 – . 10 symmetry and which may otherwise mediate dimension five rapid U(1) LLLL 2 Z S 56 , -symmetric R 4 R − FSU(5) GeV, we obtain , with ] this is discussed in a no-scale supersymmetric 10 U(1) field and with no ij 16 δ symmetry which makes these operators highly suppressed as the 32 × i S 3 ) 10 . R 1 . In general, these color triplets can contribute to proton decay via opera- γ ) (6 × 1 − H 4 . U(1) = ( 1 diag 10 , For proton decay via dimension five and dimension six operators we mainly focus on ij ' 1 H ) ' 1 1 γ proton decay. the mediation by color(10 triplets in thetors conjugate of pairs chirality of types Higgs superfields, expected to acquire athat vev these of operators order are also TeV scale forbiddenbreaking by from the operators the soft at susy nonrenormalizableGUT breaking level scale terms as mass [ discussed termsallowed in for due Higgs the to 5-plets, previous section. The Without the decay incompatible with theby the experimental observations. The presence of operators that mediate rapid protondecay mediated decay. through For the example, color thevia triplet, dimension the four following rapid operators, proton studied in the pastrecent [ paper [ with an approximate proton decay in an emphasized earlier the the inverse seesaw mechanism described above.in This hybrid scenario inflation can be models naturally with incorporated successful reheating and4 nonthermal leptogenesis [ Proton decay inProton FSU(5) decay with in U(1) ( hierarchy of theM light neutrinos, γ significantly larger than the corresponding estimate of heavy neutrino masses obtained in for the rightobtained handed via neutrinos the assuming standard seesaw mechanism [ and is diagonalized by ain unitary this matrix second model of neutrino masses we adopt the basis where where the third term in the superpotential JHEP02(2021)181 by (4.7) (4.5) (4.6) (4.3) (4.4) H LLRR 10 , H 10 , . † † . , c c c h † E D E D U † † c c c ) h e E ( U U D D φ y ) ) c ) ∗ e † L ( D D U y DU DN † L ∗ , = / )) relevant for proton decay me- ∂ ψ φ 2 h U + + ) 5 e ψ ( VP 2.1 c , -charge zero at the nonrenormaliz- ) 2 P 2 1 U , given by R UD , y UE − h ) m u,ν † e 5 ( D + ( D y V can be expressed in terms of mass eigen- (eq. ( ⊃ = ( = ( h . Therefore, the corresponding effective ¯ , y ) † ) θ † † c ) d ). c 2 h c µ d ( D W 5 ( U D σ E QD D , ΦΦ ). Therefore, the dimension of chirality non- ) 4.4 † 2 † † + x , y P y c – 7 – c † − h ( ) h ∗ ψ V ΦΦ (5 D u,ν V µ ) 1(b) ( D θ d , ∂ 4 y ( of D Q = 2 ) and ( QLU QQU d θ ) ) ) h 2 d P y 1(a) ( Z 1 ⊃ ⊃ 4.3 with two scalars and two fermions is six. √ ∗ D u,ν i † † 2 P ( D , , y V 3 5 1 ) y ). The dashed and solid lines represent bosons and fermions h 1 symmetry with renormalizable interactions only allows the m − − 2 L † 5 D ) u,ν Q 3 ( ( D † U R x 1 − y LLRR 1 2 ( 5 L ⊃ L − 10 1 = , we finally obtain the four Fermi effective operators of the kind θψ ). Here the dot represents the effective dimension five operator once 3 ) and ( ) 2 10 ⊃ − 1 involving two scalars and two fermions. These are actually dimension √ 1 5 u,ν -symmetry breaking terms with ( 1 1(d) SUSY W y 10 R 10 ) + M x ( LLRR φ ) and ( ]. In our model ⊃ 1(c) 33 already expressed in eqs. ( Φ The Yukawa terms in the superpotential A word of caution is in order regarding the proper use of the terminology for the oper- The Feynman diagrams for dimension five and dimension six proton decay operators with the diagonal Yukawa couplings, diated by the color triplets states as operator with two scalarsthat and the two chirality fermions non-flipping always fermionfor contains propagator the a picks diagram up derivative. given theflipping It in momentum operators is of figures of the also ( type fermion clear six operators which can be seen from the following Kähler potential term, where the heavy color tripletbreaking mediating scale, fields hasLLRR been integrated out. Below supersymmetry ators of type are shown in figures ( respectively. In dimension fivelines diagrams can be the interchanged bosonicdiagrams at and form each fermionic a vertex. loop character The with offigures chirality external ( external nonflipping dashed higgsino lines or in gaugino the mediation as dimension shown five in Later we also discussallowing the explicit proton decay mediationable by level. the color triplets from chirality nonflipping modes whichchirality reduce generated to via the color following triplet four exchange Fermi from operators of model [ JHEP02(2021)181 ] † ) 3 5 † 57 iϕ , (4.8) , e 32 5 10 2 iϕ 10 , e 1 iϕ , e ( † N U ∗ L diag U are shown in panels (a) = = h P 5 (b) ⊂ (d) h PMNS U D , ]. ! 32 [ and and U h VD 5 !

= 0 ⊂ N i = h ϕ i E D Q P PMNS U – 8 –

with } = c L N c N with ,U c o D L ∗ T L , c (a) E (c) ,U c c E U Q, V P U n = 3 { 3 5 1 3 1 − mediated by fermionic color triplets 10 ¯ 5 supermultiplets are expressed in terms of the following mass eigenstates [ † 1 † is the Cabibbo-Kobayashi-Maskawa (CKM) matrix and 5 . Dimension five proton decay diagrams corresponding to effective operators V FSU(5) As the amplitude of dimension five diagrams involves loop factors their contribution is 10 10 where is the phase factor matrix with the condition generally expected to be suppressedwill as include compared to the dimension contribution six of diagrams. color Therefore, triplets we only from dimension six diagrams which are the dot represents the effectivehave dimension been five operator integrated once out. the heavy color triplet mediating fields The and and (b) respectively. Thebosonic dashed and fermionic and character solid ofdiagrams. lines external represent lines The can bosons be generic andmediation interchanged fermions loop at by each respectively. diagrams higgsino vertex The or in for the gaugino dimension first are five two shown proton in decay the with last chirality two nonflipping panels (c) and (d) respectively. Here Figure 1 JHEP02(2021)181 † 5 † (4.9) . The (4.10) (4.11) (4.12) (4.13) 5 10 h 5 10 and their ⊂ h θW D ) is generated 2 (c) d . The first term ) which has been R , 2(a) and , † l , h . c 2(a) λ M c 5 ! ) . E l lk ) , ⊂ = ) † l ). This contribution has † k ) N ¯ h λ c c ) + h u,ν D c M ( E D U 2(b) y D † k ∗ j L . PMNS c Q U and 2 ¯ λ U ! ( i ( U VP ji M k kl ( Q ) ) ) ) ) X e † λ M ( D ijkl 6(2) ijkl ) 6(2) u,ν ], y VD ( D C C = Q y T L ) 32 † j λ + ( + U ( U l V l M ( 2 i ) + ( λ ij L E ) ) (b) L k k † – 9 – + M T U L 2 Q V ( ∗ U kj † j VD ) ( d V ( D c 2 ijkl 6(1) li X are given by + ( ) † D C M ) j L P y † j 2) c ) , † i U ∗ c U ( c ( ijkl 6(1 V D

( U VD − C j (

† i color triplet exchange diagram ( i 5 iϕ c − e g U h ijkl 6(1) 2 U D = = C +( gauge vector superfield. The combined effects of the superheavy √ ] for an inflation based model. The contribution of the first term = ⊃ ⊃ 32 ijkl ijkl 6(1) 6(2) C C eff 6 K L SU(5) mediated by gauge bosons and scalar color triplets † (a) 1 † is the 5 . Dimension six proton decay diagrams corresponding to effective operators arises from the is the contribution from the gauge boson exchange diagram ( X gauge boson and color triplet mediation below their mass scales are described by ijkl ijkl 6(2) 6(1) 10 10 C C Here the color tripletin masses are written as studied recently in [ in where the Wilson coefficients where SU(5) the dimension six effective operators, generated from a combinationHermitian of conjugates. the Yukawa Similarly, termsfrom the in the gauge the following Lagrangian boson part of exchange the diagram Kähler ( potential [ Figure 2 and wavy, dashed and solid lines represent vector bosons, scalars and fermions respectively. JHEP02(2021)181 ]. 2) = , are 10 L ijkl 6(1 2 decay ) A (4.16) (4.17) (4.18) (4.19) (4.20) (4.14) (4.15) C π to be of ¯ ν (4 ¯ λ + / 2 i K M are encoded ]. The effect , , g 0 0 Z 64 (1) i (1) i = (1) i (2) i b b and c c 2 2 i M . , ] which naturally λ α ] ] . The triplet mass ) ) 4 38 T 8 3 M ) ) − − M ) are run down to low color triplet exchange Z Z , , only the charged lepton SUSY SUSY h 9 2 M M 3 , ( ( 4.13 D i i M M − G − ( ( model [ 0 α α i i , , M , α α 1 23 10 23 15 , 3 5 × × ) and ( − − SU(5) (2) i (2) i (1) i (2) i D b b c c 4.12 2 2 of order = = n 0 ¯ λ ) ) + ) with equal mass (2) i (2) i ) ) M , of the gauge couplings D arises from the T T and the electroweak scale 1 n , (1) i SUSY SUSY , M M in eqs. ( G b 0 ( ( = 7 ijkl i i 6(1) ]: M M , , doublets, MSSM and SM content respectively. M ( ( – 10 – − T 2) , c , c α α C i i 2 5 , n 3 and 66 , α α D , 4 8 3 n , and are given as − 6 T 19 − = 1 65 , (2) i − × × n -symmetric flipped b 1 , − n ,[ , , R , , ( 9 2 + n 3 SUSY (3) i (3) i i (2) i (1) S b b ) 5 c c (3) i − − 2 2 10 33 41 n M b A (2) i , , ijkl b 6( ) ) ) ) C T T G G 11 10 11 15 = = = M M M M triplets and − − ( ( ( ( (3) i (1) i (2) i i i i i b b b T α α α α n = = electroweak doublets are required to achieve MSSM gauge coupling 3 3 =1 =1 0 ) are the coefficients of one-loop RGEs for Wilson coefficients Y Y i i 0 depending upon the color triplet that makes the dominant contribution D (1) i (1) i 2) c n , ¯ λ = = c (1 i 1 2 ) and is crucial for making a nonvanishing prediction for the M c to be of intermediate scale. With S S ( A A λ or 2(c) 2) , λ M (1 i c M = The Wilson coefficients symmetry, as described in the next section, naturally predicts both T 4 The additional unification as discussed incolor the triplets next and section. electroweak doubletsM For ( simplicity, we take theto same proton numbers decay. of The renormalization factors are suplemented by another factor for MSSM plus light The one-loop beta coefficients, given by where above (below) the SUSY scale, in the renormalization factors, Z intermediate scale. This leadsneutral to lepton distinctive decay channels proton as lifetime described predictions below. especially forenergy the scales using the Renormalizationof Group one-loop Equations (RGEs) RGE given between in the [ GUT scale predicts channels are predicted to lieThe in the contribution observable range of of the futurediagram experiments second ( at term Hyper-K [ in channel which is usually assumed to be suppressed. The present model with an additional been studied more recently in an JHEP02(2021)181 β with ] tan ] and (4.21) (4.22) (4.23) (4.24) 9 , β 8 10 2 , . from lat- — — — — — , 6 cos     2 υ to calculate ml

, with 2 DUNE [

T λ 2 2 λ 2 years) = λ 1 M 1 ] M 34 d M i 10 l d υ i 10 = l ( Sensitivities d the contribution of , υ ], Hyper-K [ . m 8 7 2 υ . . . 2 m ) — — — ¯ λ 7 7 3 ∗ s us d d ! d 42 d and υ V m , M 2 p υ 1 2 K /v

m 6 i Hyper-K [ ) β

) – 2 m m = 2 2 S 1 L e,µ 2 S ( A U sin − ] T ( A m i υ 1 42 l + ) M 0 + 6 1

2 years) 77 59 16 . . 039 . . .

= K d,s 2 1 . 0 0 0 0 (

2 L -factors are respectively deﬁned as 0 2 34 ¯ λ T u 2 Super-K π ¯ 1 λ A m 10 C bound [ υ ( 1 M p = 32 M √ 2 i ( m l 2 0 u K = u u u υ m λ < K υ m - and the . = k 049(2)(5) + . Z 131(4)(13) 118(3)(12) 134(4)(14) 186(6)(18) . – 11 – 103(3)(11) 099(2)(10) . . . . ) , k + ¯ . . 0 λ ,C are the masses of proton, pion, kaon and charged 0 0 0 0 2 2 0 0 2 2 − M ∗ 2 − − − − ) 1 ud ! µ 1 M V

M 2 p 2 π 1 . The gauge boson contribution becomes dominant i

R u u 2 = ( ) u u ) π remains dominant over other contributions. We can use the | R + i /v R R ) R R l i 2 L ) ) boson mass, ) T + i ) ) ud us l 0 l λ ( ( π ). Therefore, the decay rates for charged-lepton channels with ud | | tan us us A 0 ( ud ud GeV. Finally, the K ( ( m | 2 h e,µ Z + + = p | | ( ( π M ( | | = Matrix element (GeV 0 0 32 + 5 C 0 0 i K K C ≤ l π | m h h π π K K as a function of the triplet mass ml | h m 0 h h h h T ) π K = = π ] and the corresponding Super-K bounds [ 2 and = 3 = = = = = k k ) ) ¯ C ¯ ¯ ν ν ν d,s and = 174 + + + + 68 ( + + + π K e e µ µ π 0 0 = = 0 2 0 K K k υ 0 00 ( ( π T m π K K − h T T T T T + + i i T become, 5 l l √ , m 0 0 ( π ) π K + & → respectively. The MSSM parameters are → p , m 0 p ] sensitivities are given in table 0 ] which is the perturbative QCD renormalization factor below the electroweak 0 0 i , µ + λ p Γ + l 7 π π . The Super-K bounds, Hyper-K and DUNE sensitivities and values of relevant matrix K Γ K + 67 + + m + + ¯ [ ν π e ¯ ν K = e µ e µ channel Decay The numerical estimates of partial proton lifetime for charged lepton decay channels The dimension six proton decay is mediated by vector gauge bosons and the color ¯ λ = ( 247 . + i are shown invalues ﬁgure in the range for color triplet with mass experimental bounds on the various partial proton lifetimes depicted in table For convenience, the recentlytice updated computation values [ of hadronicDUNE [ matrix elements electroweak vev, and where leptons triplets (from l elements for various proton decay channels. 1 scale represented by the Table 2 JHEP02(2021)181 . . u × ∼ 16 60 16 0 10 /v . 10 ), we u 1 ≤ λ (4.25) SK HK m 0 0 β . SK & π π 0 6 + + 4.21 K μ μ 60 15 . − + τ τ 60 T , μ 15 tan , τ 10 λ 50 10 M 10 in the range 50 , . The bottom ≤ , × 40 β 2 60 40 , 0 ] , 0 π 24 14 ] 30 . ≤ and eq. ( 30 + K , 14 tan 3 10 , µ GeV + 0 [ can be boosted by a β 20 10 GeV µ T 20 [ , π → , T λ M β 10 p → M + 10 tan , 2 . As expected from the = , p e 2 13 2 , it is instructive to con- 4 (b) ≤ ¯ 13 λ 10 ¯ λ in the range (d) → = β 10 2 corresponds to = β 8) β p . tan 0 tan 1 + tan π 91 12 tan q , + 12 10 6 e . , where the contribution from the 10 for ¯ λ & 20 4 37 36 35 34 → 8) ,

39 37 35 33 λ

−

10 10 10 10

π μ p

] [ τ

years

+ 0 .

10 10 10 10

in the range,

] [ τ

K years = 10 91 + 0 6 , , β × λ 6 4 9 . . . 1 tan 20 16 ), the weak dependence on – 12 – − , 16 2)(1 10 6 4 / . 10 − ) SK HK 6 with 4.22 0 0 SK β π π , 0 10 + + λ K 15 4 e e + GeV or . × τ 15 τ e 60 M 10 τ 2 , 10 60 . 10 , 7 ((tan 50 = 2)(1 , ) and ( 10 50 / ¯ λ , 14 & ) ∼ 40 14 × ] , 40 β 10 M λ 0 0 , ] 10 ¯ λ 4.21 30 π K ) 56 becomes dominant over the other contributions for 30 , = = . GeV + u + [ , e 4 e GeV ¯ λ T as discussed in the subsection below. 20 [ λ symmetry these relatively tiny values of 13 T 20 T , ((tan M 13 4 → , 4 → /m M 10 M M ) 10 10 p β p Z 10 , 10 , 2 2 e,µ 2 ) on proton lifetime for the decay channel ( (a) (c) ∼ does not exhibit any spread in the proton lifetime predictions shown in 12 λ 12 GeV or 2 m = β 2 = β 10 ) ) 10 60 12 tan tan d,s ( 1 + tan 10 /M ≤ 11 m P 11 q × . The partial-lifetime estimates of proton for charged-lepton decay channels as a function β 10 7 √ 10 m . ( 35 34 34 36 35 2 )( & 34 33 38 37 36 35 tan 10 10 10 10 10 β contribution in eqs. (

As the contributions from both color triplets become comparable for −

T × × × × ×

10 10 10 10 10 10

] [ τ K e years

0 11

λ 1 1 5 1 5

≤

τ ] [ e years

+ 0 M sider the limit color triplet of mass The numerical prediction forM this scenario is2 depicted in ﬁgure Thus, the Super-K bound10 on the decayWith channel an additional factor (tan K bound (table obtain the lower bound, of 5-plet triplet mass dashed-lines represent the experimental limitsK from limits. Super-K and top dashed-lines represent Hyper- the corresponding lower bounds on the color triplet masses. For example, using the Super- Figure 3 JHEP02(2021)181 ) 16 16 4.21 10 (4.28) (4.26) (4.27) 10 SK HK SK 0 0 GeV, is 0 π π K + + 15 + μ μ 15 11 μ τ τ τ 10 , 60 10 60 2 10 ,

, 50 1 . 50 , 14 j 14 with eq. ( , based on the , ) i 40 10 T 10 0 ¯ 40 0 ν , ] V 0 , 2 π π ] ( 2 ¯ + λ M j 30 K + 30 ) , 13 ij µ + + GeV , 13 K . M [ ) GeV e µ 20 V 10 2 T [ . The lower dashed lines 20 10 , (

2 → ¯ ∗ T λ contribution quoted in N M , 1 → . M p 10 60 ij U → j λ 5 10 M , 12 p ) ( ) and , 2 − p 12 ≤ 2 ∗ 10 j M N (b) i V ) 10 (d) ( 2 ¯ ¯ β ν λ 10 ) U = β ( u = β u + ij ( 11 M υ × tan j ) π tan ) tan 10 11 m ∗ 2 ) N . ( u u 10 ≤ 5 U ( υ ( 10 2 j m j & X 10 ( ) ) 37 36 35 34 u λ u u u 39 37 35 33 j

(

10 10 10 10

υ m

μ X

[ τ ]

years 10 10 10 10 + 0

μ [ τ ]

K years + m 0 u ( u ud υ V m j X + )), are expressed as in the range ud 16 ) u – 13 – 16 1 u β ∗ i 10 3.4 υ ) 2 10 m V SK HK ∗ SK N ( 0 0 GeV with 0 tan M π π us ¯ K ν U + + + ) 15 15 e e ( + 60 e 10 ∗ 60 τ τ τ ,

(eq. ( 0 10 10 , K 1 V 50 10 with T 2 S ( 50 , 1 ¯ , ν λ A I × 14 HN 40 + 14 iϕ 2 , 40 | M 4 10 e 00 , K W . ] ¯ 10 ν

30 0 ] 0 7 1 T , + 30 K π 2 S 2 , π 20 13 GeV + + ¯ λ & [ T A , iϕ e GeV e 20 13 | T [ 10 e , λ T M 10 π K M 10 → → , M k k + 10 2 M p p = , 12 2 = = T 10 = β 12 (c) (a) i i M ¯ ¯ = β ν ν 10 tan + + 11 tan π K 10 → 11 → p p 10 Γ Γ 10 . Estimates of proton partial lifetime for charged lepton decay channels as a function of ). However, a potentially observable range of this bound, with 10 35 34 34 36 35 . In this case the Super-K bound for the decay channel 33 38 37 36 35 34 10 10 10 10 10

10 10 10 10 10 10

× × × × × τ ] [

K e years 4.25

+ The proton decay rates for neutral lepton channels, 0

5 1 1 5 1

π τ ] [

e years + 0 neutrino model described in which is somewhat smaller thaneq. the ( corresponding estimate of in contradiction with Super-K bounds on neutral lepton channels described below. represent the experimental limits fromfuture Super-K Hyper-K and limits. the upper dashed lines in (a) and (b) represent ﬁgure gives the following lower bound, Figure 4 5-plet triplet mass JHEP02(2021)181 λ M ] by with 16 = (4.29) PMNS 32 10 + ¯ λ U 60 M , SK DUNE ν K + + K K 50 = _ _ ν ν , channel than τ τ 15 → T 40 ¯ ν 10 , p track with high limit the proton M + 30 , to lie within the + , T 6 K ] + 20 − K M , ] the contribution of ¯ . The lower dashed-line νK GeV → 10 14 10 [ , T 32 . 60 → 10 p 2 4 M × p ≤ − = β 24 β channel lies far beyond the . (b) 10 tan 3 ¯ ν . We particularly include the × tan 13 + 2 6 β 10 . channel [ π ≤ 2 2 i 2 ¯ ν model recently presented in [ & + λ π 46 41 36

1 + tan

10 10 10

] [ τ years K

+ _ ] is expected to provide us the world’s best SU(5) p 7 in the range & – 14 – 16 β 10 is shown in ﬁgure > λ GeV with tan 4 SK 60 + 12 − π _ ν 15 ≤ τ 10 10 with 60 , 10 β λ × × equal to the unit matrix. In the large 50 , M 2 2 . . N has been ignored so far. The numerical results are displayed 40 tan 3 ] 2 = + , ) where we have used the recently updated values of U 14 λ ¯ λ 30 ¯ symmetry. This bound also allows the charged lepton channels, νπ & ≤ & , GeV 10 [ M M λ 4 T λ 2 → 20 5(b) M , Z p = M mode by the mid-2020s, before DUNE/Hyper-K catch up with the 10 ] with T , ¯ (a) ν 2 13 41 M with + 10 = β ) and ( 3 K ]. This is mainly due to fact that the DUNE collaboration will exploit tan 9 → , 5(a) 12 8 in the range p 10 . Estimates of proton partial lifetime for neutral lepton decay channels as a function of ) gives the following lower bound, β 35 34 33 32 36

10 10 10 10 10

4.28

π ν ] [ τ

years In order to make a comparison of proton partial lifetime predictions among various It is important to note that DUNE is more sensitive to the For neutral lepton channels the Super-K bound for the decay channel + _ tan GUT models the estimatesvariation of of branching various branching fractions fractions playfor with a respect pivotal to role. color tripletcorresponding For mass predictions this purpose from a the unflipped Hyper-K [ liquid Argon time-projection chamberefficiency technology as which compared can to identify JUNO, a which water is Cherenkov detector a liquid likelimit Super-K scintillator for detector or [ Hyper-K. However, JUNO’s limit. accessible value with shown in figure observable range of Hyper-Kreach whereas of future the detectors. prediction of eq. ( This is the largest bound among the neutral and charged lepton channels with a naturally in figures ( parameters from [ lifetime of the first channelsecond is decay dominated channel by lifetime theboson gauge increases contribution. boson without contribution bound whereas for due the to the absence of the gauge 5-plet triplet mass in (b) represent theDUNE experimental limit. limit from Super-K and the upperApart dashed-line represent from future thecolor gauge triplet boson with contribution mass in the Figure 5 JHEP02(2021)181 ] , ) G . a 39 S 1 H M 2.1 (5.1) 10 (4.30) (4.31) 1 i models 10 a S ] where this ! ! 38 1 ]. Also see [ SO(10) ) , 1 − H , relevant for the 72 − H 32 2 P [ ai 10 γ 10 m 1 H 1 , still remains intact. model makes a very 2 h 10 5 h.c (10

] and for P 1 SU(5) · + − H ) -charge assignments: m ai 33 -symmetry breaking terms 4 b 1 4 γ 10 S R − H Z 1 FSU(5) a + − H , S 10 , for the gauge singlet ﬁelds 2 1 † P 1) b 10 − h Σ m , S 5 ! a 1 (10 3 ab 1 S , − j

y − H 5 ab (1 5 i 2 P P ⊃ µ 10 , and the couplings, 1 S symmetry breaking coming from the soft ) 2 m m 1 H → e ) K ( ij 2 R the present ) P 4 y 10 . η a channel plays a key role in making distinctive 6 S

! + + , 5 + U(1) 2 h λ – 15 – 1 H ! 16 5 M/m 3 symmetry is spontaneously broken during smooth 3 ( 3 − 10 − j + 4 ν K − 5 5 , 5 2 Z 1 i 1 -charge. This modiﬁes the superpotential in eq. ( λM 1 P H − h 4 − H 5 10 10 m = Z P ) 1 H (10 10 1 H λ m 4 1 ! u,ν 10 Z ( ij − H 10 2 M q y 1 H 1 H 10 M + 10 10

and 2 − !

− h P 2 . The explicit mass term, 2 1 ) S 5 2 P − H m ) 1 j 1 P S ]. For a comparison with 4-2-2 model see [ 3 2 P − H m 8 10 10 η 2 P 1 1 i /M 20 m 8 1 H 10 η symmetry can be employed to make the color triplets naturally light for + – m P 10 1 H M/m 4 10 ) ]. As is obvious from ﬁgure ( m 13 ⊃ d Z ( 10 (

ij 71 y violation and proton decay – λ W λM

16 1 8 R 69 TeV [ symmetry and color triplet masses = -symmetry, + + κS 4 λ R Z 100 = M 4 It is important to note that both color triplets are now naturally light relative to Z W by the 5 U(1) In this section we firstsusy discuss breaking the terms. effect of For example, consider the following nonrenormalizable terms allowed realization of light neutrino masses viaby a the double seesaw factor, mechanism have alsogenerated been effectively enhanced from the KählerNote potential, that we do noton consider the this symmetry standard in seesaw theat mechanism second nonrenormalizable arising model level. from of the neutrino masses explicit based This superpotential can befor employed a to relevant realize model smooth of inflation.hybrid hybrid inflation inflation The and [ the domain wall problem is thereforewith avoided. with all other fields carryingas follows: zero channel is highly suppressed. 4.1 An additional observable proton decay. This is achieved with the following order see refs. [ distinctive predictions of various branching fractionsK. within Especially the the observable branching range fraction ofcomparison of Hyper- of the current model with the other models of flipped ignoring the dimension five contribution of color triplets with large sfermion masses of JHEP02(2021)181 , λ 16 16 M 16 10 10 = , from 10 LLLL LLRR FSU(5) (5) SU ], these ) ¯ (5) SU λ (5) SU 60 0 ) 7 ) 0 , 0 π 15 15 K 63 + π M 60 15 60 e + + 50 , e 10 10 e , , /Γ 10 = 0 /Γ /Γ 50 + 50 + K 40 , K + π , T , _ e _ ν ν 14 14 40 14 40 (Γ (Γ M 30 , (Γ 10 , 10 , 10 30 ] ] 30 ] 20 , , , 13 20 13 20 13 GeV GeV 10 , [ GeV [ , 10 [ , 10 T T 10 T 2 10 10 M M (d) (b) , M (f) , 2 2 12 = β 12 12 10 10 = β 10 tan = β tan tan 11 11 11 10 10 ]. The solid line curves of 10 32 10 10 10 10 6 5 4 1 10 10 9 4 1 10 -6 10 -11 10 10 10 100 0.1

10 10

1000 0.10 0.01

π π

) ( π ν ) ( 5 FSU 5 FSU

) Γ / Γ ( ) Γ / Γ (

e K e

e 10

+ + + 0 0 0

) ( ν 5 FSU _ ) Γ / Γ (

K e

K + 0 + _ , from the soft susy breaking terms [ 2 / – 16 – 16 16 16 3 10 10 10 renders such decays from operators of type . For comparison the corresponding predicted values κ m (5) SU (5) SU (5) SU 60 ) 12 ) ) 0 + 0 15 15 15 π 60 60 − π π + _ , ν , + ≤ μ 10 10 10 e 60 i ∼ /Γ 10 50 /Γ 50 , + /Γ 0 0 S , β , K K _ π h ν 50 + GUT are included from [ . + 14 14 40 14 40 μ , μ (Γ , , ) (Γ 10 10 tan 10 (Γ 40 30 30 P , ] ] ] , , ≤ 30 20 13 20 13 , 13 /m SU(5) GeV GeV , , GeV [ [ 2 [ 2 10 10 20 10 T T T / 10 10 , M 3 M , M (c) (e) , (a) 2 2 10 12 , 12 12 2 κm 10 10 = β = β 10 ( = β tan tan 11 11 11 tan relatively suppressed in comparison to proton decay operators of type 10 10 10 in the range . Estimates of various branching fractions as a function of triplet mass 10 β 10 attains a nonzero vev, 10 10 10 4 1 1 10 -8 -4 5 1 S 10 -12 10 tan 10 100

RRRR

10 10

0.01 0.10

0.50 0.10 0.05 1000 10

μ π μ μ π π

) ( ) (

5 FSU 5 FSU π ν ν ) ( 5 FSU Γ ( ) Γ / Γ ( ) Γ /

Γ ( ) Γ / e

+ + + + 0 0 0 0 + +

_ _ interactions can lead to dimensionmixing five of proton the decay color-triplets diagrams,of as from a shown small the in factor Higgs figure and 5-plets and 10-plets.discussed earlier. However, the The presence suppression of these dimension five operators is a distinctive fea- with of branching fractionpredictions for are consistent with the Super-K bounds shown with red dots. Once Figure 6 JHEP02(2021)181 and (5.2) 5 are shown in 10 10 10 ) h 5 , H c H 10 ( D c ⊂ D ) . h ) (d) (b) ) ∗ D c ). This feature is absent in H , D c H VP c 2.1 3 D γ E ( ( -symmetry breaking allowing only ) c c R E U and U ) + 4 h γ 5 c H ( , c H U QD (10 ) )), relevant for proton decay mediated by the + 3 – 17 – ⊂ γ c H 3.4 , can be expressed in terms of mass eigenstates as ) L ) D h U ] where the above mentioned potentially dangerous H ( Q ,D 38 10 ) L (eq. ( , 2 ( c H γ )), where an explicit H D ( ( II HN P 3.4 Q (10 M 1 2 ] and [ m W − 32 ) lead to both dimension five and dimension six proton decay (eq. ( from ⊃ (c) (a) 5.2 ) -symmetric model described in eq. ( II c H HN II HN R D W -charge is assumed at the nonrenormalizable level. The effective Yukawa W , R c H D ( . Dimension five proton decay diagrams corresponding to effective operators mediated via color triplets Finally, we briefly comment on the second model of neutrino masses described in 5 1 5 The operators in eq. ( the superpotential operators of zero terms in the superpotential color triplets ture of the present the models considered indimension [ five proton decay operatorsof can the lead couplings to involved. rapid proton decay with natural values Figure 7 10 panels (a), (c) and (b), (d) respectively. The dashed (solid) lines represent bosons (fermions). JHEP02(2021)181 , 2 23 / ) = TeV M (6.1) (6.2) (5.3) G lying gauge 10 models M G FSU(5) ( X GeV, we ) = 1 2 ∼ M v g 2 12 5 / ( obtained by 3 U(1) 10 couplings are R . 4 ) = m 2 v gauge coupling − 3 FSU(5) G ∼ , 5 g 1 10 M T 10 − v ) = ( v 1 ∼ M model, the diagrams ∼ 10 g and (5 1 2 and the SU(5) ) β − H 2 R 1 P g at some scale g 10 tan 2 SU(5) X 4 λ g . With . M/m ( ) 2) + = , 2 23 2 5 − v , in a way similar to the recent dis- g GeV, the M 24 5 1 ( ) 1 v , 2 13 5 H years, with g 10 , unify with the 10 . 3 10 1 H 5 33 + . This implies that , g − − ) ) 10 10 ) = (1 H GeV. Therefore, we need to take care of G 1 v 10 23 10 . With two pairs of color triplets the 4 × λ 5 17 M 5 and (10 10 1 M ( 10 . ( 5 , 2 + 10 & 4 g v – 18 – of g 2 , X 1 > g ∼ 3 g × + , ) − v of the additional vectorlike 5-plets can be achieved 10 1 2 5 c H , g 5 γ ) = ¯ = νK 10 as v τ 1 v D G 5 ) M ∼ , with odd matter parity and , 23 , 1 10 M 23 v c H ( − v M 10 M 25 X (5 D ( 10 4 ( g M 2 1 at Z ]. However, in the present flipped . The third MSSM gauge coupling g + q 5 + GeV with g 1 v 33 with the color triplets of intermediate mass the proton lifetime M 2 v 5 18 5 4 . The relevant superpotential terms for these additional multi- 10 2 g , with 2 10 − v / couplings, G 5 = 5 4 − ]. , M 3 , 23 M 17 2 = 38 , γ ) = 1 M ⊃ . A similar order of suppression is also expected in the 10 v 23 1 GeV with the help of the suppression factor 32 and the string scale 10 M W are related to 12 ( ∼ 23 R make the dominant contribution due to the absence of a suppression factor 10 X λ g M symmetry a common mass , and 10-plets, 7 the two MSSM gauge couplings, . To avoid rapid dimension five proton decay from the mixing of the color triplets ∼ v 2 , which is an attractive feature of MSSM. Next we consider vectorlike 5-plets, charge assignments, 4 = ) 5 P ) = Z 4 G v λ To remedy the above mentioned problem we first make a simple choice for the unifi- + Z /m at the scale, M 2 i ( (10 − v 5 3 S With around the effectively obtain vectorlike 5-pletsdoublets of and color intermediate triplets mass and with this the automatically guarantees same gauge number coupling of unification. light 5 R plets are between reconciling potentially observable proton decay and gauge coupling unification. cation scale, g With a single colorunified triplet around pair ofunification mass scale goes beyond thebased Planck on scale. a For a simple possible gauge GUT group model beyond we ultimately require FSU(5) g coupling 6 Gauge coupling unification As emphasized in section is predicted to liepling within unification the which is observable otherwise range. achieved naturally This with brings the forth MSSM the matter content. issue In of gauge cou- Here we use the experimental bound, and considered in [ cussion for 4-2-2 model inof [ figure h in the Higgs 5upper and bound 10-plets, on we obtain the following order of magnitude estimate for the mediated by the color triplets JHEP02(2021)181 with Q 16 16 GeV GeV GeV GeV 16 GeV GeV GeV GeV GeV 16 16 14 13 13 13 16 15 13 10 10 10 10 10 10 10 10 10 ======of the vec- = G G for one or 14 14 D T D T G 14 D T 5 8 M 3, M 2, M 2, M 2, M 6, M 2, M ======12 12 T T D D M 0.04, M 0.04, 12 T D M 0.04, n n n n = ] ] = n n = ] G G G α α α GeV GeV GeV / / 10 10 / 10 Q Q Q [ [ [ additional light doublets 10 10 (f) (d) 10 (b) 8 8 D 8 Log Log Log n GeV. With one light color TeV. The values of the unified versus the energy scale -1 -1 -1 -1 -1 -1 1 2 3 1 2 3 13 6 6 -1 -1 -1 1 2 3 6 α α α α α α 2 i α α α 10 = 10 π/g 4 4 4 GeV and 13 = 4 are also displayed in all panels. SUSY 2 2 1 couplings the color triplets in the ad- 2 G − i

2 60 50 40 30 20 10 60 50 40 30 20 10

60 50 40 30 20 10 α α

i , α

= 10

α M

- - 1

1 1 - 1 T λ M – 19 – GeV and ) 15 GeV 16 GeV GeV 16 16 GeV GeV GeV GeV GeV GeV 16 16 10 15 13 16 13 14 13 13 10 , 10 10 10 = 10 10 10 10 10 = = = G × = = = = G 14 14 D T 14 14 D T D T symmetry we can assume a common mass 10 0.79 = M 3, M 1, 4 , M 1, M 1, M 2, M 1, G ======12 T D Z 12 12 13 T T D D M 0.038, n n at the uniﬁcation scale M 0.039, ] n n n n ] ] = = G G G α α GeV GeV GeV π / 10 / / 10 10 M 0.039, 4 Q = (10 Q Q [ = [ [ / G 10 (a) 10 10 2 (c) (e) D α g 8 8 8 Log Log Log M = G -1 -1 -1 1 2 3 6 -1 -1 -1 -1 -1 -1 2 3 1 2 3 1 6 6 α α α α α α α α α α GeV. With natural values of the 4 4 4 15 . The evolution of inverse gauge couplings 10 2 2 2 − For a scenario without

20 10 60 50 40 30

color triplet pairs of intermediate scale mass

20 10 60 50 40 30 20 10 60 50 40 30 α

α α

i i

1 13 - - 1 1 T torlike electroweak doublets to10 lie below theditional GUT vectorlike scale, 5-plets, formechanism. however, example achieve This within GUT can the scaletwo lead range, light masses to color gauge via triplets coupling with the unification a missing as typical partner mass shown value in of figure order Figure 8 n with common mass gauge coupling JHEP02(2021)181 , in in for 0 π + GeV), + ]. ¯ νπ µ 8(f) 15 can play ¯ ν 10 → Phys. Rev. → , + p p SPIRE K IN symmetry. The and and GeV ( ][ 4 0 0 and figure Z 14 π ¯ νπ + . The gauge coupling 10 e = 1 D → ∼ n → D n and other GUTs. p n = = T n T n SU(5) arXiv:1305.4391 [ for ]. 8(e) models. The decay channel SPIRE -symmetric model where the color triplets of IN R (2014) 121802 – 20 – ][ SU(5) Review of nucleon decay searches at Super-Kamiokande Search for proton decay via Search for nucleon decay via 113 symmetry employed in the first model makes the color 4 . The issue of gauge coupling unification with intermedi- ), which permits any use, distribution and reproduction in Z G ]. M model of supersymmetric hybrid inflation supplemented by a collaboration, collaboration, collaboration, SPIRE arXiv:1610.03597 IN Phys. Rev. Lett. [ [ , CC-BY 4.0 SU(5) This article is distributed under the terms of the Creative Commons model and other flipped GeV). In proton lifetime estimates, for simplicity, we assume same mass symmetry. Two distinct models of neutrino masses with normal hierarchy 15 . R 10 (2017) 012004 SU(5) = 2 megaton-years exposure of the Super-Kamiokande water Cherenkov detector D 95 U(1) 31 n . D Super-Kamiokande Super-Kamiokande Super-Kamiokande arXiv:1605.03235 Super-Kamiokande 0 -symmetry. An additional GeV ( = R 14 T [2] [3] [1] any medium, provided the original author(s) and source are credited. References This work is partially supported by the DOEOpen grant Access. No. DE-SC0013880 (Q.S). Attribution License ( color triplet masses. Comparisonunflipped with other GUTs isa discussed pivotal role with in special discriminating emphasis various on models of flipped Acknowledgments to triplets suitably lighter than ate mass scale color tripletsthese is 5-plets can resolved naturally with attain additionalpredicted masses vectorlike range of 5-plets. of intermediate various The scale decay doublets rates due in and to branching a fractions is presented as a function of are briefly discussed. Hereintermediate mass we from discuss the the Higgs 5-pletsable mediate range proton of decay with future lifetimethe experiments. in color the triplets Rapid observ- in proton 5second decay and mediated neutrino 10-plet through mass Higgses mixing model. severely between constrains This the decay relevant couplings is in adequately the suppressed in the first model due 7 Conclusion Proton decay with lifetimesplored accessible in at a Hyper-K flipped global and other future experiments are ex- whereas for two light color10 triplets we require threevalues (six) for vectorlike both doublets color at triplets massunification and scale plots electroweak for doublets this with n case are shown in figure triplet we need two (three) vectorlike doublets at a mass scale JHEP02(2021)181 ] , , , 76 ] SU(5) ] (2005) 260 72 ]. ]. in using 0 ]. arXiv:1408.1195 + ]. K SPIRE [ arXiv:1805.04163 ]. SPIRE + IN Eur. Phys. J. C hep-ph/9808255 TeV with the ATLAS [ , µ νK , IN [ SPIRE Beyond the CMSSM (2016) 030401 ][ , Tokyo University, Tokyo, Supersymmetric proton → → SPIRE arXiv:1205.6538 IN Phys. Rev. D [ p p SPIRE = 13 , IN ][ 43 The Super-GUT CMSSM s IN ][ ]. √ ][ ]. ]. SuperGUT in pure gravity (2014) 072005 SPIRE Decoupling Can Revive Minimal (1999) 115009 SPIRE 90 IN Ph.D. thesis SPIRE , IN J. Phys. G ][ 59 IN SU(5) (2012) 012006 , ][ ]. arXiv:2002.03005 ][ , arXiv:1912.04888 86 ]. [ grand unification in pure gravity mediation arXiv:1304.3651 [ SPIRE arXiv:1902.09084 IN [ arXiv:1608.05370 – 21 – [ Phys. Rev. D SPIRE ][ SU(5) A minimal ]. , IN [ Phys. Rev. D Hyper-Kamiokande design report Search for proton decay via Search for proton decay via Search for nucleon decay via modes favored by , (2020) 332 ]. (2013) 038 Phys. Rev. D arXiv:1502.00034 hep-ph/0108104 SPIRE 80 , arXiv:1711.01901 (2019) 490 [ [ 07 IN [ ]. 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