Jhep05(2020)021

Home , Split supersymmetry

JHEP05(2020)021 ) ), + + e e 0 0 π ), and π + f,g,h → µ Springer → 0 p May 6, 2020 π p : April 11, 2020 April 19, 2020 Γ( March 12, 2020 Γ( : : → / ) / : p ) ¯ , ν + + Γ( µ / π 0 ) π + Revised Published → µ Accepted Received p 0 → K p → p , Dimitri V. Nanopoulos ) and Γ( e + e 0 π ) and Γ( Published for SISSA by https://doi.org/10.1007/JHEP05(2020)021 + → e 0 p π Γ( [email protected] / → , ) p + µ Γ( 0 / π ) Natsumi Nagata, + d e → 0 p [email protected] K , → p ), Γ( . + 3 e 0 i π 2003.03285 → The Authors. Marcos A.G. Garcia, p c Supersymmetry Phenomenology

Γ( We analyze nucleon decay modes in a no-scale supersymmetric flipped SU(5) , / a,b,c ) [email protected] ¯ ν + π → p [email protected] [email protected] Mitchell Campus, Woodlands, TX 77381,Academy of U.S.A. Athens, Division ofWilliam Natural I. Sciences, Fine Athens Theoretical 10679,University Physics Greece of Institute, Minnesota, School of Minneapolis, MN PhysicsE-mail: 55455, and U.S.A. Astronomy, Instituto (IFT) de Teórica UAM-CSIC, F´ısica Campus deDepartment Cantoblanco, of 28049, Physics, Madrid, University Spain ofGeorge P. Tokyo, Bunkyo-ku, and Tokyo Cynthia 113–0033,Texas W. Japan A&M Mitchell University, Institute College for Station,Astroparticle Fundamental TX Physics Physics 77843, Group, and Houston U.S.A. Astronomy, Advanced Research Center (HARC), Theoretical Particle Physics and CosmologyKing’s Group, College Department London, of London Physics, WC2RTheoretical Physics 2LS, Department, United CERN, Kingdom CH-1211National Geneva Institute 23, of Switzerland Chemical Physics and Biophysics, Rävala10, 10143 Tallinn, Estonia i b c e g d a h f Open Access Article funded by SCOAP to explore both GUT and flavour physics in protonKeywords: and neutron decays. ArXiv ePrint: GUT models makeΓ( very different predictions forthat the predictions for ratios the Γ( also ratios differ Γ( in variantsneutrino masses. of Upcoming the large flipped neutrino experiments SU(5) may have model interesting opportunities with normal- or inverse-ordered light Abstract: GUT model, and contrastoperators in them a with standard the unflipped predictions supersymmetric for SU(5) GUT proton model. decays We via find dimension-6 that these John Ellis, and Keith A. Olive Proton decay: flipped vs. unflipped SU(5) JHEP05(2020)021 ], 17 1 ] — in 6 induced by 11 ] — in which + e in the minimal 8 0 , 7 π ¯ ν + → K p → p ]. The rate for dimension-5 12 ] induced by dimension-5 oper- , 10 11 10 , 9 [ ¯ ν ], whereas in minimal supersymmetric + 15 K , ], and the rate for 14 → 5 p – – 1 – 1 3 ]. ] — will also open up new prospects for searches 6 15 5 13 ) 17 + 17 ) 12 e ) ) + 0 + + µ π e e 18 0 0 0 π π → π p → → → p Γ( p p / ) Γ( Γ( i Γ( / / ¯ ν / ) ) ) + + + + π µ e µ 1 0 0 is mediated by dimension-5 operators [ ¯ 0 ν → K π K + ¯ ν p + K → → → ]. The prospective sensitivities of the new generation of neutrino detectors to Γ( ] and Hyper-Kamiokande [ K i p p p 4 → – 12 P p , → 2 p 11 As is well known, the difference between the dominant nucleon decays in the minimal 5.1 Γ( 5.2 5.3 Γ( 5.4 Γ( 5.5 supersymmetric and non-supersymmetric versionstween of their SU(5) respective is decay linkedSU(5) mechanisms. to is the mediated Proton difference by decay be- dimension-6SU(5) in operators minimal [ non-supersymmetric dimension-6 operators — andthe the dominant minimal decay supersymmetric mode SU(5) isators expected GUT [ to [ be these decay modes hassupersymmetric been SU(5) documented GUT [ has recentlyuncertainties been in re-evaluated, including the an lifetime assessment estimate of [ the magnitude beyond current experiments. Thisdecay motivates modes a in re-evaluation different of grand possiblethat unified nucleon theories may (GUTs), discriminate and analyses betweentinction of that the specific can different signatures be models. drawnwhich between the the A most minimal characteristic well-known nonsupersymmetric proton example SU(5) decay GUT is mode [ the is expected dis- to be 1 Introduction The advent of a new generationDUNE of [ high-mass underground neutrino detectorsfor — proton JUNO [ (and neutron) decays into an array of channels with sensitivities an order of 6 Discussion and prospects 3 Nucleon decay in flipped4 SU(5) Dimension-six proton decay in5 unflipped SU(5) Comparison of proton decay rates in flipped and unflipped SU(5) Contents 1 Introduction 2 The no-scale flipped SU(5) model JHEP05(2020)021 2 ]. 2, . ). 3 + 17 e , 17 in 0 ]. It is ∼ 16 π ∼ 23 with 57 [ ), the IO → ], in which ∼ ¯ ν ) would be + p e + + 43 0 e This issue has π – Γ( π 0 / 1 π 37 ) ¯ ν and → ] now required by + → + π p 29 ]. e p – 0 003, whereas the ratio Γ( 008, whereas this ratio → 56 . π . Γ( / – 0 0 ) / p 18 , ) ¯ ν → 54 + ∼ + ∼ 13 p e π 0 ) ] in the combined framework of + K e → we review relevant features of the 52 0 – 2 π 02 vs. → p . ] suppresses dimension-5 proton de- 0 49 we study proton (and some neutron) p → 46 3 – ∼ p Γ( 44 , we present predictions for ratios of proton / ) 5 40 + ] appearing in some stringy extra-dimensional ), and the other is Γ( – 2 – µ 95 and the NO model predicts a ratio + 0 36 e π 0 ∼ ] no-scale supergravity [ π 53 → . → 02 in the flipped SU(5) model, as opposed to p 6 . p 0 ], and the purpose of this paper is to update the available . In section Γ( 4 ], and dimension-5 proton decay may be suppressed by a ∼ 74 / – ) ) 35 + + – 68 µ µ 0 30 0 π ], stressing the connection between the flavour structure of nucleon π 52 ], and a unified cosmological scenario for inflation, dark matter, neu- → – → p p 48 49 , Γ( / 47 ) , ] that may also cast light on the mass-ordering of light neutrinos. One signa- ] for proposed diagnostic tools for other GUT models. + 43 52 µ 67 – 0 – – 41 K 1 in flipped SU(5) with normally-ordered (NO) light neutrinos and 58 49 . 0 → The outline of this paper is the following. In section We identify two primary proton decay signatures of the no-scale flipped SU(5) The dominant final states for proton decay in supersymmetric flipped SU(5) are not ex- See [ Here and subsequently, the sum over the three light neutrino species is to be understood. ∼ 1 2 p SU(5) are discussed indecay section rates in the flippeddiscuss future and prospects unflipped in SU(5) section GUTs, and we review our conclusions and final state could discriminate betweenfor underlying neutron GUT decays models, may and also we show play that an searches important role. no-scale flipped SU(5) GUTdecay model, and modes in in section thiselements model, and discussing giving their expressions uncertainties. in terms The of corresponding the expressions relevant in hadronic unflipped matrix whereas the minimalheadline SU(5) signatures, model we allowslarger also values in find as flipped thatΓ( SU(5) the low than ratio as in Γ( 0.4.minimal minimal SU(5). SU(5), It In is addition clear therefore, to that measurements these of proton decay in more than one ture is the ratio Γ( In minimal SU(5) oneis expects Γ( inversely-ordered (IO) neutrinos.flipped SU(5) In model the predicts case a of ratio Γ( diagnostic kit in theposed framework previously of [ the unifieddecay operators cosmological and framework the that pattern we of have mixing pro- between neutrinosmodel and [ their mass ordering. pected to contain strange particles,similar with to many those of the in favoured minimaltherefore decay supersymmetric important modes to SU(5), expected assemble including a to kit be use of diagnostic to tools discriminate that between the the upcomingbeen flipped experiments discussed can and previously unflipped [ SU(5) GUT models. an economical missing-partner mechanismcay. [ This model istheory [ also of interesttrino because masses and it baryogenesis can hasflipped easily been SU(5) be constructed and [ accommodated string-motivated [ within string This problem isfruitless mitigated LHC by searches the [ discrete higher symmetry sparticle such as massesmodels the [ with hexality local [ grandthe motivations unification. for considering Nevertheless, the the supersymmetric proton flipped decay SU(5) issue GUT [ has added to proton decay is high enough to put pressure on minimal supersymmetric SU(5) [ JHEP05(2020)021 i ¯ f , and and i (2.1) F h H , respectively, , which reside c i u ` H ¯ h ¯ Higgs fields, and H -flat direction along i ¯ acquire VEVs in this 5 ¯ . and H f D 5 b , ¯ d i λ H φ ) with the inflaton mass , playing the role of the a F H 0 and ], the three generations of ] that solves naturally the + φ φ 2.1 5 52 symmetry, under which only 40 - and ab and . These fields develop VEVs – µ 2 F ¯ H Z H + HHh 49 ) in ( 40, the U(1) charges of the , symmetry forbids some unwanted 4 c P λ φ 2 √ = 1 supergravity, which we assume 43 and b 3). The vacuum expectation values / M Z – + φ gauge symmetry down to the U(1) of 3 has an N ). On the other hand, the electroweak a H h 37 √ φ j c Y ]. After V U(1) gauge group down to the SM gauge ` ,..., 2.1 i (3 ¯ abc 8 f / 50 × λ s 3 ij ], as is motivated by the low-energy structure U(1) in ( = 0 λ m + 5 a 56 × a − ( – 3 – λ + L a ¯ h = ¯ hφ ], this model offers the possibility of successful j φ ¯ h f and i a 7 , respectively, a pair of 49 000 8 λ 4 F ¯ H λ λ ij 2 do not acquire masses from the flaton VEV — this is + λ a U(1) GUT model [ ¯ h symmetry also forbids a vector-like mass term for + acquires a VEV. This and , which would cause baryon/lepton-number violation as well × 2 ¯ h ¯ h Hφ 2 and j i H Z H / and H F F s i i ¯ f ia 6 h fields break the SU(5) F m λ ij 1 ¯ = H in λ + and ] inflation, with one of the singlet fields, u ]. In this case the potential 00 representations of SU(5), which we denote by = µ H 75 and 1 53 via the couplings Hh 3, and +5, respectively. The assignments of the quantum numbers for the W i ¯ h − 3 is the generation index. In units of 1 H F Higgs fields, , and 2 , and , ]. For d 10 ¯ and , respectively, break the SU(2) 5 H , ¯ h 76 h = 1 field is odd while the rest of the fields are even. This symmetry is supposed to are +1, 10 i and c i and -parity violation. The ` As discussed in detail in ref. [ We embed the flipped SU(5) model in minimal The renormalizable superpotential in this model is given by In addition to these matter fields, the minimal flipped SU(5) model contains a pair H R 10 h , which is advantageous for suppressing rapid proton decay induced by colour-triplet , respectively, and four singlet fields, ¯ ¯ an economical realization ofdoublet-triplet the splitting missing-partner problem. mechanism [ Starobinsky-like [ inflaton [ of string theory [ a linear combination ofin the this singlet direction, components as in ‘flaton’ discussed direction, in the detail coloured in componentsthose ref. in in [ these fields formdoublets vector-like multiplets with as H Higgs exchange. to have a Kählerpotential of no-scale form [ We assume here that thethe model possesses an approximate be violated by somedomain walls Planck-scale when suppressed the field operators,terms, such which as prevent the formation of h (VEVs) of the group, and subsequently thein VEVs of theelectromagnetism. doublet Higgs fields and right-handed leptons, up- and down-type quarksSU(5) are assignments, “flipped” giving with the respect model to its the standard flippant name. of In the no-scale flippedthe SU(5) minimal supersymmetric extensionembedded, of together the with Standard three Modelsets right-handed of singlet (MSSM) neutrino matter chiral fieldswhere superfields, are into three 2 The no-scale flipped SU(5) model JHEP05(2020)021 - ]. ]. R , is 68 77 s = 0, (2.3) (2.2) ] and n 2. To 3 i 6 , 1 λ 79 , , 78 and expressed = 0 i ¯ f ]. a . Nevertheless, 77 -parity, generates and 000 8 6= 0 (Scenario B), i λ R 0 F , i 6 2 are real and diagonal. λ coupling; i.e., = 0 for φ ab 6 a 7 − µ λ , λ 1 ] by scanning over possible φ , − and o 52 , , c j 0 ij 2 ν φ λ 51 ij − ) . , c c i 3 ν ` U − ( coupling in this model controls i) infla- , ¯ h, φ i 6 , ¯ , f c j λ = 0 (Scenario A) or -parity, under which the fields in this model d using the field redefinition of − o 0 j R , i 6 6 ¯ 2. We also assume , does not have the ji i H, h, λ iϕ , the reduced Planck mass, the measured ampli- ) 3 U = 1, where these matrices are as defined in ref. [ l F , just as in the next-to-minimal supersymmetric – 4 – 1 − φ 2 , i , φ U . ], the e H, / 3 can couple to the matter sector via the couplings c j ( 1 U ij ij e φ j 0 δ 52 − h i → − → ) – = 0 φ = 3 7 ij iϕ ) ,L ,V 2 3 λ N , which is allowed by the modified i e a c c 49 c i l 3 7 u , φ Q u = = λ U ]. U 1 πG ¯ h, φ n n 3 ij µ 52 ]: ) = (8 , φ , 3 3 = ( 7 – 0 3 and -parity is slightly violated by the coupling ¯ h u i i U 68 c i ≡ ¯ , ¯ ` f 49 U F R , φ H, h, 2 c i P , as ( and , ` H, 7 M i ¯ ], so the LSP can be a good dark matter candidate. We also note ), the inflaton U h f = 1 ], two distinct cases, , i i can acquire a VEV without spontaneously breaking the modified 2.1 81 , which we denote by 49 ]. In particular, we showed in refs. [ F , 0 3 , well within the range allowed by the Planck results and other data [ φ 82 50 3 φ − GeV and term for ]. The predicted value of the tilt in the scalar perturbation spectrum, 10 6= 0 for that the observed values of neutrino masses, the dark matter abundance, and -parity-violating effect is only very weakly transmitted to the matter sector, 13 µ 80 × 6 R ia 6 . In ref. [ 10 3 λ 7 λ λ × ' 3 r Without loss of generality, we adopt the basis where As discussed in detail in refs. [ As seen in eq. ( ' We use the basis in which and 3 s 6 Moreover, we have removed the overall phasethe factor diagonal phase matrix In this case, theSU(5) MSSM representations matter as in fields [ and right-handed neutrinos are embedded into the via leptogenesis [ values of baryon asymmetry can bebreaking explained simultaneously, scale together in with ascenario the soft developed multi-TeV supersymmetry- in range. refs. [ In this paper, we study nucleon decays in the an effective extension of the SM. ton decays and reheating; ii)abundance of the the gravitino LSP; production iii) rate neutrino and masses; therefore and iv) the the non-thermal baryon asymmetry of the Universe since this the lifetime of the lightestof supersymmetric the particle Universe (LSP) is [ that still much the longer singlet than theparity. age In this case, the coupling transform as We note that this modified λ were studied. We focus onfields Scenario B other in than this work.whereas In this scenario, one ofrealize the this three scenario, singlet we introduce a modified tude of the primordial powerratio spectrum is successfully reproducedThis and prediction the can tensor-to-scalar beLiteBIRD tested [ in futurealso CMB within experiments the such range favoured as by CMB-S4 Planck [ and other data at the 68% CL [ m JHEP05(2020)021 ]: ]. 92 ∗ MNS 68 – (2.6) (2.7) (2.8) (2.9) (2.4) (2.5) U , and l (2.10) 84 = U , = 0 [ c ν i PMNS U ϕ U i ., c . P 4 , + h ) 2    i c a , µ j ], and that ˜ 1 ν ν φ , 83 ) is only an approximate    , µ . ! s j 2    2.5 ν m i i 0 ij ¯ c , H ) ¯ h ˜ ν a 2 h i 0 h i , so that 2) are given by µ e diag( jk , ¯ ν c H ja 6 ) 1 1 2 ˜ ν U c λ . , PMNS k h ν satisfy the condition . ) c i U = i U c ν † ja 6 ( ¯ ν i 0 ( c D ν H a = 0 U λ ˜ ¯ h ν U ϕ

ik c µ are written as 0 h m h ν ia 6 ) ν ( i c 2 ij λ -ﬂat direction of the singlet components of = ja 6 m a, b ν : m λ L i ( 2 λ ∗ c c ν D , U , µ T ν ν 1 ( i U , ) j 2 ab U 0 U – 5 – t corresponds to the fermionic superpartner of the X λ µ ¯ and h =0 = i 2 0 0 h a = 0 i , m λ ij 2 ˜ ν D - and c φ c ,L Q = D ν λ F and m k ! ij m    , m X j ) ij 2 u c d i λ ν GeV in the following analysis. We diagonalize the mass = a . The ﬁrst equation in eq. ( u m ij ˜ 2, and φ m V , 16 ij ab ( ) GeV (see above). In what follows we express these matrices c j 1 δ

ν , a ν 13 diag( µ i m = i ( ν = 10 i 10 0 = 0 = 1 i ¯ h Q a × h ¯ 1 2 c H (10)% and depend on the mass spectrum of the theory, we neglect ab ]. ˜ ν − h µ ' O 68 = 3 2 = ) using a unitary matrix λ s 3 and . The mass matrix of the right-handed neutrinos is then obtained from a , and m 0 2 2.7 are the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements, is the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix. , mass φ ij denotes the VEV of the L ij δ i 2 V i = 1 : we take λ ¯ c H ¯ PMNS ˜ H ν = h U i, j ij 2 The neutrino/singlet-fermion mass matrix can be written as The diagonal components of are unitary matrices, and the phase factors We define the PMNS matrix as in the Review of Particle Physics (RPP) [ to deviate from the up-type Yukawa couplings at low energies. However, since these λ and 4 c l 2 This mass matrix is diagonalised by a unitary matrix in the notation of ref. [ The light neutrino mass matrix is then obtained through a second seesaw mechanism [ where H matrix in eq. ( where inflaton field first seesaw mechanism: them in the following analysis. where we take as expression, since in generalλ renormalization-group effects and thresholdeffects corrections are cause at most where U The components of the doublet fields where the JHEP05(2020)021 5 , via . and c (3.2) (3.3) (3.4) (3.5) (3.1) 2 . 3 (2.11) µ g ): + h and aβ 2.10 1 Q µ α a , X c β l † ν L matrices, as well as in eq. ( U c , † c ν † ) ν α k c l c indices. U e ν m Q c ( C G † 3 c k αβ is non-zero, and is given by g 2 u and ) and e G , + ν 3 ijkl 6(1) g c c 2.3 B m 2 0 , C d g 6(2) † are the SU(5) gauge vector super- ijkl − j 2 3 are SU(3) e O iϕ α a − e . e VP B X b 0 ∗ kj α T ijkl b 6(2) g ν in eq. ( † V 2 3 l C U a, b, c c j X li e ∗ † l d U ) ) l + bβ j U a U aα Q ( = † Q 6(1) ijkl c i 2 X aα i – 6 – 2 ( 5 u ], the main contribution to nucleon decay is due O g Q M abc gauge vector superfields, respectively, and 40 αβ ijkl PMNS 6(1) = αβ Y indices, and + U C β abc L abc ijkl = 6(1) L , the eigenvalues of the , are uniquely determined as functions of ¯ ¯ C θ θ T ν l ia 6 6 eff 2 2 λ U U L α θd θd a and U(1) 2 2 plays an important role in determining the partial decay widths X d d † l and C ) are SU(2) vanishes in flipped SU(5), we retain it in the following formulae so that it can U c a Z Z c ν u ( = = ijkl 6(2) U α, β C αβ , are non-zero, but in flipped SU(5) only 6(1) 6(2) ijkl ijkl ij − δ 2) O O , the SU(3) i 5 ). The PMNS matrix is given by iϕ g ijkl 6(1 B e 2 is the SU(5) gauge boson mass. The Wilson coefficients are run down to low C 2.9 . The matrix √ ≡ ν is the SU(5) gauge coupling constant, the X )–( U and = ij 5 M g P 2.7 G Below the GUT scale, the effects of SU(5) gauge boson exchanges are in general de- However, although gauge from the corresponding gauge couplings. In the unflipped SU(5) GUT both of the Wilson 5 l 0 K where energy scales using the renormalisation group equations. Thealso renormalisation be factors used for for the unflipped case. with g coefficients scribed by the dimension-six effective operators where where fields, We are now ready todimension-5 discuss nucleon contribution decay mediated in by our colouredmechanism model. Higgs in In fields the view thanks flipped of toto SU(5) the the exchanges GUT suppression missing-partner of [ of SU(5) the gauge bosons. The relevant gauge interaction terms are U of proton decay modes, as we will see in the3 subsequent section. Nucleon decay in flipped SU(5) the mixing matrices eqs. ( Using the measured values of the PMNS matrix elements, we can use this relation to obtain We note that, given a matrix JHEP05(2020)021 247. (3.7) (3.6) . ]. The indicate , ]. = 1 97 µ ) 93 L Ri l A and R c 7 e u )( , , , are evaluated at L b n d 82 23 82 11 S 23 1 L a 198 18 − − A u − − ( ) ) ) ) . ) ) ) ) abc i Z Z ϕ ) ] SUSY SUSY m m i ( ( 2 GUT GUT µ µ SUSY SUSY 1 1 ( ( µ µ µ µ ). The subscripts 1 1 α α ( ( ( ( 1 1 α α 1 1 udul ... α α ( α α )( 38 27 38 27 ] at the two-loop level: LR 049(2)(5) 3 2 3 2 134(4)(14) 186(6)(18) 118(3)(12) 131(4)(13) ... . . . . . C − − 3) the gauge coupling constants of the 0 96 ) ) 0 0 0 0 , 0.099(2)(10) 0.103(3)(11) ) ) 2 + ) ) − − ) ) Z Z , ) SUSY SUSY m m ( ( µ µ Li ]. The relevant hadronic matrix elements are GUT GUT = 1 2 2 SUSY SUSY l ( ( µ µ 2 2 α α µ µ L c µ e 97 – 7 – ( ( i − i − µ e A ( ( i i -boson mass, the SUSY scale and the GUT scale, i − α α u 2 2 i i ( p p 2 2 p p p | | p p Z | | α α | )( α α | | L L 7 2 7 2 A L L L R b L L s d g u u d d 4 9 4 9 u u R R ) ) R R R ) a R ) R R ) ) ) ) ) ) ) u ) ) ) ) Z Z ( ud us us us ud ( ( ud ud ( ( | | SUSY SUSY m m ( | | ( ( | abc ) with ( ( | | µ µ + + 0 0 GUT GUT 3 3 SUSY SUSY + 0 0 π ( ( denote the ]: µ µ 3 3 α α µ µ ) K K K K π π π ( ( i (4 ( ( h h h h h h h Matrix element Value [GeV α α 3 3 3 3 / 95 α α , 2 α α A GUT g × × µ udul 94 ( ≡ = = as [ 1 2 RL A S S C 6 α , and A A . The relevant effective operators below the electroweak scale are 1 ) = 2) between the GUT scale and the electroweak scale, , SUSY + i l . µ 0 , + π = 1 Z e . Hadronic matrix elements used in our analysis, which are taken from ref. [ 0 n m → ( π p ) In the following we summarise the partial decay widths for the proton decay modes ( The two-loop RGEs forWe these note coefficients that above these the partial SUSY-breaking decay scale widths are do given not in depend ref. on [ the phases n 6 7 → L ijkl 6( p corresponding hadronic matrix elements,QCD for lattice which simulation we performedlisted use in in ref. the table [ results obtained from the that we discuss in this paper, as well as two relevant neutron decay modes. respectively, and SM gauge groups. We givein the what electroweak-scale follows. matching conditions Below forrenormalization the each factor, electroweak decay scale, which mode we is takeWe computed into then in account calculate ref. the the perturbative [ partial QCD decay widths of various proton decay modes by using the where that the matrix elements are evaluated at the corresponding lepton kinematicC points. the one-loop level Table 1 statistical and systematic uncertainties are indicated by ( JHEP05(2020)021 ) for (3.8) (3.9) (3.12) (3.14) (3.15) (3.13) (3.10) (3.11) 3.9 mode as , , 2 2 , + e . . ). µ e 0 2 i i | π p p ) 5.9 | | yrs yrs + i , L L l 2 → 2 u 0 u 2 ) vanishes for this | p R π R ) ) 11 8 . ) 3.7 l 5 → 5 4 ud ud ( α U ( α 0378 , . 0378 p | . | . . ( ( i i 0 | 0 0 0 p p R ) π π | | h h ' Z L L 4 4 |A u u 2 1 m 1 ) and the rate in eq. ( 2 S ( 2 S R R 2 S + i ) ) A A 2 A | 11 3.7 2 2 L j 2 L ) 6(2) ud ud ) X GeV X GeV 2 ( ( 2 A + i | A | | C | l 2 2 0 0 M M 0 16 11 16 ud π π ) π h h l V ) + 2 p 2 π 2 p 10 2 π 10 | ) ) U i i Z ( m → m m m | , 2 2 m 1 p ) ( − − 2 S − ( − i | | Z udul udul 1 L 1 A ( ( 1 j 21 11 m + (1 + 1 6(2) ) ) ( – 8 – |A l l 1 i RL LR 2 C 2 mode is | 2 S | U U 2 C C ( ( 1 111 21 A 6(1) + 11 j ) ) µ l 2 p C V l 2 π × | × | 0 ' ) = ) = U U 4 m m 4 X π ( X ( + + i i | 35 35 | l l 2 ) = ) = 2 0 0 | − i i | → 10 8 10 πM π π πM 1 ]. ud ud p × × V 32 V flipped 32 | | → → 7 ) udul udul unflipped 9 p p π 70 . p . ( ( ) , p p + 9 7 ( ( m m + e m 32 4 5 e 0 RL LR 4 L 5 R 68 ' ' ), we obtain 0 g g , π )) C C A A = 0 in flipped SU(5), the second term in eq. ( π ) = = 3.9 = → 57 4.2 + i , → denote the masses of the proton and pion, respectively, and here and l ), we can readily compute the partial lifetime of the ijkl p 6(2) 0 flipped flipped ( p in specific flipped SU(5) GUT scenarios are discussed later: see eq. ( 17 π ) ) C ( π τ + + τ m 11 flipped By using the effective Lagrangian in eq. ( 3.11 flipped ) e µ ) l ) → 0 0 + U + π p . π e µ and 0 0 + Γ( = 1 in eq. ( → → π π p µ i 0 p p m ( ( → → π τ τ From eq. ( p p Values of ( 8 → Γ( Γ( = 2, we have and the partial lifetime of the p i factor (see also eq. ( as found in refs. [ We note that this tends to be longer than the lifetime predicted in unflipped SU(5) by a where subsequently the subscript onat the the hadronic corresponding matrix lepton element kinematic point. indicates that it is evaluated Setting at the hadronic scale: where Note that, since model. The partial decay width can be expressed as follows in terms of these coefficients where JHEP05(2020)021 (3.23) (3.24) (3.16) (3.17) (3.18) (3.21) (3.27) (3.22) (3.25) (3.26) (3.19) (3.20) . , and those 2 ) i . , e ¯ ν i Ri 0 2 l p 2 | π | c R i ) L u p | + u i → l )( L R 0 b L ) n d s K R , , a L us ) , . ( 2 u | | . ) . i i → ( ) 0 ud i p p i ( Li | | p ) p | ¯ ν abc K | ( ν L L Z h + + L c L R u u π π d d ) m 1 R R h i , . , ( |A R ) ) 2 S )( i ) ) ) ) → i 1 b R A 11 Z + + i us us ¯ 2 S ν j d l 6(2) p 2 L ( ( ud 2 usul 0 | | ( + m ( | A a R ( C | A 0 0 ( ) π π 2 L u 2 + |A ( ji + i K K LR A l 2 π h h → 11 2 ) + 0 → 6(1) h ) ) C abc 2 p ) i i 2 K Z p ) through SU(2) isospin relations: C i p K 2 p 2 π + 1 , m p 2 2 π + m i m ) j l ) m Γ( m i ( ) 0 V m i → m − Z usul usul 1 2 1 π Li ( ( 1 uddν − j − p l m ( 1 − ( 6(2) c L ( – 9 – 1 ) = 2Γ( L RL LR uddν i → 1 C 2 u ) = ( | RL i + i ) = C C p l i 2 )( ¯ |A ν p 121 π C 6(1) 11 j 2 0 RL − b R | ) 2 C V l m 1 s π 32 π C i ) = ) = U ) a R 4 4 X ) = X l ( + + i i uddν i | u p 2 l l → → ( 2 K ¯ ) = ) = U ( ν 2 0 0 i i ) = ) = | ( m i n i n + m | πM πM RL ¯ K K ¯ ν ν p abc us π Γ( C Γ( + + 32 V − 32 m | usul usul → → π ) π 4 5 p 1 → ( ( i g p p p m ( ( → → ( RL LR 4 5 p π L = R g p p C C A usul ( m A A 32 ( = Γ( L RL = 1, we have flipped ) = C i ) i + i l flipped ¯ We note in passing that the rates of neutron decay modes that include a ν The effective interactions in this case are given by 0 ) ) = The relevant effective Lagrangian term in this case is There is a relation between the partial decay widths for + + given by isospin: + i K π . l . e i is the kaon mass and 0 . 0 . + ¯ + ν i i l → → e ¯ K ν K + ¯ K ν − 0 p p 0 π + m π → → π K π Γ( Γ( p p → ( → p → L Γ( → → In particular, for We then obtain the partial decay width where with which applies to both the flipped andp unflipped SU(5) models. We then have n of with with the following matching condition at the electroweak scale The partial decay width is then computed as charged lepton can be obtained from Γ( which applies to both the flipped andp unflipped SU(5) models. n JHEP05(2020)021 , 13 (4.2) (4.1) ) are (3.28) (3.29) (3.30) (3.31) (3.32) . 3.2 . , 2 2 ) 2 S i µ 100 TeV [ A ν i 2 p c L ) & | s 2 L | )( u ud b R R d V ) | ]. For more detailed a R us u ( ( 46 | – 0 abc +(1+ K 44 1 h 2 S ) i 1 A , . 2 S ) ) . 2 A Z Z , jl 2 L udsν e ) ( m m i , ∗ A ( ( p , 2 | jl = 0 ji ji V RL L δ ( C 12 11 6(1) 6(1) ji u 2 p ik ik 2 K C C R δ δ 11 6(1) + ) = 0 i i ) m 1 2 m i j j C ¯ iϕ iϕ ν ) 2 V V ud − i e e j + ( ν 1 − − | V ]. 2 2 X X 2 2 5 5 c L 0 K g g – 10 – d π = 2 M M h | 99 ) = ) = )( , → i i ji − − 21 b R ), we have 2 L p ) s 12 6(1) l 98 = = A , a R U C Γ( 2 4 X ( u usdν udsν 1 | 3.25 ( j ( ( 24 2 ijkl ijkl 6(2) 6(1) , | V 2 p 2 π πM abc RL RL C C us m m 20 C C V 32 , | ) − i p 1 13 m 4 5 4 g X usdν p = 2 in eq. ( ( i = m 4 5 πM RL ]. g C 32 68 With flipped The low-energy effective interactions for this decay mode is given by ) = ) i ) = ¯ + ν . + . µ + e i . + 0 0 ¯ ν K µ + π K + 0 e 0 → → → ] or if a suitable missing-partner mechanism is invoked [ K K π p p In unflipped SU(5), the Wilson coefficients of the effective operators in eq. ( p ( 24 – Γ( → → L Γ( → expression for each partial decay width. p The rest of the calculation is exactly the same as before, so we just summarize the resultant 18 discussions of the calculationunflipped of SU(5), proton see decay refs. induced [ by SU(5) gauge bosongiven exchange by in In this section we reviewthat briefly the proton proton decay decay calculation isi.e., in that dominantly unflipped the SU(5), induced assuming dimension-5 byassumption contribution is dimension-6 of valid, e.g., colour-triplet SU(5) when Higgs the gauge sfermion exchange masses boson is are negligible. sufficiently exchange, large, i.e., This as found in ref. [ 4 Dimension-six proton decay in unflipped SU(5) We note that the unitarity of the CKM matrix leads to in the case of flipped SU(5). As a result, we have with p p JHEP05(2020)021 . 2 (4.7) (4.3) (4.4) (4.5) (4.6) 2 S A . 2 ) . 2 . 2 | . 2 i | 2 us 2 | p V | ∗ us | ∗ i us L V p | V d L ud R ud ) d V | V R | us ) + (1 + ( 2 | 2 1 ud e + 2 S ( i µ | i A p K | + p h | L π 2 L h u 2 | u µ R i i ) R 1 ud p ) | δ V us 1 | L i ( ud 2 S | u 2 ( 0 | δ A R 0 ) 2 L K π + h h A us 2 2 ( | 2 2 i 0 2 S 2 S p 2 p 2 π 1 | A K A 2 S L h m m 2 L 2 L s A – 11 – A A 2 L R − 2 L 2 2 ) A 1 A 2 2 ud 2 p 2 p 2 π 2 K ( | 2 | m m 2 p m m ), 2 K + 2 p 2 K 4 X ud + m ), m m K − − m e ), V ), h | + 0 1 + 1 p + πM − µ π − e 2 e 0 0 | 1 m 0 1 32 π 4 4 5 4 X π X → π p us p g V 4 p which may enable future experiments to distinguish these two X m 4 | → m X p → i p → 4 5 πM 4 5 πM 1 9 p Γ( g p m g ) = p δ m / i 4 32 5 πM 32 4 5 πM ) Γ( ¯ ν g Γ( i Γ( g / / ¯ 32 ν + / 32 × ) ) ) + π ) = + ) = + + . π + e µ + µ ) = 0 0 → e ) = µ i ¯ 0 ν 0 0 → ¯ ν + p K K π + . π . µ + . p K . 0 Γ( + K . + + → → → ¯ K ν → Γ( µ e ¯ ν K → µ i p p p + 0 0 → p + 0 p → p P → K K K π π Γ( p Γ( Γ( Γ( Γ( p We assume here that the contributions of dimension-5 operators are suppressed, either by large sparticle i) 9 Γ( v) ii) → → → → → iv) iii) Γ( and/or triplet Higgs masses, or by some missing-partner mechanism. As we now discuss, the predictionsGUT for proton model decay branching are fractions different inunflipped the from flipped those SU(5) SU(5) generated GUT, byGUT dimension-6 scenarios. operators in To thepredictions this for standard end, them we in focus flipped and on unflipped the SU(5) following GUTs: five quantities and compare the 5 Comparison of proton decay rates in flipped and unflipped SU(5) p p p p p JHEP05(2020)021 and 10 (5.4) (5.5) (5.1) (5.2) (5.3) ), we ν U 5.1 is a generic )): , 6 2 4.3 M ) = 1 we have: 2 | A . ud 2 , R (1). | V 2 2 | | ∗ O | ) and ( 01 us . . 0 V 21 11 6 ) 41 ) 4.2 l l ' ud U , we generate 1000 random V U ) | ( (10)%, which mainly comes 6 ( | | + (1 + ) r c + 1 O 2 ), many of the factors in these θ 2 . Z 2 A 2 ) 2 + ) e µ SUSY m c R e i i ( θ 0 µ cos p 008 p , which is determined from 1 2 ( | . | l c ]. We first write the Yukawa matrix π 2 2 1 α 0 θ L L ) U ] , the SU(5) gauge coupling constant, α 2 e µ u u 52 → i i ' , X R R p p sin 100 , ) p ) | | 2 33 M 51 6 L (1 + cos L ud , are cancelled, which makes the prediction ud u u ) ( ( 1 M ) = | | R R 6 S 0 ) 0 ) r A π π unflipped unflipped h h with a logarithmic distribution over the range ud ud ) GUT ) = SUSY ( – 12 – ) and Γ( ( ( | µ + | + µ 6 6 + ( 0 0 ( e ). We also note that by taking the ratio between r µ and λ 1 = µ 1 0 π 0 π nonstrange 0 h h α L α π π ( π ) X A 2.11 2245. . + → → ) = 0 ) → = e flipped flipped p p 0 X X ) ' p ) 1 2 π S S c + Γ( + + Γ( + θ e µ A A + + 0 with each component taking a value of 0 e µ → ]. via eq. ( π π ), we find that this ratio in the flipped SU(5) is given by 6 ≡ unflipped unflipped p → → ) 19 ) M A → → p p + + Γ( 3.14 R p e p µ Γ( Γ( / 0 0 PMNS ) Γ( π Γ( π U + µ → → 0 ) and ( p p 1 in a typical supersymmetric mass spectrum, and for π 3 matrix. We then scan 3 matrices Γ( Γ( ' 3.11 × → × A is a real constant, which plays a role of a scale factor, and p R 6 is the Cabibbo angle: sin r 1) choosing a total of 1000 values. For each value of c θ , 4 To determine the predicted value of the ratio in flipped SU(5) given by eq. ( In unflipped SU(5), on the other hand, we obtain (see eqs. ( This result is consistent with the formula given in ref. [ − in the form , and the renormalization factors, 10 6 5 complex 3 where λ where complex 3 (10 color-triplet Higgs exchange to theseby decay small modes Yukawa in couplings, supersymmetricsfermion and mass SU(5) thus matrices is is [ suppressed negligible unless there isperform a flavor parameter violation scan similar in to the that in refs. [ Hence, the branching fractionthe of electron mode the by muon approximately modeThis two orders is prediction of predicted magnitude is in to againfrom the be the unflipped rather errors SU(5) smaller robust: in GUT. than the the that hadronic matrix uncertainty of elements. is We note also that the contribution of the We find where the PMNS matrix the two partial decay widthsquantities Γ( such as theg SU(5) gauge boson mass, for this ratio rather robust. From eqs. ( We see that this ratio depends on the unitary matrix 5.1 Γ( JHEP05(2020)021 l 2 U µ = 2 and (5.6) , are ` ν 2 ` and m , m 1 µ    − 3 2 matrices for ) in the NO 2 3 α 0 i + 6 m e e λ 0 -fit 4.0 given in 2 2 π ≡ 2], the Dirac CP ν α i ` e 2 3 → , π/ 1 00 0 0 0 m p 2 . As for the Majorana    Γ( δ 10 = [0 /    ) iδ ij + 13 13 NO IO ) − θ and ∆ c c µ e + 0 2 1 , and = 1 for the NO case and 23 23 e 13 , the eigenvalues of the π 1 c s 0 ` 6 s 13 m θ π 10 λ → iδ − , iδ e p 2 2 e → 23 13 θ are obtained as functions of 13 m s p , s 0 ν 23 Γ( 12 23 ≡ c U ) in the flipped SU(5) GUT model for the NO 10 θ s 13 / parameters by requiring that the observed 3 matrix + 12 c ) e 12 s 2 21 + 0 × µ 12 s and − is chosen without loss of generality. Again we π s m µ − 0 – 13 – ] for c 2 23 1 ν → π 23 parameter sets sampled. This difference indicates s − c U p 6 12 10 102 12 c with the mixing angles → , Γ( < m c = 0 in this analysis since, as we shall see below, the / − 1 p ) ij 3 θ + 101 iδ m α 2 Γ( µ iδ e ], for each 3 − 0 Unflipped SU(5) e sin 13 π = determined in this manner, we then compute the matrix 10 13 s 52 s , 2 ≡ ν → 23 0 α

23 U s 51

p ij c 400 300 200 100 500 13 Number s 12 c ). We parametrise the PMNS matrix elements following the RPP 12 c c 12 − c − and 2.11 23 23 c , we set s ij 3 12 θ ], and the order 12 s α we display histograms of the ratio Γ( π s ]: ]. By using − 2 cos 1 , 83    and [0 ). We then determine these two ≡ 102 = , 2 . Histograms of Γ( ∈ ij 2.5 α c δ 101 matrices and the mixing matrices In figure As discussed in refs. [ PMNS c U ν cases, respectively, out of asome total preference of for 10 the NO case in our model. and IO scenarios in blue andpredicted green, value respectively. in The unflipped vertical SU(5). black solid As line we represents see, the the flipped SU(5) Model predicts this ratio where phase use the values obtainedphases in refs. [ result scarcely depends oneach these mass phases. ordering, We and generate find the solutions same for number 2399 of and 180 matrix choices for the NO and IO convention [ values of the squaredreproduced within mass the differences, experimentalfor uncertainties, ∆ where the IO case.refs. [ For theusing experimental the input, relation we ( use the results from prediction. m in eq. ( Figure 1 and IO cases in blue and green, respectively. The vertical line corresponds to the unflipped SU(5) JHEP05(2020)021 has . On (5.7) (5.8) (5.9) ν (5.10) (5.11) (5.12) 2 U ), , 2.9 10 . , 0 , 9 . ' in eq. ( 2 IO NO 22 ν | iδ m ' − 2 e ) 13 008. We may therefore be 12 , 2 s . 2 13 c ) iδ , 0 s 23 −    23 13 s e s c 2 ' ( 2 12 13 , 12 √ NO IO c 2 s 2 √ c ) in flipped SU(5) are much larger / ) ( / 1    + 23 e + µ i θ i s e − θ p 0 p 23 | | 12 c π 2 2 1 c L L 3 sin 2 12 α u cos √ √ u 3 i 2 − s − → / / α R | R − i ) ) will not be changed even if we take different θ 2 p θ e ) predicted in the NO and IO flipped SU(5) 23 2 − ) c 1 00 1 0 1 0 + 1 ud ud iδ 12 13 e µ Γ( e ( ( c e i c 12 i |    | / – 14 – 0 p s p 0 ) 0 | 23 13 | 12 π 1 00 cos 0 sin 0 π    π + s c s − L L h 38, and h . µ    u ( u → 0 = = = = 0 R R π ) ' ) p ' ∼ 0 1 0 0 0 1 1 0 0 11 13 21 23 ν ) ) ) ) ud Γ( ud →    U ( ( / | | ) p 0 0 ' + π π flipped ∗ ∗ ∗ ∗ flipped ν PMNS PMNS PMNS PMNS h h µ ) ) ( 0 U U U U U + + ( ( ( ( π e µ ' 0 0       π → π ' ' p is found to be 23 for the NO and IO cases, respectively. To understand the origin of → → 11 21 flipped flipped ) ) θ p p . As a consequence, although we have fixed these phases to be zero in our ∼ l l ) ) 3 + U U + Γ( Γ( are then given by α ( ( e µ 0 ν 0 π π U and 10 and . → → 2 0 α p p . We also note that these two expressions do not depend on the unknown Majorana ∗ PMNS ∼ 1 Γ( U Γ( The predicted values of Γ( The values of Γ( = l lightest neutrino mass. Itexperiments may to be measure challenging anybut for deviation the the NO from envisaged and next-generation the IO neutrino central predictions values remain of well separatedthan the and model the hence predictions, standard distinguishable. unflipped SU(5) prediction, which is phases, analysis, we expect that thevalues results for in these figure phases. scenarios are rather insensitive tothe the other mass of hand, the we lightest also neutrino, as see seen there in that figure the spread in predicted values increases with the for IO. Thesefigure approximate estimates are in good agreement with the results given in which leads to for NO, and for IO, where themass eigenvalues first in matrix accordance with inU the the RPP convention. right-hand The side relevant matrix arranges elements the of order of the neutrino for NO, where sin these values, we first notea that, simple due form: to the hierarchical structure of to be JHEP05(2020)021 yrs (5.15) (5.14) (5.13) 34 ) imply 10 × 3.11 4 . . , 2 2 ]. | ) 5 ) and ( 2 | 11 ) l 2 ud | 3.21 U V 1 | ( 1 ud | − V 2 | | 10 NO IO 2 A ud V R . | 4 + (1 + ) to obtain 4 ) in the flipped SU(5) GUT model − ). Eqs. ( 2 2 . 2 A ) + ) + 0 10 ] which can be compared to the e i e 4.2 e R i 0 p 0 | p ' ), respectively. This makes the ratio π | π 2 L 2 L 7 ) ) 104 d e − , → i u [eV] i R 3.15 → p R ) p 10 | p ) and ( ) | p L 103 L Γ( ud d / ud ( u 4.4 unflipped Γ( ) | ) from Super-Kamiokande is 2 R ( ) 10 lightest / | R ) i unflipped + + + − ) ) 0 ) ¯ ) interesting for testing the prediction of flipped ) and ( ν i yrs [ e µ m π π ¯ – 15 – + ud ν 10 0 0 + h h ud ( e ( π ( + | 5.1 π π ( 34 0 | ) 3.12 π + 0 π → + = 13 10 π → π → − h e h p → ( p → ( 0 × 10 p p p ( π 6 Γ( = . τ i Γ( flipped Γ( 16 ) i → i flipped P − ) ¯ ν 1 2 2 1 0

p P

− − + +

) is 1 →

10 10 10 →

e ) given in eq. (

Γ( 10 10 + 0

) e π p Γ( / ) µ π p Γ( +

unflipped / µ + 0 + 0 π e ) ) 0 → i unflipped 0 i ) ¯ ν π π ¯ ν p → + + + e p π Γ( → 0 → π i π Γ( p → p P ( → Γ( p → τ / p ) p Γ( = 1 again, we find + i Γ( Γ( A µ . Scatter plots of values of Γ( 0 i P R π P → p Setting whereas for unflipped SU(5) we can use eqs. ( 5.2 Next we consider thethat ratio for the flipped SU(5) we have lifetimes: the currentand limit that on on predicted partial lifetimes given inΓ( eq. ( SU(5) in future proton decay experiments such as Hyper-Kamiokande [ able to distinguishthe these partial two lifetimes models of inordering these in two future the decay proton case modes. of decayboth We flipped of experiments can SU(5). these by also decay Proton modes, determine measuring decay leading the experiments to neutrino are the mass relatively strongest sensitive available to constraints on proton partial Figure 2 for the NO and IO cases (blue and green, respectively), as functions of the lightest neutrino mass. JHEP05(2020)021 ], 105 (5.16) (5.17) tend to yrs [ ¯ ν 0 ) should be 33 from Super- π 10 ¯ ν , , 5.15 × + → 8 ]), which can be . 1 π 15 . . n 1 94 3 23 , → > ) in the flipped SU(5) ' ' 3 ) + p and 2 20 ¯ e ν 2 13 10 ) , 0 0 s ¯ ν ) 13 2 π π | 1 c + + NO IO 19 , e π ud 12 ): → → 0 c 1 V 13 | ( p π n ) in standard unflipped SU(5). 2 2 5.9 ( → 2 | 2 + ) Γ( τ ) 10 e e i / → ud p i 0 ) p V . i | p π | p | ¯ ν L 3 2 L 2 + d ) ) in flipped SU(5) for the NO and IO cases Γ( ) → u e π i R + / i R 1 ) p p e ) ) | p i 0 | 10 → ¯ L ν Γ( π ud L ud d ( / p + ). Therefore, the value in eq. ( | u ( ) | R → i π + R ) 0 Γ( ¯ ν ) – 16 – ). First, due to the potential contribution of p π 4.4 ; the present bound on i π h + h + ud yrs and that on + ( → Γ( ( ud ( e π 0 P | ( µ / 0 | ) 32 p 0 10 + i 0 π = → π ¯ ν π π 10 Γ( h

−→ Unflipped SU(5) h + p ( i (see, for instance, refs. [ ( × π → → Γ( 9 ¯ ν . P = i p → flipped p 1 + 3 ) − p i flipped π P Γ( ) ¯ ν > 0 / 10 Γ( + ) + )

i 800 600 400 200 e →

¯ + π ν flipped 1200 1000 0 Number ) histograms of P µ + i p flipped π 0 ) ¯ ν 3 π → π + + p → e π → 0 p Γ( → π p i ( → Γ( p τ p → P p Γ( i Γ( . Histograms of P This ratio is, however, less powerful for distinguishing the flipped and unflipped SU(5) We show in figure prediction. Sincelimit, the predicted the values unflippedtions. in SU(5) the prediction Secondly, flipped canbe the SU(5) in much sensitivities are principle worse of larger mimic thanKamiokande is than the experiments that flipped this to to lower SU(5) predic- for IO, which agree with the results shown inGUTs figure than Γ( the colour-triplet Higgs exchange, we have only a lower limit on the unflipped SU(5) for NO, and regarded as a lower limit on model for the NOlower and limit IO indicated by cases the inestimate vertical blue this solid and line. ratio green, using As the respectively. in approximation the Unflipped given previous SU(5) in subsection, has we eq. can the ( again vertical solid line. We note, however, that inexchange the also supersymmetric induces standardmuch SU(5) larger GUT than the colour-triplet contribution Higgs in eq. ( Figure 3 in blue and green, respectively. The unflipped SU(5) prediction has a lower limit shown as the JHEP05(2020)021 ) ) ) ), , , 3 7 4.5 4.3 − . 3.27 3.32 (5.20) (5.21) (5.18) (5.22) (5.19) 10 16 × ' 3 . . ) and ( is negligible 3 2 2 ) − . 4.6 + ' 2 is small unless | e 10 0 02 + 2 us 2 . )

µ K 0 × V 2 | 0 ∗ | us 8 . ' 2 V → K ud | 1 2 V 2 ∗ p ud

us | | ], so this value can be → ' V V

us ud p 2 + (1 + 2 V V ud

| 20 |

, 2 A V | us ud 2 +(1+ 2 R 19 V V ) | 2 A

µ µ i 2 i 2 2 R 2 ]. Therefore, this ratio can again p ) p ) | | e µ e µ 2 L i i L 2 i i 20 p ) p u p p . On the other hand, the value of u . , | e | | e | . In unflipped SU(5), we use eqs. ( R i + R i L L l L L ) p ) 19 ), and thus we can in principle also use p µ u u | U u u | 0 . In unflipped SU(5), eqs. ( L R R us l L R R ud π ) ) ( ) = 0 ) ) u ( u | U i | 5.18 0 R ¯ R 0 ν us us ) ud ud ) ( ( → π + ( ( K | | | | h h 0 0 us ( 0 0 ud p K ( ( – 17 – | π π K | K 2 2 0 h h h h 0 ) ) ( ( ) in flipped SU(5) is computed from eqs. ( → π 2 π K 2 K + h p 2 2 h 2 2 ) e ( ) ) ) m ) ) m 0 ) to distinguish between these two GUT models. + Γ( 2 π 2 π 2 K 2 K 2 2 + π + − ) − ) µ e m m e m m 2 p π 2 0 2 p 2 K 0 0 ]. In flipped SU(5), on the other hand, as seen in eq. ( → − − π π − − m π m m m ( ) predicted in the flipped SU(5) model in the IO case is so ( 2 p 2 p 12 2 p 2 p p − , − + → → p 2 m m 2 p → m m e = ( ( Γ( ( 11 ( ), we have 0 p m p m / p π ( ( ) Γ( = = + Γ( / Γ( 3.14 = e ) → / / 0 flipped ) + flipped ) ) p ) e + K + 0 + + flipped Γ( µ e flipped µ ) µ / ) 0 0 K 0 0 unflipped ¯ ) → ν ) and ( unflipped + + i ) ) π unflipped e K K K unflipped e ¯ + ν → ) + 0 p ) + 0 + µ + p µ + ] π → K 3.28 K → 0 π e → → 0 e 0 p 0 ) to be p 68 π K → p p → ) to find π → K → Γ( = 1. The contribution of colour-triplet Higgs exchange to = 1. The contribution of the colour-triplet Higgs exchange to Γ( p p → p → p → 4.2 3.11 → A A p Γ( p Γ( p p Γ( R R i Γ( Γ( Γ( Γ( This process tends toflipped be SU(5) GUT the model [ dominantwe decay have mode [ in the supersymmetric standard un- for flavour violation occurs in sfermionbe mass used matrices to [ distinguish between the flipped5.5 and unflipped SU(5) GUTs. Again, this ratio does not dependlead on to the matrix 5.4 Γ( From eqs. ( unless flavour violation occursregarded as in a sfermion prediction mass ofmuch lower unflipped matrices than SU(5). the [ As flippedthe we SU(5) ratio see, prediction Γ( this ( unflipped SU(5) prediction is for As we see, this ratio doesand not ( depend on the matrix large that this might be detectable. 5.3 Γ( The ratio Γ( and ( which areP much lower than the limit on JHEP05(2020)021 ) ) ), → + + + e yrs, µ e 0 p 0 0 π π π 34 Γ( ). / of about ) 10 → → → 3.16 + ], are weaker p p + × p µ e 0 0 Γ( 4 Γ( 106 Γ( . π / / π / Unflipped Flipped (NO/IO) Flipped (NO) Flipped (IO) ) ) ) ]. We have pre- + + + → yrs [ → µ µ µ 0 68 0 p 0 33 , p π π K 10 6 and 2 43 3 . – → × 10 → → 5 p 37 . p p 3 > 2 ) 10 + µ − ) and Γ( π 1 ) vanishes. + ¯ ν e with a sensitivity to a partial lifetime 10 → 0 + + π n K ( µ τ 0 0 → → π p 10 p −→ → Γ( – 18 – / p yrs and 1 ) are less constraining; the present limits on the lifetimes of − + 33 e + 10 l 0 10 − K π × 2 3 − . → → ) we also see clear distinctions between the NO and IO 5 10 n + p > e In all cases we see clear differences between the predictions 0 ) 3 π ], which builds upon earlier studies [ + − e ), Γ( We expect that future proton decay experiments such as Hyper- 11 ) ) ) ) − 52 . 10 + + + + + → π – e e e e 12 µ 4 0 0 0 0 0 ]. p 49 π π π → π π , while the predicted partial decay widths are larger, as shown in eq. ( n Γ( + → → → l → ( / → 0 104 τ p p p ) p , π ¯ p ν Γ( Γ( Γ( ] should have an order of magnitude greater sensitivity to both these decay Γ( + / / / / 5 → Γ( ) ) ) ) 103 i π / + + p ¯ + ν e ) µ µ + 0 0 ¯ 0 ν π → π K yrs. . Compilation of the ratios of proton decay rates predicted in standard unflipped SU(5) + K yrs would constrain the model as much as a sensitivity to π → p → → 36 → p p 35 p ). We recall that Super-Kamiokande has similar sensitivities to these two decay p The ‘Golden Ratio’ from the point of view of our analysis is Γ( → 10 Γ( We note also theThe flipped corresponding SU(5) prediction searches that for Γ( + Γ( Γ( i Γ( e p 11 12 × 0 P of 10 2 these decay modes, than those on modes, and hasrespectively established [ limits onKamiokande [ their partial lifetimesmodes, of and hence 1 havetions. a Indeed, window in the of IO opportunity case to a probe search for both the NO and IO predic- of flipped SU(5)and and Γ( standard SU(5),predictions. and in the cases of Γ( π sented flipped SU(5) predictions(NO) in and inverse scenarios ordering with (IO),unflipped both and SU(5) GUT. compared normal-ordered them Our neutrino results withΓ( for masses the the predictions ratios of of decay the ratesare standard Γ( compiled in figure proton decay experiments, flipped SU(5) is excluded. 6 Discussion and prospects We have explored inmodel this developed paper in various [ nucleon decay modes in the flipped SU(5) GUT (dashed black lines), no-scaleblue shading) flipped or SU(5) inverse-ordered withindependent (IO, of neutrino green the boxes). masses neutrino Cases that mass where are ordering the are normal-ordered flipped indicated (NO, SU(5) by predictions solidThis blue are is lines. a distinctive prediction in flipped SU(5) — if this decay mode is discovered in future Figure 4 JHEP05(2020)021 ] ¯ ]. ν → + 19 yrs. 105 p K 34 → yrs [ 10 currently p yrs. fields. For × ¯ ν 33 search. We + 5 35 ¯ 10 H π + ), is still useful > e 10 × 0 + ) → e × 1 π . + and 0 p 5 1 e π 0 ]. If this decay mode → ), so the current limit H > π > of the upcoming large by the same factor as 13 → + ) p ) e ¯ ¯ ν ν + → ¯ p ν 0 e + + 0 π 0 p Γ( π π π ( π / τ ) → → ]. In fact, it was found in the → → + → p ( µ p n ), involve mixtures of first- and p p 0 ]. However, in the IO model this τ ( 10 ( + , π τ τ 9 µ 0 105 → π p ), does not involve identifiable second- + yrs [ → e ) is predicted to be larger, as shown in the 0 p 32 + π e Γ( 0 yrs. Hence the search for 10 , whereas their contribution to the decay modes / π – 19 – ¯ → ν ) × 34 + + p 9 → π . µ 10 0 p Γ( × / → K ) Γ( 7 p ¯ / ν with a sizable rate [ . ) 3 + → ¯ ν ¯ ν yrs. The current limit π ] or a missing-partner mechanism. If present, dimension-5 + + > p ] would constrain the IO model as much as a sensitivity to π 36 35 ) is only 3 ) K 5 → ¯ [ ν + 10 → + e p → + 0 π p × e π p 0 3 π → ) and Γ( → > + p e ( → p 0 ( τ , while the other ratios remain unchanged in the presence of the dimension- p π τ 4 → p Γ( / ) These examples show that if the upcoming large neutrino experiments do discover We have focused in this paper on the contributions of dimension-6 operators to nucleon The fourth ratio, Γ( Our results highlight the importance of targeting proton decay modes involving final- + e 0 Acknowledgments The work of J.E. was supportedand partly by partly the United by Kingdom the STFCM.A.G.G. Grant Estonian ST/P000258/1 was Research supported Council by via the a Spanish Mobilitas Agencia Pluss grant. Estatal de The Investigaciónthrough work the of 5 operators. In particular,to discriminate the the ‘Golden flipped Ratio’, and Γ( unflipped GUT models. nucleon decay, they willphysics. have interesting opportunities to explore both GUT and flavour is indeed discovered indimension-5 the operators also future, induce thenwith the a flipped charged SU(5) lepton GUTAs is model a negligible result, unless is the there excluded. ratioarrow is in Γ( flavor The figure violation in sfermion masses [ contribution to nucleon decay, thoughsymmetry these such may as suppressed hexality inoperators [ the may presence induce ofconstrained a minimal discrete supersymmetric Standard Modelcould that well the be partial lifetime within of the reach of future proton decay experiments [ order-of-magnitude improvement would correspond to decay. This issuppressed sufficient due for to the theunflipped SU(5), flipped absence on of SU(5) the other a case, hand, vector-like dimension-5 since operators mass dimension-5 can term in proton general for give decay the a is significant neutrino experiments, but increasinganticipated for the sensitivity to the latter mode of constrains the IO modelAgain, even we more, are since unaware it of corresponds any to estimate of the sensitivity in a future experiment, but an values of the ratiocurrent limit predicted on in thelifetime would two be flipped two orders SU(5)corresponds of scenarios to magnitude we shorter than havesets studied. a tighter The constraintare on unaware of the estimates IO of model the than improved that sensitivity to set by the state particles from different generations,K since our ‘Golden Ratio’second-generation and leptons two and others, quarks. 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