Low Scale Trinification Symmetry Through Effects of Dimension-5

arXiv:2108.05564v1 [hep-ph] 12 Aug 2021 † ∗ asscale mass ihboe -aiy h cnrocagsie ihteipc fgr of impact the with i.e. changes solely scenario is the conservation D-parity, D-parity the broken that with mention to noteworthy is It rnfiainsmer aebe icse 1–6 ihphenomeno ( with [12–16] discussed been have symmetry trinification ymty( sc symmetry intermediate witho the and lower with can possibility, mentioned corrections centr above gravitational the the of explore at inclusion to been tempting have is GUTs it supersymmetry(SUSY) Since effect. oki osac o viable Keeping for constant. search coupling to gravitat GUT initiate is and operators scale work P the mass of Thus unification remnants gravity. the the with affect show model operators GUT these the of of presence the fact In pt lnksae edsustegnrlfaeokfrtemode the s In for section-III. framework in general constant the coupling discuss GUT We inverse mass, scale. unification Planck to up fbt togadeetoekitrcin,wieicroaign incorporating Many while interactions, models. electroweak (LRSM) and strong both of nemdaemass intermediate nfiainmass unification o-eomlzbeoeaos eetepeitdvleof focu value main predicted our the Thus Here untestable. it operators. seesaw making non-renormalizable thereby neutrino the probes that observab collider disadvantage with and the scale intermediate has low scale permits intermediate bonus a as correction, iedtcino akmte addt,utaih etioadt e to and neutrino ultralight the candidate, in maximally matter embedded dark of detection like a oepansm fteeprmna rdcin fLC h trin The LHC. of predictions experimental the of some SU explain to way ovninlGTmdl like models GUT conventional gemn ihtepoo ea on n swti h pe ii se limit upper the within is and bound perturbative. decay remain proton the with agreement o cl rnfiainsmer hog ffcso dimensio of effects through symmetry trinification scale Low g lcrncades [email protected] address: Electronic lcrncades [email protected] address: Electronic 3 oee h rsn td sa tep orvsttemdl ihb with models the revisit to attempt an is study present the However thsbe salse hog hoe 1,1]ta nalgrand all in that 16] [15, theorem a through established been has It h tnadMdlS) h oteeatter fpril phys particle of theory elegant most the Model(SM), Standard The h ae sognsda olw.I h etscinw ieabifa brief a give we section next the In follows. as organised is paper The L (3) 6= C g ⊗ 3 R SU a eahee yqatmgaiyeettruhnon-renorma through effect gravity quantum by achieved be can ) M G ffc infiatylwr h nemdaesae en as mass being intermediate scale, the intermediate of oper the choice dimension-5 lowers through significantly effect effect gravity quantum by achieved rnfiainsmer ( symmetry trinification h cesbelmto h rtnlftm npeec fad of presence in lifetime proton (8 the like of limit accessible the osataeuaetdb h ffcs h rdce au of value predicted The effects. the by unaffected are constant oo ce clrwt aso h re fTVmysprs t suppress may TeV of The order the fusion. of gluon mass with through scalar octet color ihDaegnrtd ic h iceeDprt sboe a broken is D-parity discrete the stri no since that generated, expected are is D it with issue, cosmological to reference I 333 (3) eexamine We aevnsigcnrbtosdet n to opadgravitationa and loop (two) one to due contributions vanishing have D L M stehgetitreit ymty h lcrwa iigang mixing electroweak the symmetry, intermediate highest the as ) ⊗ , M 2 U SU , I 2 1 safe aaee.A motn bevto ftepeetwo present the of observation important An parameter. free a as swl steGTculn constant coupling GUT the as well as r(8 or ) (3) E eateto hsc,BraprUiest,Odisha-7600 University, Berhampur Physics, of Department 6 nfidter ihadwtotsupersymmetry without and with theory unified R E U oeswt -aiy(D D-parity with models GUT E 6 17 sa neetn xeso fteSadr oe ocml wit comply to Model Standard the of extension interesting an is [1–7] , E 1 6 U oes(ohSS n o-UY ihDprt violating D-parity with non-SUSY) and SUSY (both models GUT 6 , 8 ]gaduie hoy h oe uoaial otisalthe all contains automatically model the theory, unified grand 9] [8, SU )adeetoektiltfrin(1 fermion triplet electroweak and 0) U oeswt nemdaetiicto ymty npeec fg of presence in symmetry, trinification intermediate with models GUT 5 and (5) SU (3) hniiDash Chandini SU C (2) SO ⊗ M L SU I h ag nfiainmass unification gauge the , 1)ec h oiainfrtiicto isi h nfiddescription unified the in lies trinification for motivation The etc. (10) .INTRODUCTION I. rpe emo a eaelk tbedr atr With matter. dark stable a like behave may fermion triplet (3) L ⊗ ∗ SU n ngh Mishra Snigdha and (3) LR R M en iceelf-ih ymty 1,1]conserving 11] [10, symmetry) left-right discrete a being ( ) ac cl hsc,wihi unipyteunification the imply turn in which physics, scale lanck U α g G 3 g rwlsbuddb tig associated strings by bounded walls or ngs ssont aif wnrqieet hti sin is it that requirements twin satisfy to shown is c etrso h ovninllf-ih symmetric left-right conventional the of features ice L , eae e hsc swl idnfr o energy low form hidden well is physics new related l,bign tee oterneo e TeV. few a of range the to even it bringing ale, 3 fatninfranme fatatv features, attractive of number a for attention of e epnil o hsvnsigcreto.However, correction. vanishing this for responsible eeet.I skonta,GTmdlwt high with model GUT that, known is It effects. le r nffce ygaiainleet,wt the with effects, gravitational by unaffected are iinlprilsie oo ce scalars octet color i.e. particles ditional , 6= oa orcint h ag opig n hence and couplings gauge the to correction ional oia prediction. logical o s1 e.Hwvrwt given a with However TeV. 10 as low cinI,w ietenmrcletmtosfor estimations numerical the give we ection-IV, pantemte-niatraymty Being asymmetry. matter-antimatter the xplain )a h oe cls h rsneof presence The scales. lower the at 0) h U scale. GUT the t vttoa correction, avitational g saogwt h nltclepeso fthe of expression analytical the with along ls hsi iw h rm betv ftepresent the of objective prime the view, in this tr.Icuino h gravitational the of Inclusion ators. M c,nest eetne nawl motivated well a in extended be to needs ics, st civ o nemdaesaethrough scale intermediate low achieve to is s 3 ayi o h eurmnso perturbativity of requirements the for nalysis R ytePac asa ela l couplings all as well as mass Planck the by t tSS.I h rsn ok eso that show we work, present the In SUSY. ut U epouto fteHgsboson Higgs the of production he M nfidtere(Us aigtrinification having theories(GUTs) unified hr h -aiybekn is breaking D-parity the where ) fiainmdlbsdo h ag group gauge the on based model ification ssont ecmail with compatible be to shown is U oe -aiy h -aiybreaking D-parity The D-parity. roken ial ieso- prtr 1,17–22]. [15, operators dimension-5 lizable † swl steGTcoupling GUT the as well as - prtr in operators n-5 7 India 07, swl strsodcorrections. threshold as well as l esin le ksosta,tegravitational the that, shows rk 2 θ intermediate M W I n h intermediate the and hne.Tegauge The changes. t novdissues unsolved its h iefaue fthe of features nice E ravitational 6 grand 2 both SUSY and non-SUSY models. The last section is devoted to a discussion on the phenomenological implications of the numerical results.

II. PERTURBATIVE CRITERIA FOR AN E6 GUT MODEL

The Higgs sector of a GUT model plays a very crucial role in accomplishing two important tasks like breaking the GUT symmetry down to the Standard Model and to give mass to the matter sector in tune with the current experimental data. In general one can have many choices of Higgs, but the minimal choice is always better, provided the obtained unification mass is consistent with proton decay constraint and the GUT coupling constant remains perturbative till Planck scale. In the present context of E6 GUT model, which is rather a larger group with many exotic particles, unlike the conventional SO(10) and SU(5) GUTs, the perturbative requirement [23, 24] needs to be examined for a successful modeling. It is obvious that, for sufficiently large particle content, the non-perturbative regime is reached at relatively low scales. Specifically, addition of new (non-singlet) matter/scalar particles in a given model for the sake of unification, always affects the perturbative condition. It is particularly problematic in non-minimal SUSY models. For the sake of completeness, in the present context of SUSY and non-SUSY E6 GUTs, 1 we analyse this by locating the the Landau pole(µ0), where the inverse GUT coupling constant αG− vanishes. Here, we limit to one-loop R.G. equation to find out the pole, for a given set of gauge, matter and Higgs particles belonging to E6. To derive this constraint, we note the (one-loop) RG equation for the unified gauge coupling constant within the mass range MU and µ0 where MU is the unification mass scale and (µ0) is the position of the Landau pole.

1 1 b µ0 α− (µ )= α− (MU ) ln (1) G 0 G − 2π M U By using the standard formula, the one-loop beta coeﬃcients b is determined by the particle content of the model 1 with contribution from bgauge, bHiggs and bmatter of the GUT group E6. Putting αG− = 0, at the Landau Pole µ0, we have,

2π 1 µ0 = MU exp αG− (MU ) (2) " b #

Representations beta coeﬃcients (b) 27 3 78 12 351 75 ′ 351 84 650 150

TABLE I: Beta coefficient of E6 representations for the GUT gauge coupling evolution . Using the values of beta coefficients from Table-I, we can locate the pole position corresponding to various choices of Higgs sectors. For E GUTs with three generations of matter fields, we note that, bgauge = ( 44) and ( 36) and 6 − − bmatter = 6 and 9 for non-SUSY and SUSY cases respectively. Thus we have,

1 b = 38+ bHiggs; non SUSY − 3 − b = 27+ bHiggs; SUSY (3) − In the present context of E6 GUTs (with one intermediate symmetry) for diﬀerent possible choices of the Higgs sector, the Landaue pole position(µ0) can be estimated. For a rough estimate one can have a limit for choosing the 18.38 16 17 appropriate Higgs in the GUT models. Putting µ = 10 GeV (the reduced Planck scale), MU = 10 10 GeV 0 − in equation (2), the perturbativity till µ0 demands, b 1 =1.146 1.977 (4) α−G (MU ) −

where b is given by equation (3). Usually, in SUSY E6 models, due to large particle contents with high beta coefficients, it is difficult to push the Landaue pole closer to the Planck scale. However in presence of gravitational correction, it may be possible to obtain high values of the unification scale and perturbative gauge coupling up to the Planck scale by appropriate choice of the correction parameter. In the subsequent section we focus on these possibilities. 3

III. THE MODEL FRAMEWORK

We consider here an E6 GUT (with and without SUSY) with D-parity violating intermediate triniﬁcation symmetry SU(3)C SU(3)L SU(3)R where D-parity is spontaneously broken at the uniﬁcation mass scale MU . The corresponding breaking⊗ pattern⊗ is given as,

MU E ( SUSY ) (G )SU(3)C SU(3)L SU(3)R(g L = g R)( SUSY ) 6 ⊗ −→ 333 ⊗ ⊗ 3 6 3 ⊗ MI (G )SU(3)C SU(2)L U(1)Y ( SUSY ) −→ 321 ⊗ ⊗ ⊗ MZ (G )SU(3)C U(1)Q (5) −→ 31 ⊗ The ﬁrst step of spontaneous symmetry breaking from E6 GUT to G333–is achieved by giving a GUT scale VEV to D-parity odd singlet scalar φ(1, 1, 1) contained in 650H E leading to g L = g R. The next stage of symmetry ⊂ 6 3 6 3 breaking i.e. from G333 G321 is done by assigning a non-zero VEV to the G321 neutral component of either 27H or ′ → 351H . The last stage of symmetry breaking i.e. SM to low energy theory (G31) is done by assigning a non-zero VEV to SM Higgs doublet contained in 27H E6. At the uniﬁcation mass scale we introduce⊂ non-renormalizable dimension-5 operator

η µν LNRO = T r Fµν Φ650F (6) −4MG which induces gravitational corrections. This LNRO may arise due to spontaneous compactification effects of extra 19 dimensions or due to the quantum gravity. Here in equation (6), the scale MG MPl 10 GeV, if it is due to ≃ ≃ quantum gravity effect. However, if it emerges as a result of compactification of extra dimensions then MG MPl. ≤ η is a dimensionless parameter and Fµν is the field strength tensor. As has been mentioned before, Φ650 is the odd singlet scalar which takes the VeV at the mass scale MU .

φ0 Φ650 = h idiag ǫ3C , ǫ3L , ǫ3R h i √6    9 9 9 

0 1 η φ |{z} |{z} |{z} with ǫ = h i , we have ǫ3C =0,ǫ3L = ǫ = ǫ3R, which breaks D-parity. This assigned VeV modify the boundary √6 MG − condition of the gauge couplings α3C , α3L and α3R given by,

α C (MU )=(1+ ǫ)α L(MU )=(1 ǫ)α R(MU )= αG(MU ) (7) 3 3 − 3 where αG is the GUT coupling constant. Now using this modified boundary conditions for the gauge couplings, we can express the RG equation at different mass ranges MZ MI and MI MU . As has been mentioned before, − − within the mass scale MI to MU , the LNRO will be operative in the RGE. We may note here that in order to achieve unification in E6 GUT models with one intermediate symmetry with the check point of proton decay constraint, additional light particles may be required. In the present frame work, corresponding to different models, additional light scalars/fermions (with masses M1 and M2) are introduced at lower scales within the mass MI to MZ . These additional particles may be correlated with consistent phenomenology that may be accessible in future collider, which will be discussed in the later section. With the above input the evolutions of the Standard Model gauge couplings can be written as

1 2 U 1 1 b3C M1 b3C M2 b3C MI b3C MU α− (M )= α− + ln + ln + ln + ln (8) 3C Z G 2π M 2π M 2π M 2π M Z 1 2 I

1 2 U 1 1 b2L M1 b2L M2 b2L MI b3L MU α− (M )= α− (1 + ǫ)+ ln + ln + ln + ln (9) 2L Z G 2π M 2π M 2π M 2π M Z 1 2 I

1 2 1 3ǫ 1 bY M1 bY M2 bY MI α− (MZ )= 1 α− + ln + ln + ln Y − 5 G 2π M 2π M 2π M Z 1 2 1 bU + 4 bU M + 5 3L 5 3R ln U (10) 2π M I 1,2 where bi and bi with i =3C, 2L, 1Y are the one-loop beta coeﬃcients between the mass range MZ to M1, M1 to U M2 and M2 to MI respectively. Similarly bi with i =3C, 3L, 3R are the one-loop beta coeﬃcients between the mass range MI to MU . 4

We now follow the standard procedure to obtain the analytical expression for the uniﬁcation mass scale MU , the 1 2 inverse GUT coupling constant αG− and electroweak angle sin θW , given as,

MU 1 3 1 1 ln = 16π αem− (MZ ) αs− (MZ ) MZ 4 bU + bU 2bU " 8 − 3L 3R − 3C α α α Mα 8∆3C 3∆2L 5∆Y ln − − − MZ α=1,2 ( ) X U U U 2 2 2 MI + 4 b3L + b3R 2b3C + 8b3C 3b2L 5bY ln (11) ( − − − ) MZ #

1 1 U U 1 U 1 αG− = 4 b3L + b3R αs− (MZ ) 3b3Cαem− (MZ ) 4 bU + bU 2bU " − 3L 3R − 3C 1 U U α U 5 α α Mα + 4 b3L + b3R ∆3C 3b3C ∆Y + ∆2L ln 2π − 3 MZ α=1,2 ( ! ) X 1 U U 2 U 5 2 2 MI 4 b3L + b3R b3C 3b3C bY + b2L ln (12) −2π ( − 3 ) MZ #

1 2 1 U U U U αs− (MZ ) sin θW = 3b3L 3b3C(1 + ǫ) 4 b3L(1 ǫ) b3R(1 + ǫ) 1 4 bU + bU 2bU " − − − − αem− (MZ ) 3L 3R − 3C n o α (M ) em Z 4 bU + bU 2bU ∆α (5 3ǫ) 5∆α (1 + ǫ) − 16π 3L 3R − 3C 2L − − Y α=1,2 ( X h U U α α α Mα +4 b3L(1 ǫ) b3R(1 + ǫ) 8∆3C 3∆2L 5∆Y ln + − − − − MZ ) i αem(MZ ) U U U 2 2 + 4 b3L + b3R 2b3C b2L(5 3ǫ) 5bY (1 + ǫ) 16π ( − − − U U 2 2 2 MI +4 b3L(1 ǫ) b3R(1 + ǫ) 8b3C 3b2L 5bY ln (13) − − − − ) MZ # α(1,2) Here ∆i(3C,2L,Y ) denote the increment of the corresponding beta coeﬃcients due to addition of particles, such that 1 1 2 2 1 1 1 ∆i = (bi bi) and ∆i = (bi bi ). αem− and αs− are the inverse coupling constant for the electromagnetic interaction and strong− interaction respectively.−

It is observed that for a given choice of intermediate scales MI and M1,2 (mass of additional particles), unification mass scale MU and the corresponding GUT coupling constant are unaffected by the gravitational correction parameter 2 ǫ. However sin θW value is controlled by this parameter. Thus for viable phenomenology, we fine tune ǫ and the free 2 mass parameters M1, M2 and MI , such that sin θW is in agreement with the accepted value 0.23129 [25]. Hence in our numerical estimation, we strictly follow this constraint in order to obtain the unification mass MU , the inverse 1 GUT coupling constant αG− and the proton lifetime τp. Here we confine ourselves to the contribution from gauge dimension-6 operator for calculating the Proton lifetime p e+π0 due to superheavy gauge boson exchange. By → using dimensional analysis τp is given by 4 MU τp = C 5 2 (14) mpαG

where C is O(1) which contains all information about the ﬂavor structure of this theory and mp is the mass of the proton. The∼ experimental bound [26–28] on the lifetime for the above mentioned channel is as follows:

+ 0 34 τp(p e π ) > 1.6 10 yrs → SK × + 0 34 τp(p e π ) > 9.0 10 yrs → HK,2025 × + 0 35 τp(p e π ) > 2.0 10 yrs (15) → HK,2040 ×

In the next section we focus on the quantitative estimation for various viable models both non-SUSY and SUSY inspired possibilities.

IV. NUMERICAL ESTIMATION FOR UNIFICATION MASS, GUT COUPLING CONSTANT AND PROTON DECAY LIFETIME

It is noteworthy to mention that in E6 GUT, with intermediate D-parity conserving trinification symmetry, the mass scale MI is shown to have vanishing multiloop, gravitational as well as threshold corrections. With D-parity violating models both in SUSY and non-SUSY cases in presence of gravitational correction, we can get a range of solutions for the free parameters M1, M2 and MI consistent with unification. However as mentioned before, it is observed that in the E6 GUT, even with gravitational correction, to obtain phenomenologically consistent MU for comparatively low MI , additional light particles are necessary. In the present section we analyze those possiblility with numerical estimation so as to have testable predictions. We refer to the Appendix for more details about the particle contents and the corresponding beta coefficients for the models.

A. NON-SUSY Models:

It has been pointed out [14] that, the non-SUSY E6 GUT with G333 intermediate symmetry (both D-parity conserving and broken) do not admit consistent unification with only (1, 3, 3) of 27H . In presence of gravitational correction, it is possible, but with high MI closer to MU . However to achieve comparatively low MI with consistent MU , gravitational correction alone is not sufficient. The consistency of the present framework can be materialised by introducing additional particles either scalar or a fermion around TeV scale or higher. Four different models, which allow low 9 MI 10 GeV in comply with experimentally accessible proton decay, are analysed. The viability of the models is shown≤ below with a quantitative estimation in Table-II.

−1 Models Particles Content Particle Particle ǫ MI MU αG τp (MI − MU ) with mass with mass (GeV) (GeV) (years) M1(GeV) M2(GeV)

(8, 1, 0)S −0.352147 106 1016.00 33.1212 3.15 × 1035 105 −0.188465 109 1016.05 35.3933 5.70 × 1035

27F ⊕ (1, 3, 3)27 Model-I H − {(1, 8, 8) ⊕ (8, 1, 1)} 650H

−0.238286 108 1015.77 34.9779 4.22 × 1034 107 −0.186661 109 1015.78 35.7353 4.83 × 1034

(1, 3, 0)F (8, 1, 0)F −0.558954 106 1015.93 29.0173 1.27 × 1035 105 −0.418357 108 1016.10 30.6664 6.78 × 1035

27F ⊕ {(1, 8, 1) ⊕ (8, 1, 1)} 3 Model-II 78F 10 (1, 3, 3)27H ⊕ (1, 8, 8)650H

−0.476418 107 1015.75 30.2816 2.63 × 1034 106 −0.345651 109 1015.92 31.9307 1.40 × 1035 , , , , ′ 27F ⊕ (8 3 3) ⊕ (1 6 6) 351 1 H Model-III (8, 2, 2 )S (8, 1, 0)S (1, 3, 3)27H ⊕ (1, 8, 8)650H 3 4 10 10 −0.831056 109 1016.22 8.98183 1.76 × 1035

27F ⊕ (1, 8, 1)78F ⊕

Model-IV (1, 3, 3)27H ⊕ (1, 3, 0)F − ′ (1, 3, 3) ⊕ (1, 6, 6) 351 H 3 10 −0.332393 108 1015.82 38.2902 8.02 × 1034

−1 TABLE II: Numerically estimated values for MI , MU , αG , τp for non-SUSY E6 GUTs . 6

We now summarise the observations with relevant phenomenological implications. The Model-I utilises only 650H 27H with additional light multiplets (1, 8, 8) (8, 1, 1) of 650H . The model can accommodate a color { ⊕ } { 5 ⊕7 } { } 6 9 octet scalar, with possible mass M1 ranging from 10 10 GeV, in tune with MI -10 10 GeV, such that the proton decay will be experimentally accessible in future. These− color octet scalars are phenomenologically-rich− example of physics beyond the SM. Model-II utilises 650H 27H along with a non-standard fermion 78F , so as to allow a TeV scale electroweak triplet (1, 3, 0), accompanied{ ⊕ by a} color octet (8, 1, 0) fermion. The triplet fermion with even matter-parity may act as a stable dark matter candidate. The accompanied colored fermion′ may have mass ranging 5 6 6 9 10 10 GeV, for a permissible MI within 10 10 GeV. In Model-III we use the 351 Higgs along with the − − { } conventional scalars 650H 27H . Through gravitational correction we can have low MI with successful gauge uniﬁcation. The most{ interesting⊕ } feature of the model shows the presence of an electroweak color octet scalar [29] (8, 2, 1 ) along with a (8, 1, 0), which can be very informative from the collider physics point of view. In Model-IV we 2 ′ use the 351 Higgs along with the conventional scalars 650H 27H and the 78F for successful gauge uniﬁcation. A TeV scale{ electroweak} triplet fermion can be accommodated,{ ⊕ which}may be a stable dark matter candidate. The 8 intermediate scale MI = 10 GeV, such that MU is consistent with respect to accessible proton decay. Model-I and II shows the Landau pole beyond the Planck scale. The Model-III is perturbative till 1016.82 GeV and Model-IV has Landau Pole position at 1017.95 GeV.

B. SUSY Models:

SUSY E6 models with D-parity conserving G333 intermediate symmetry have been discussed by some authors [14, 15] which accommodate high intermediate scale > 1015 GeV. However gravity induced D-parity broken models can trigger low MI scale as low as TeV scale. In the present framework we observe that low MI may not always comply with experimentally accessible proton decay constraint. Additional light multiplets are required to solve the purpose as has been mentioned in case of non-SUSY models. We now consider three diﬀerent models to analyse the phenomenological viability. For simplicity, we assume that the SUSY scale MS coincides with MZ in all models. The numerical estimation is given in Table-III.

−1 Models Particles Content Particle Particle ǫ MI MU αG τp (MI − MU ) with mass with mass (GeV) (GeV) (years) M1(GeV) M2(GeV) −0.839354 104 1016.34 10.7103 7.54 × 1035 5 16.34 35 , , 27 , , 27F ⊕ 2 (1 3 3) H ⊕ (1 3 3)27H −0.699153 10 10 11.8097 9.17 × 10 Model-I − − 6 16.34 36 ⊕(1, 1, 8)650H −0.582832 10 10 12.9091 1.10 × 10 −0.48477 107 1016.34 14.0085 1.29 × 1036 −0.887257 104 1016.24 10.7103 3.00 × 1035 27F ⊕ {(1, 8, 1) ⊕ (1, 1, 8)} 78F 5 16.14 35 (1, 3, 0)F −0.78604 10 10 11.8097 1.45 × 10 Model-II ⊕(1, 3, 3)27 ⊕ (1, 3, 3) − H 27H 103 −0.702064 106 1016.04 12.9091 6.91 × 1034 ⊕(1, 1, 8)650 H −0.631268 107 1015.94 14.0085 3.24 × 1034

(8, 1, 0)S

27F ⊕ (1, 3, 3)27 Model-III H − ′ ⊕(1, 6, 6)351 ⊕ (8, 1, 1)650H H 106 −0.55056 1012.45 1016.04 8.95951 3.33 × 1034 107 −0.450654 1012.65 1015.95 10.3765 1.95 × 1034 108 −0.356779 1013 1015.96 11.8485 2.79 × 1034

−1 TABLE III: Numerically estimated values for MI , MU , αG and τp for SUSY E6 GUTs .

We now discuss the qualitative implication of the SUSY models. The Model-I uses the 650H 2(27H 27H ) { ⊕ ⊕ } for successful gauge unification, where we introduce additional light multiplet (1, 1, 8) of 650 at MI . The Model 16.34 { } ensures gauge unification at a fixed point i.e. MU = 10 GeV for all possible values of MI , making it more predictive. However in order to satisfy the proton decay check point, the intermediate scale MI is constrained to 4 8 take a value within 10 10 GeV. The Model-II utilises the 650H (27H 27H ) and a non-standard fermion 78F − { ⊕ ⊕ } 4 7 to achieve gauge unification within the accessible proton decay for low MI in the range 10 10 GeV. The Model accommodates a TeV scale electroweak triplet, which may be a dark matter candidate. The Model-III− has the scalar ′ 6 8 contents 650H 27H 351 which includes a color octet (8, 1, 0) scalar with permissible mass 10 10 GeV so as { ⊕ ⊕ H } − to comply with the proton decay. However, it is observed that the high intermediate scale MI is inevitable in presence 7 of color octet scalar unlike the case of non-SUSY model. In Models I and II, we note that for given MI , the inverse gut couplings are same, even if the models are distinct. Accordingly, we quote an interesting remark for the SUSY GUT models with G333 intermediate symmetry, It states that: “For all E6 supersymmetric Grand Unified Theories (GUTs) with intermediate D-parity violating trinification symmetry, in presence of the conventional MSSM (minimal SUSY standard model) particles along with or without additional color singlets at the lower scale (within MZ MI ) − the inverse GUT coupling attains a constant value corresponding to a given MI .” It is estimated to be,

1 1 3 MI α− = α− + ln (16) G s 2π M Z 1 Thus αG− solely depends on MI , leading to a constant value for all such models. However the above remark is valid with one-loop renormalisation eﬀects only.

As has been mentioned before all the above models are perturbative till a point slightly above the gauge unification scale MU . This is due to large particle contents leading to high contribution to the corresponding beta coefficient. The early non-perturbative region [30] is mainly due to the large representation 650 , which is unavoidable in case of trinification symmetry at the intermediate scale as well as for inclusion of gravitational{ } correction.

V. CONCLUSION

We have investigated the effect of gravitational correction through non-renormalizable operators in E6 GUT models (with and without SUSY) with the main objective to achieve low intermediate scale MI . The analysis is confined to D-parity violating trinification symmetry at the intermediate scale. It is nice to mention that it is indeed possible 9 to achieve low MI 10 GeV for the models with successful gauge unification in comply with the verifiable proton ≤ 1 lifetime. As far as phenomenological implication is concerned, a TeV scale color octet electroweak doublet (8, 2, 2 ) [29] scalar can be predicted in E6 GUT (non-SUSY Model-III), which can be very interesting from the collider physics point of view. These colored scalars, would be produced in pairs with a large rate at the LHC, which may affect the Higgs boson production cross section in gluon fusion [31, 32]. Based on the predictions SUSY inspired models-I and II, we establish a very interesting remark for E6 SUSY GUTs with trinification G333(g3L = g3R) symmetry at 1 6 the intermediate scale MI , such that αG− remains constant for a given MI , irrespective of the models chosen, even in presence of additional particles. The models also give interesting predictions on dark matter candidates (1, 3, 0)F and occurrence of color octet scalar (8, 1, 0). These GUT models with low MI may contribute to matter asymmetry through non-thermal or resonant leptogenesis, thermal leptogenesis without gravitino overabundance. However the SUSY models suffer from obvious problem of non-perturbativity beyond the unification scale. We hope the problem can be solved with the concept of Seiberg duality [33, 34]. With reference to the cosmological issues [35–37], since the discrete D-parity is broken at the GUT scale, the models are not plagued by the generation of topological defects like strings or domain-walls bounded by strings. The subsequent breaking to the standard model may or may not generate any monopoles depending on the Higgs sector used. It is worth noting that the monopoles associated with the spontaneous breaking of the gauge symmetry SU(3)L SU(3)R SU(2)L U(1)Y may have low-lying masses, which will be interesting to investigate from a cosmological⊗ perspective.→ ⊗ Acknowledgments Chandini Dash is grateful to the Department of Science and Technology, Govt. of India for INSPIRE Fellow- ship/2015/IF150787 in support of her research work. 8

Appendix A: Fields and one-loop beta coeﬃcients for diﬀerent Models

1. NON-SUSY CASES