331 Models and Grand Unification: from Minimal SU(5) to Minimal SU(6)

Physics Letters B 762 (2016) 432–440

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Physics Letters B

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331 models and grand uniﬁcation: From minimal SU(5) to minimal SU(6) ∗ Frank F. Deppisch a, Chandan Hati b,c, , Sudhanwa Patra d, Utpal Sarkar e, José W.F. Valle f a Department of Physics and Astronomy, University College London, London WC1E 6BT, United Kingdom b Theoretical Physics Division, Physical Research Laboratory, Ahmedabad 380009, India c Discipline of Physics, Indian Institute of Technology Gandhinagar, Ahmedabad 382424, India d Center of Excellence in Theoretical and Mathematical Sciences, Siksha ‘O’ Anusandhan University, Bhubaneswar 751030, India e Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India f AHEP Group, Institut de Fisica Corpuscular – C.S.I.C./Universitat de Valencia, Parc Cientiﬁc de Paterna, C/ Catedràtico Josè Beltràn 2, E-46980 Paterna (Valencia), Spain a r t i c l e i n f o a b s t r a c t

Article history: We consider the possibility of grand unification of the SU(3)c ⊗ SU(3)L ⊗ U(1)X model in an SU(6) Received 30 August 2016 gauge unification group. Two possibilities arise. Unlike other conventional grand unified theories, in Received in revised form 26 September SU(6) one can embed the 331 model as a subgroup such that different multiplets appear with different 2016 multiplicities. Such a scenario may emerge from the flux breaking of the unified group in an E(6) F-theory Accepted 1 October 2016 GUT. This provides new ways of achieving gauge coupling unification in 331 models while providing the Available online 5 October 2016 ⊗ ⊗ Editor: A. Ringwald radiative origin of neutrino masses. Alternatively, a sequential variant of the SU(3)c SU(3)L U(1)X model can fit within a minimal SU(6) grand unification, which in turn can be a natural E(6) subgroup. This minimal SU(6) embedding does not require any bulk exotics to account for the chiral families while allowing for a TeV scale SU(3)c ⊗ SU(3)L ⊗ U(1)X model with seesaw-type neutrino masses. © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction all the three generations of fermions are included the theory becomes anomaly free. As a result, different multiplets of the The discovery of the Higgs boson established the existence of SU(3)c ⊗ SU(3)L ⊗ U(1)X group appear with different multiplicity spin-0 particles in nature and this opened up the new era in look- and as a result it becomes difficult to unify the model within usual ing for extensions of the Standard Model (SM) at accelerators. It is grand unified theories. For this reason string completions have now expected that at higher energies, the SM may be embedded in been suggested [20]. In this article we study how such a theory larger gauge structures, whose gauge symmetries would have been can be unified in a larger SU(6) gauge theory that can emerge from broken by the new Higgs scalars. So, we can expect signals of the a E(6) Grand Unified Theory (GUT) [21]. We find that the anomaly new gauge bosons, additional Higgs scalars as well as the extra free representations of the SVS 331 model can all be embedded in fermions required to realize the higher symmetries. One of the ex- a combination of anomaly free representations of SU(6), which in ⊗ ⊗ tensions of the SM with the gauge group SU(3)c SU(3)L U(1)X turn can be potentially embedded in the fundamental and adjoint [1] provides strong promise of new physics that can be observed at representations of the group E(6) motivated by F-theory GUTs with the LHC or the next generation accelerators [2–7]. Recently there matter and bulk exotics obtained from the flux breaking mecha- has been a renewed interest in this model as it can provide novel nism [22–25]. ways to understand neutrino masses [8–19]. Interestingly, the SVS 331 model can also be refurbished in an ⊗ ⊗ The SU(3)c SU(3)L U(1)X model proposed by Singer, Valle anomaly free multiplet structure which can be right away embed- and Schechter (SVS) [2] has the special feature that it is not ded in a minimal anomaly free combination of representations of anomaly free in each generation of fermions, but only when SU(6) as an E(6) subgroup. We refer to this new 331 model as the sequential 331 model. This scheme is particularly interesting since its embedding in SU(6) does not require any bulk exotics to ac- Corresponding author. * count for the chiral families; and in that sense it provides a truly E-mail addresses: [email protected] (F.F. Deppisch), [email protected] (C. Hati), [email protected] (S. Patra), [email protected] (U. Sarkar), minimal unification scenario in the same spirit akin to the minimal valle@ific.uv.es (J.W.F. Valle). SU(5) construction [26]. http://dx.doi.org/10.1016/j.physletb.2016.10.002 0370-2693/© 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. F.F. Deppisch et al. / Physics Letters B 762 (2016) 432–440 433

The article is organized as follows. In Section 2 we discuss the exotic ones) carry baryon number (B = 1/3) and no lepton num- basic structure of the SVS SU(3)c ⊗ SU(3)L ⊗ U(1)X model whereas ber (L = 0), while all leptons carry lepton number (L = 1) and no Section 3 describes the sequential SU(3)c ⊗ SU(3)L ⊗ U(1)X model. baryon number (B =√0). Notice that in Ref. [10] the lepton number In Section 4 we then analyze the resulting renormalization group is defined as L = 4/ 3 T8 + L, where U (1)L is a global symmetry running of the gauge couplings in the SVS model with and with- and a Z2 symmetry is introduced to forbid a coupling like ψL ψL φ0, out additional octet states, and discuss necessary conditions for in connection with neutrino masses. Since the charge equation gauge unification. In Section 5 we then embed the different vari- given in Eq. (2) remains the same for this assignment, the follow- ants of the SU(3)c ⊗ SU(3)L ⊗ U(1)X model in an SU(6) unification ing discussion regarding Renormalization Group Equations (RGE) in group and demonstrate successful unification scenarios. Section 6 the SVS model remains valid for this assignment as well. concerns the experimental constraints from achieving the correct For the symmetry breaking and the charged fermion masses, electroweak mixing angle and satisfying proton decay limits. We the following Higgs scalars and their vacuum expectation values conclude in Section 7. (vevs) are assumed, ∗ ∗ ⊗ ⊗ φ ≡[1, 3 , 2/3] and φ ≡[1, 3 , −1/3], 2. The SVS SU(3)c SU(3)L U(1)X model 0 ⎛ ⎞ 1⎛,2 ⎞ ⎛ ⎞ k0 0 0 ⊗ ⊗ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ The SU(3)c SU(3)L U(1)X extension of the SM was origi- φ0= 0 , φ1= k1 , φ2= k2 . (4) nally proposed to justify the existence of three generations of 0 n1 n2 fermions, as the model is anomaly free only when three genera- ∼ tions are present. Such a non-sequential model, which is generically Here we assume k0,1,2 mW to be of the order of the elec- ∼ referred to as the 331 model, breaks down to the SM at some troweak symmetry breaking scale and n1,2 M331 to be the ⊗ ⊗ higher energies, usually expected to be in the TeV range, mak- SU(3)c SU(3)L U(1)X symmetry breaking scale. We shall not ing the model testable in the near future. The symmetry breaking: discuss here the details of fermion masses and mixing, which can be found in Refs. [9,10]. SU(3)c ⊗ SU(3)L ⊗ U(1)X → SU(3)c ⊗ SU(2)L ⊗ U(1)Y allows us to identify the generators of the 321 model in terms of the genera- 3. The sequential SU(3) ⊗ SU(3) ⊗ U(1) model tors of the 331 model. Writing the generators of the SU(3)L group c L X as ⎛ ⎞ In this model the fields are assigned in a way such that the 1 2 00 anomalies are cancelled for each generation separately. The multi- 1 ⎝ 1 ⎠ T3 = I3 = 0 − 0 and plet structure is given by 2 2 ⎛ ⎞ 000 ⎛ ⎞ uaL 10 0 Q = ⎝ d ⎠ ≡[3, 3, 0], u ≡[3, 1, 2/3], 1 1 ⎝ ⎠ aL aL aR T8 = √ I8 = √ 01 0 DaL 2 3 2 3 00−2 daR ≡[3, 1, −1/3], DaR ≡[3, 1, −1/3], = [ − ] = [ − ] ⎛ ⎞ ⎛ − ⎞ with I3 diag 1, 1, 0 and I8 diag 1, 1, 2 , (1) − E eaL aL = ⎝ ⎠ ≡[ ∗ − ] = ⎝ 2 ⎠ ≡[ ∗ − ] we can readily identify the SM hypercharge and the electric charge ψaL νaL 1, 3 , 1/3 ,ξaL NaL 1, 3 , 1/3 , as N1 N3 ⎛ aL ⎞ aL 1 1 N4 Y = √ T8 + XI3×3 = I8 + XI3×3 and aL 6 = ⎝ + ⎠ ≡[ ∗ ] 3 χaL EaL 1, 3 , 2/3 . (5) + 1 1 eaL Q = T3 + Y = I3 + I8 + XI3×3 , (2) 2 6 It is straightforward to check that each family is anomaly free. where X is the U (1)X charge and I3×3 is the 3 × 3identity matrix. In order to drive symmetry breaking and generate the charged This allows us to write down the fermions and the representations fermion masses, we assume a Higgs sector and vevs similar to the in which they belong as SVS 331 model.1 The Yukawa Lagrangian for the quark sector can ⎛ ⎞ ⎛ ⎞ be written as uiL bL ⎝ ⎠ ⎝ ⎠ ∗ ∗ i ∗ i ∗ Q iL = diL ≡[3, 3, 0], Q 3L = tL ≡[3, 3 , 1/3], L = y Q u φ + y Q d φ + y Q D φ + h.c. , quarks ua aL aR 0 da aL aR i Da aL aR i D T iL L (6) uiR ≡[3, 1, 2/3], diR ≡[3, 1, −1/3], DiR ≡[3, 1, −1/3] with a = 1, 2, 3, i = 1, 2 and where we neglect any flavor mix- ≡[ − ] ≡[ ] ≡[ ] bR ⎛3, 1, ⎞1/3 , tR 3, 1, 2/3 , T R 3, 1, 2/3 , ing. After the chain of spontaneous symmetry breaking the up-type quarks obtain a mass term eaL ⎝ ⎠ ∗ ψaL = νaL ≡[1, 3 , −1/3], eaR ≡[1, 1, −1]. (3) = mua yua k0, NaL while the down-type and vector-like down-type quarks form a The generation index i = 1, 2 corresponds to the first two gener- mass matrix in the (d, D) basis given by ations with the quarks uL,R , dL,R , D L,R and cL,R , sL,R , S L,R . For the 1 2 1 2 = y k1 + y k2 y k1 + y k2 leptons, the generation index is a 1, 2, 3. a = da da Da Da mdD 1 2 1 2 . (7) There are several variants of the model that allow slightly dif- y n1 + y n2 y n1 + y n2 da da Da Da ferent choices of fermions as well as their baryon and lepton number assignments. Here we shall restrict ourselves to the one which 1 contains only the quarks with electric charge 2/3 and 1/3 and no A model with similar fermion content and with k1 = n2 = 0in the scalar sector lepton number (L). In this scenario all quarks (usual ones and the was discussed in Ref. [27] using the trinification group SU(3)c × SU(3)L × SU(3)R. 434 F.F. Deppisch et al. / Physics Letters B 762 (2016) 432–440

1 2 1 2 Note that in the cases y = y = y = y ≡ y ; or k1 = k2 = k in the basis (ν, N1, N3, N2, N4), where N1, N3 are SU(2)L isosin- da da Da Da d and n1 = n2 = n the determinant of the above Yukawa matrix van- glets and ν, N2, N4 are components of doublets. Next, we rotate 1 2 1 2 = ishes giving m = 0 and mD = y k1 + y k2 + y n1 + y n2. the above mass matrix by an orthogonal transformation m da a da da Da Da νN T However, in the absence of any symmetries forcing the above con- R mνN R, where ditions, the down quarks obtain mass as a result of the mixing ⎛ ⎞ with the vector-like quarks. One can determine it perturbatively − 1 1 1 1 0 2 2 2 2 by expanding the Yukawa contributions in terms of ki /ni 1so ⎜ ⎟ ⎜− √1 1 1 00⎟ as to obtain ⎜ 2 2 ⎟ ⎜ 2 ⎟ = ⎜ √1 1 1 ⎟ 1 2 R 2 2 00. (11) y n1 + y n2 ⎜ 2 ⎟ 1 2 1 2 da da ⎜ 1 1 1 1 ⎟ md = y k1 + y k2 − y k1 + y k2 +··· , 0 − a da da Da Da 1 + 2 ⎝ 2 2 2 2 ⎠ yD n1 yD n2 a a 000√1 − √1 = 1 + 2 + 1 + 2 2 2 mDa yd k1 yd k2 yD n1 yD n2 a a a a This yields the rotated mass matrix m given by a 1 2 νN − M / y n + y n +··· , dD Da 1 Da 2 where mνN ⎛ − − ⎞ 00 0√u − x z √u + x z a 1 2 1 2 2 2 2 2 M = y k + y k y n + y n (X−Z)√+(x+z) (X−Z)√+(x+z) dD da 1 da 2 Da 1 Da 2 ⎜ 0 −u 0 − ⎟ ⎜ 2 2 2 2 ⎟ ⎜ (X−Z)−(x+z) (X−Z)−(x+z) ⎟ = 00 u − √ √ , − y1 k + y2 k y1 n + y2 n ⎜ 2 2 2 2 ⎟ D 1 D 2 d 1 d 2 . x−z (X−Z)+(x+z) (X−Z)−(x+z) X+Z a a a a ⎝ √u − √ − √ − √ 0 ⎠ 2 2 2 2 2 2 2 − − + + − − + + √u + x z − (X Z)√ (x z) (X Z)√ (x z) 0 X√ Z This structure can be used to account for the SM quark masses 2 2 2 2 2 2 2 as well as the heavier vector-like quark mass limits from the LHC. (12) Turning now to the lepton sector, the relevant Yukawa terms are given by where T −1 i Lleptons = αβγ ψ C y1ξβL φ0γ + y χβL φiγ αL 2 = = 1 + 2 = 1 + 2 u y1k0, X y2n1 y2n2 , x y2k1 y2k2 , + T −1 i + ξαL C y3χβL φiγ h.c. , (8) = 1 + 2 = 1 + 2 z y3k1 y3k2 , Z y3n1 y3n2 . where α, β, γ are the SU(3)L tensor indices ensuring antisymmet- ric Dirac mass terms, C is the charge conjugation matrix, and Now we recall that k0,1,2 ∼ mW is of the order of the electroweak i = 1, 2. After the symmetry breaking, these Yukawa terms give symmetry breaking scale, while and n1,2 ∼ M331 is of the order of rise to the mass matrices for charged and neutral leptons. In the the SU(3)c ⊗ SU(3)L ⊗ U(1)X symmetry breaking scale, and hence basis (e, E) the mass matrix is given by one expects that X, Z u, x, z. If we further assume X + Z − 1 2 1 2 X Z, then we can identify the 44 and 55 entries as the heaviest − y k1 + y k2 y n1 + y n2 m = 2 2 2 2 , (9) in the mass matrix given in Eq. (12) and these rotated isodoublet eE − 1 + 2 1 + 2 y3k1 y3k2 y3n1 y3n2 states form a pair of heavy quasi Dirac neutrinos with mass of the order of the SU(3)c ⊗ SU(3)L ⊗ U(1)X symmetry breaking scale. We with the eigenvalues given by can now readily use perturbation theory to obtain the masses for 1 2 the three remaining lighter states. Up to second order in perturba- y n1 + y n2 m =− y1k + y2k + y1k + y2k 2 2 +··· , tion theory we obtain two Dirac states with mass of the order of e 2 1 2 2 3 1 3 2 1 + 2 y3n1 y3n2 the electroweak symmetry breaking scale ±u =±y k and a light 1 0 1 2 1 2 a 1 2 seesaw Majorana neutrino with mass 2u(z − x)/(X + Z). With this m = y n + y n − y k + y k − M / y n + y n E 3 1 3 2 2 1 2 2 eE 3 1 3 2 we see that the model has enough ﬂexibility to account for the +··· , observed pattern of fermion masses. It is not our purpose here to present a detailed study of the structure of the fermion mass spec- where trum, but only to check its consistency in broad terms. While a detailed description of the collider phenomenology is a = 1 + 2 1 + 2 MeE y2n1 y2n2 y3k1 y3k2 beyond the scope of this work, we would like to comment that there is a rich set of phenomena associated to the production of − 1 + 2 1 + 2 y2k1 y2k2 y3n1 y3n2 . new fermionic states at high energies. If the heavy gauge bosons are accessible at the LHC, the exotic quarks and leptons can be ef- For the case of neutral leptons the mass matrix can be written ﬁciently produced with gauge coupling strength. For example, the as: heavy quasi Dirac neutrinos could be pair-produced at the LHC

through a Z resonance, pp → Z → Ni Ni . They would then decay mνN ⎛ ⎞ through SM currents via their admixture with the light neutrinos, − 1 + 2 00y1k0 0 y2n1 y2n2 (∗) in a manner similar to [28], Ni → 1 W → + 2jets, 2 + Emiss. ⎜ 000−y k y1k + y2k ⎟ ⎜ 1 0 2 1 2 2 ⎟ The produced signal would be lepton number conserving due to 1 + 2 = ⎜ y1k0 00 0y k1 y k2 ⎟, ⎝ 3 3 ⎠ the quasi Dirac nature of the heavy neutrinos. These SM scale 0 −y k 00− y1n + y2n 1 0 3 1 3 2 heavy neutrinos may also be searched for through production via − y1n + y2n y1k + y2k y1k + y2k − y1n + y2n 0 2 1 2 2 2 1 2 2 3 1 3 2 3 1 3 2 SM W and Z processes but this is suppressed by their admixture (10) with the light neutrinos. F.F. Deppisch et al. / Physics Letters B 762 (2016) 432–440 435

1 4. Renormalization group equations and gauge coupling 2 = + 2 nY nX , (20) unification 3 and the normalized couplings are related by In this section we study the SVS model RGEs to explore if unifi- − − cation of the three gauge couplings [29] can be obtained in the 1 1 − 1 1 ⊗ ⊗ n2 αN = α 1 + n2 − αN , (21) SU(3)c SU(3)L U(1)X theory at a certain scale MU , without Y Y 3 3L Y 3 X any presumptions about the nature of the underlying group of grand unification [10]. Using the RGEs we express the hypercharge where (and X) normalization and the unification scale as a function of ⊗ ⊗ 1 SU(3)c SU(3)L U(1)X breaking scale. Next we study the allowed N = 2 N = 2 − = αY nY αY , αX nY αX , α3L α2L . (22) range of SU(3)c ⊗ SU(3)L ⊗ U(1)X breaking scale such that one can 3 obtain a guaranteed unification of the gauge couplings. First we Now using Eqs. (13), (21), (22) we obtain discuss the SVS model discussed in section 2. Then, we study the impact of adding three generations of leptonic octet representa- [ ] −1 1 −1 2 1 −1 tions 1, 8, 0 that can give gauge coupling unification for a TeV α = α (M Z ) cos θw (M Z ) − α (M Z ) U 2 − 1 em 3 2L scale SU(3)c ⊗ SU(3)L ⊗ U(1)X breaking while driving an interest- nY 3 ing radiative model for neutrino mass generation [10]. UN − 1 UN b b2L M X b MU The evolution for running coupling constants at one loop level − Y 3 ln − X ln , is governed by the RGEs 2π M Z 2π M X ∂ g b − − b2L M X b3L MU i = i 3 1 = 1 M − ln − ln μ gi , (13) αU α2L ( Z ) , ∂μ 16π 2 2π M Z 2π M X which can be written in the form b M b X M −1 = −1 − 3C X − 3C U αU α3C (M Z ) ln ln . (23) 1 1 bi μ2 2π M Z 2π M X = − ln , (14) αi(μ2) αi(μ1) 2π μ1 Here, the SM running is described by the SU(3)C coefficient b3C , = 2 the SU(2)L coefficient b2L and the U(1)Y unnormalized coefficient where αi gi /4π is the fine structure constant for i-th gauge UN b . Likewise, in the unbroken SU(3)c ⊗ SU(3)L ⊗ U(1)X phase, the group, μ1, μ2 are the energy scales with μ2 > μ1. The beta- Y gauge running coefficients for the SU(3)C , SU(3)L and unnormal- coefficients bi determining the evolution of gauge couplings at ized U(1) components are b X , b and bUN, respectively. The one-loop order are given by X 3C 3L X scale M corresponds to the Z boson-pole, the 331 symmetry Z 11 2 breaking scale is denoted by M and M is the scale of unifi- b =− C (G) + T (R ) d (R ) X U i 2 f j f cation for the normalized gauge couplings. From the above set of 3 3 = R f j i equations the unification scale M can be obtained as a function U 1 of M X , + T (Rs) d j(Rs). (15) 3 = b −b Rs j i − 3C 2L −1 −1 X − M X b −b α (M Z ) α (M Z ) M = M 3C 3L exp 2π 3C 2L . Here, C2(G) is the quadratic Casimir operator for the gauge bosons U X X M Z b − b in their adjoint representation, 3C 3L (24) N if SU(N), C2(G) ≡ (16) 2 0ifU(1). Similarly, nY can be expressed as a function of M X , On the other hand, T (R f ) and T (Rs) are the Dynkin indices of 1 1 2 = + −1 2 − −1 the irreducible representation R for a given fermion and scalar, nY αem (M Z ) cos θw (M Z ) α (M Z ) f ,s 3 3 2L respectively, ⎧ UN − 1 − bY 3 b2L M X UN 1 b3C b2L M X ⎨1/2ifR f ,s is fundamental, − ln + b ln 2π M X 2π b X − b M T (R ) ≡ N if R is adjoint, (17) Z 3C 3L Z f ,s ⎩ f ,s −1 − −1 0ifR f ,s is singlet, α (M Z ) α (M Z ) − b2L M X − 3C 2L α 1(M ) − ln X − 2L Z and d(R ) is the dimension of a given representation R under b b3L 2π M Z f ,s f ,s 3C all gauge groups except the i-th gauge group under consideration. − An additional factor of 1/2is multiplied in the case of a real Higgs 1 b3C b2L M X + b3L ln 2 X − M representation. π b3C b3L Z The electromagnetic charge operator is given by − − −1 α 1(M ) − α 1(M ) − 3C Z 2L Z = + = + √1 + . (25) Q T3 Y T3 T8 X, (18) b X − b 3 3C 3L where the generators (Gell-Mann matrices) are normalized as X = X − The above two relations are valid provided b3C b3L and (b3C = 1 = − Tr(Ti T j) 2 δij. We define the normalized hypercharge operator b3L ) (b3C b2L ), which are satisfied in the cases that we shall 17 Y N and XN as discuss below. Furthermore, we take M X ≤ MU ≤ 10 GeV and assume that 331 is the only gauge group (in other words M X is Y = nY Y N , X = nX XN , (19) the only intermediate scale) between M Z and the unification scale such that we have MU . 436 F.F. Deppisch et al. / Physics Letters B 762 (2016) 432–440

Fig. 1. (Left) Allowed range for SU(3)c ⊗ SU(3)L ⊗ U(1)X symmetry breaking scale M X for guaranteed uniﬁcation. The solid line represents MU as a function of M X . The = 17 = 2 dashed and dot-dashed lines correspond to MU 10 GeV and MU M X respectively. (Right) The hypercharge normalization factor nY as a function of the 331 symmetry 2 ≥ 1 2 = 1 2 breaking scale M X . The shaded region represents the allowed region nY 3 with the dashed line corresponding to the lower limit nY 3 . The solid line gives nY as a 2 = 5 function of M X and the red dot-dashed line shows the standard SU(5) normalization nY 3 .

16 Fig. 2. Gauge coupling running in the SVS Model with 331 symmetry breaking scale M X = 7.9 × 10 GeV, demonstrating successful gauge uniﬁcation at the scale MU = × 16 2 = 8.05 10 GeV with nY 1.3. The right plot shows the magniﬁed view of gauge coupling running around M X .

4.1. The minimal SVS model Finally, in Fig. 2 we give an example of gauge coupling running with respect to the 331 symmetry breaking scale M X = 7.9 × 1016 GeV. It demonstrates that successful gauge unification The first case of interest is the minimal scenario described 16 2 at the scale MU = 8.05 × 10 GeV with n = 1.3can be achieved, in section 2. The relevant gauge quantum numbers are given in Y albeit this requires a very high scale of SU(3)c ⊗ SU(3)L ⊗ U(1)X Eqs. (3), (4). The Higgs sector involves three SU(3)L triplets, namely breaking very near to the unification scale. the minimal set necessary for adequate symmetry breaking and generation of fermion masses. First we notice that the model de- 4.2. The SVS model with fermionic octets scribed in Ref. [10] has the same RGE evolution, since the extra gauge singlets added to the fermion spectrum to generate neu- In this model, in addition to the field content of model I, we in- trino masses do not enter the RGEs. For the SM the one-loop clude three generations of fermion octets with the assignments beta-coefficients are given by b =−19 6, bUN = 41 6, b =−7, 2L / Y / 3C under the SU(3)c ⊗ SU(3)L ⊗ U(1)X group given by × × while in the SU(3)c SU(3)L U (1)X phase they are given by ∗ =− UN = X =− ≡[1, 8 , 0]. (26) b3L 13/2, b X 26/3, b3C 5. In Fig. 1 (left) we plot the allowed range for M X . The intersec- The Higgs sector involves the same three SU(3)L triplets as tion of the line corresponding to MU evaluated as a function of before. Although this model has the same content as the one con- = = 17 M X in Eq. (24) with the lines for MU M X and MU 10 GeV sidered in Ref. [10], here we take a completely different approach gives the lower and upper bound on M X respectively such that to unification. Indeed, we do not consider the usual SU(5) normal- there is a guaranteed unification. In this scenario, the scale M X of ization for the hypercharge and the octet mass scale is the same as ⊗ ⊗ SU(3)c SU(3)L U(1)X breaking is therefore always high and very the 331 symmetry breaking scale. In this model, the neutrinos are close to the unification scale MU . massless at tree level, however at one-loop level the exchange of Next, in Fig. 1 (right) we plot the hypercharge normaliza- gauge bosons gives rise to dimension-nine operator which gener- 2 ⊗ ⊗ tion factor nY as a function of SU(3)c SU(3)L U(1)X symme- ates neutrino masses after 331 symmetry breaking [10]. For the try breaking scale M X . The dashed horizontal line represents the SM the one-loop beta-coefficients remain the same as Model I, 2 = 1 2 ⊗ ⊗ lower limit nY 3 of the allowed value for nY . As can be seen while in the SU(3)c SU(3)L U(1)X phase they are given by from the figure, for the allowed M range from the condition =− UN = X =− X b3L 1/2, b X 26/3, b3C 5. ≤ ≤ 17 2 M X MU 10 GeV the hypercharge normalization nY is almost In Fig. 3 (left) we plot the allowed range for M X for which 17 constant ≈ 1.3 and well above the allowed lower limit. unification is guaranteed at a scale M X ≤ MU ≤ 10 GeV. Inter- F.F. Deppisch et al. / Physics Letters B 762 (2016) 432–440 437

Fig. 3. Same as Fig. 1, but for the SU(3)c ⊗ SU(3)L ⊗ U(1)X SVS model with three additional fermionic octets. The purple dotted line corresponds to the lower limit of MU allowed by the current experimental limits on the lifetime of the proton decay.

5. SU(6) grand uniﬁcation

We consider the possibility of grand unification of the SU(3)c ⊗ SU(3)L ⊗ U(1)X model in an SU(6) gauge unification group. Two possibilities arise. Unlike other conventional grand unified theories, in SU(6) one can have different components of the 331 subgroup with different multiplicity. Such a scenario may emerge from the flux breaking of the unified group in an E(6) F-theory GUT. This provides new ways of achieving gauge coupling unification in 331 models. Alternatively, a sequential variant of SU(3)c ⊗ SU(3)L ⊗ U(1)X model can have a minimal SU(6) grand unification, which in turn can be a natural E(6) subgroup. This minimal SU(6) embedding does not require any bulk exotics to account for the chiral families and allows for a TeV scale SU(3)c ⊗ SU(3)L ⊗ U(1)X model. We now demonstrate how the SU(3)c ⊗ SU(3)L ⊗ U(1)X model fermions can be embedded in an SU(6) grand unified gauge group. Fig. 4. Gauge coupling running in the SVS Model adding three generations of Our main consideration is to explore whether the combinations ⊗ ⊗ = leptonic octets with SU(3)c SU(3)L U(1)X symmetry breaking scale at M X of the SU(6) gauge group representations form an anomaly free = 14.9 3000 GeV, demonstrating successful gauge unification at the scale MU 10 GeV set, which can contain all the required fermions. In the subsequent with n2 = 1.8. Y subsections we discuss how different multiplicities of the SVS version of the SU(3) ⊗ SU(3) ⊗ U(1) model can be explained when ⊗ ⊗ c L X estingly, in this model we find that for a SU(3)c SU(3)L U(1)X this SU(6) grand unified model is embedded in an E(6) F-theory symmetry breaking scale M as low as TeV it is possible to X and how the sequential SU(3)c ⊗ SU(3)L ⊗ U(1)X model can be achieve unification. Note that in contrast to Ref. [10], here we embedded in a minimal anomaly free combination of representa- do not assume another intermediate scale corresponding to the tions of SU(6) as an E(6) subgroup. For the minimal SVS version fermion octet mass scale in addition to M X . Formally, unifica- of the SU(3)c ⊗ SU(3)L ⊗ U(1)X model, gauge coupling unification tion can thus be achieved for any scale M X between M Z and can be obtained by including both the matter multiplets in the 15.5 MU , however, MU 10 GeV is disfavored by the current ex- 27-dimensional fundamental representations of E(6) as well as the perimental limits on the lifetime of the proton decay [30]. This bulk exotics from the 78-dimensional adjoint representations of 5 consequently puts a lower limit of the order of M X 10 GeV on E(6). In particular the octet of SU(3)L coming from the bulk plays a ⊗ ⊗ the SU(3)c SU(3)L U(1)X breaking scale, although we should crucial role in allowing the unification of the gauge couplings with emphasize that we here do not specify the GUT group and thus a low 331 symmetry breaking scale. On the other hand, the em- cannot predict the proton decay rate accurately. bedding of sequential 331 model in SU(6) does not require any In Fig. 3 (right) we plot the hypercharge normalization factor bulk exotics to account for the chiral multiplets and imply, by 2 ⊗ ⊗ nY as a function of SU(3)c SU(3)L U(1)X symmetry breaking adding three generations of U(1)X neutral fermionic octets, one can scale M X . In this case as well, for the allowed M X range from the obtain SU(6) unification with a TeV scale SU(3)c ⊗ SU(3)L ⊗ U(1)X ≤ ≤ 17 2 condition M X MU 10 GeV the hypercharge normalization nY breaking scale. is well above its allowed lower limit. We shall first write down some of the product decompositions In Fig. 4 we show an example gauge coupling running with of the group SU(6): SU(3)c ⊗ SU(3)L ⊗ U(1)X symmetry breaking scale M X =3000 GeV, demonstrating successful gauge coupling unification at a scale 6 × 6 = 15a + 21s, M = 1014.9 GeV with n2 = 1.8. Thus, from the perspective of a U Y 6 × 6¯ = 1 + 35, low SU(3)c ⊗ SU(3)L ⊗ U(1)X symmetry breaking scale within the reach of accelerator experiments like the LHC (∼ O (TeV)) this 6 × 15 = 20 + 70, model is the most interesting candidate leading to a successful 6 × 21 = 56 + 70. (27) gauge coupling unification. In addition to the new gauge bosons, the model can harbor a plethora of new states associated to the The SU(6) has SU(3)c ⊗ SU(3)L ⊗ U(1)X as a maximal subgroup new exotic fermions as well as extra Higgs bosons. with the same rank. For convenience we write down some 438 F.F. Deppisch et al. / Physics Letters B 762 (2016) 432–440 of the representations of SU(6) under this maximal subgroup SU(3)c ⊗ SU(3)L ⊗ U(1)X:

6 =[3, 1, −1/3]+[1, 3, 1/3], 15 =[3¯, 1, −2/3]+[1, 3¯, 2/3]+[3, 3, 0], 20 =[1, 1, −1]+[1, 1, 1]+[3, 3¯, 1/3]+[3¯, 3, −1/3], 21 =[3, 3, 0]+[6, 1, −2/3]+[1, 6, 2/3], 35 =[1, 1, 0]+[8, 1, 0]+[1, 8, 0]+[3, 3¯, −2/3] +[3¯, 3, 2/3], 56 =[10, 1, −1]+[1, 10, 1]+[6, 3, −1/3]+[3, 6, 1/3], 70 =[6, 3, −1/3]+[3, 6, −1/3]+[3, 3¯, 1/3]+[3¯, 3, −1/3] +[8, 1, −1]+[1, 8, 1]. (28) Fig. 5. Gauge coupling running in the SVS Model with three generations of lep- The anomalies for the various representations of the group SU(6) tonic octets with 331 symmetry breaking scale M X = 3000 GeV and octet mass = are scale M8 9000 GeV, demonstrating√ successful√ gauge unification at the scale 15.5 MU = 10 GeV with nY = 5/3and nX = 2/ 3. A[6]=1, A[15]=2, A[20]=0, A[21]=10, A[35]=0, − 5/6n 2 in the notation of Eq. (19). The U(1) normalization de- A[56]=54, A[70]=27. (29) Y √ Y fined in Eq. (19) is given by nY = 5/3 and using√ Eq. (20) we We now turn to two concrete model constructions. obtain the U(1)X normalization given by nX = 2/ 3, which is below the normalizations required for a guaranteed unification in 5.1. SU(6) grand unification of the SVS model Fig. 1 and Fig. 3. However, it is still possible to obtain gauge coupling unification following the prescription in Ref. [10], where ⊗ ⊗ It can be easily verified that all fermions of the SU(3)c ⊗ the octet scale is decoupled from the SU(3)c SU(3)L U(1)X SU(3)L ⊗ U(1)X model proposed by SVS (discussed in section 2) symmetry breaking scale and is assumed to lie between the can be included in the anomaly free combination of representa- SU(3)c ⊗ SU(3)L ⊗ U(1)X symmetry breaking scale and unification tions under SU(6): scale. Here, the octets belong to 35 of SU(6), which belongs to the ¯ ¯ bulk exotics coming from the 78-dimensional adjoint representa- 6 + 6 + 15 + 20. tions of E(6). There will be some extra fermions and the multiplicity of the dif- Using Eq. (15) the one-loop beta-coefficients bi can be calcu- ferent representations are now different. It is to be noted that lated for the different phases. For the phase between the elec- ⊗ ⊗ these states can be naturally embedded in an E(6) theory. We start troweak symmetry breaking scale and the SU(3)c SU(3)L U(1)X with the maximal SU(2) × SU(6) subgroup of E(6), and write down symmetry breaking scale (M Z to M X) the one-loop beta-coefficients =− = =− the decomposition: are given by b2L 19/6, bY 41/10, b3C 7. For the phase between the SU(3)c ⊗ SU(3)L ⊗ U(1)X symmetry breaking scale 27 =[2, 6¯]+[1, 15] and the octet mass scale (M X to M8) the one-loop beta-coefficients =− = 331 =− are given by b3L 13/2, b X 13/2, b3C 5. Finally, for the 78 =[1 35]+[2 20]+[3 1] , , , phase between the octet mass scale to the unification scale (M8 to 8 MU ) the one-loop beta-coefficients are given by b = 2n − 13/2, Thus the SU(3)c ⊗ SU(3)L ⊗ U(1)X anomaly free representations of 3L where n is the number of generations of the fermionic octets the SVS model can all be embedded in a combination of anomaly ∗ ( ≡[1, 8 , 0]), b8 = 13/2, b8 =−5. free representations of SU(6), which in turn can be embedded in X 3C ⊗ the fundamental and adjoint representations of the group E(6). The In Fig. 5 we plot the gauge coupling running of SVS SU(3)c ⊗ next question is how to match the multiplicity of the different SU(3)L U(1)X model with the field content given in Eqs. (3), (4) ⊗ ⊗ representations of the SVS 331 model, which is nontrivial. At this and three generations of fermionic octets with SU(3)c SU(3)L = stage we resort to the symmetry breaking at the GUT scale induced U(1)X symmetry breaking scale M X 3000 GeV and octet mass = by flux breaking through the Hosotani mechanism [31]. Assigning scale M8 9000 GeV, demonstrating successful√ gauge unifica- = 15.5 = = particular geometry to the flux breaking, we identify the different tion√ at the scale MU 10 GeV with nY 5/3 and nX states with the different algebraic varieties, and then the intersec- 2/ 3. A relative modest variation of the octet mass scale from ⊗ ⊗ = tion numbers would give us the multiplicities of the different rep- the SU(3)c SU(3)L U(1)X scale, M8/M X 3, therefore lifts the = 14.9 resentations. Adetailed study of such E(6) F-theory GUTs [22–25] scale of successful unification from the value MU 10 GeV is beyond the scope of this article and we shall rather take a found in Fig. 4 and thus relaxes the tension with proton decay phenomenological approach to the problem. We consider the re- limits, cf. Section 6. quired representations to match the low energy phenomenological requirements. The first step is to keep the known fermions light 5.2. SU(6) grand unification of the sequential SU(3) ⊗ SU(3) ⊗ U(1) model and also to have SU(3)c ⊗ SU(3)L ⊗ U(1)X symmetry breaking scale c L X as low as TeV, while at the same time requiring for gauge coupling unification. It is easy to verify from Eq. (27) that each generation of the Considering the 6¯ representation of SU(6), which contains the fermionic multiplets of the sequential 331 model written in Eq. (5) down antiquarks dc with hypercharge Y = 1/3, isospin lepton dou- fits perfectly in the anomaly free combination of SU(6) represen- L ¯ + ¯ + ¯ c ≡[ − ] ≡ blet containing eL and νL with Y =−1/2, and NL with Y = 0; tations: 6 6 15, where 6 contains dL 3, 1, 1/3 and ψL 2 = [ ∗ − ] ¯ c ≡[ − ] ≡[ ∗ − ] we can get the normalization for the hypercharge from Tr(Y ) 1, 3 , 1/3 ; 6 contains D L 3, 1, 1/3 and ξL 1, 3 , 1/3 ; F.F. Deppisch et al. / Physics Letters B 762 (2016) 432–440 439

2 αem(M Z ) sin θw (M Z ) ≡ α (M ) 2L Z − 3 5 −1 N 1 = + αem(M Z ) α (M Z ) − α (M Z ) . (32) 8 8 2L Y Finally, using Eqs. (14), (31) the above equation can be written in the form

sin2 θ (M ) w Z 8 3 5 4 b3L M8 b3L MU = + αem(M Z ) ln + ln 8 8 5 2π M X 2π M8 (b2L − bY ) M X 4 b X MU + ln − ln , (33) 2π M Z 5 2π M X 2 which can be readily used to obtain the prediction for sin θw (M Z ). For example, in the sequential 331 model taking SU(3)c ⊗ SU(3)L ⊗ Fig. 6. Gauge coupling running in the sequential 331 Model with three generations U(1) symmetry breaking scale M = 3000 GeV, octet mass scale of fermionic octets with SU(3) ⊗ SU(3) ⊗ U(1) symmetry breaking scale M = X X c L X X 7 15.5 = × 7 M = 8 × 10 GeV, and unification scale M = 10 GeV we ob- 3000 GeV and octet mass scale M8 8 10 GeV, demonstrating√ successful√ gauge 8 U = 15.5 = = 2 unification at the scale MU 10 GeV with nY 5/3and nX 2/ 3. tain sin θw (M Z ) 0.231, which is consistent with the electroweak precision data [30]. c ≡[ ∗ − ] ≡[ ∗ ] ≡ Turning to the prediction for proton decay, we note that be- and 15 contains uL 3 , 1, 2/3 , χL 1, 3 , 2/3 and Q L [3, 3, 0]. Now the fundamental 27 of E(6) branches under the max- ing a non-supersymmetric scenario the gauge d = 6 contribu- imal SU(2) ⊗ SU(6) subgroup as 27 =[2, 6¯] +[1, 15]. Thus three tions for proton decay are most important here. An analysis of ¯ ¯ ⊗ ⊗ 27s of E(6) contain three sets of 6 + 6 + 15 accommodating the all SU(3)c SU(2)L U(1)Y invariant operators [32–35] that can three generations of the fermionic multiplets of the sequential 331 induce proton decay in SU(6) is beyond the scope of this article model. However the minimal content of the sequential 331 model and will be addressed in a separate communication. Here we will + 0 does not lead to a successful gauge coupling unification. However, consider the decay mode p → e π , which is constrained by ex- expt ≥ × 34 by adding three generations of fermionic octets again leads to a perimental searches to have a life time τp 1 10 [30]. The successful gauge coupling unification. relevant effective operators in the physical basis are given by [36, For the phase between the electroweak symmetry breaking 37] scale and the SU(3) ⊗ SU(3) ⊗ U(1) symmetry breaking scale c L X c c c μ c O(e , dβ ) = c(e , dβ )ijku γ u jLe γμdkβ , (M Z to M X ) the one-loop beta-coefficients are given by b2L = α α i L α L L −19/6, b = 41/10, b =−7. For the phase between the SU(3) ⊗ c c c μ c Y 3C c O(eα, d ) = c(eα, d )ijku γ u jLd γμeα L ; (34) β β i L kβ L SU(3)L ⊗ U(1)X symmetry breaking scale and the octet mass scale (M X to M8) the one-loop beta-coefficients are given by b3L = where − = 331 =− 9/2, b X 13/2, b3C 5. Finally, for the phase between the c 2 11 αβ 1β † α1 c(e , dβ ) = k V V + (V 1 V UD) (V 2 V ) , octet mass scale to the unification scale (M8 to MU ) the one- α 1 1 2 UD 8 = − loop beta-coefficients are given by b 2n 9/2, where n is c 2 11 βα 2 † β1 † 1α 3L ∗ c(e , d ) = k V V + k (V V ) + (V V V V ) ; the number of generations of the fermionic octets ( ≡[1, 8 , 0]), α β 1 1 3 2 4 UD 1 UD 4 3 8 = 8 =− = = b X 13/2, b3C 5. α β 2. (35) In Fig. 6 we plot the gauge coupling running of the sequen- Here i, j, k = 1, 2, 3are the color indices and α, β = 1, 2; V tial SU(3) ⊗ SU(3) ⊗ U(1) model with the field content given 1,2,3,4 c L X = † = † = in Eqs. (4), (5) and three generations of fermionic octets with and V UD are the mixing matrices V 1 U C U , V 2 EC D, V 3 ⊗ ⊗ = † = † SU(3)c SU(3)L U(1)X symmetry breaking scale M X 3000 GeV DC E, V UD U D; where U , D, E are the unitary matrices di- = × 7 diag and octet mass scale M8 8 10 GeV, demonstrating successful√ agonalizing the Yukawa couplings e.g. U T Y U = Y . k = = 15.5 = C U U 1 gauge unification√ at the scale MU 10 GeV with nY 5/3 g /M and k = g /M , where M , M ∼ = GUT (X,Y ) 2 GUT (X ,Y ) (X,Y ) (X ,Y ) and nX 2/ 3. In this scenario, the octet mass scale has to be M are the masses of the superheavy gauge bosons and g ⊗ ⊗ GUT GUT detached rather strongly from the SU(3)c SU(3)L U(1)X scale in is the coupling constant at the GUT scale. The decay rate for + order to achieve successful unification. p → e π 0 mode is given by

2 + 0 mp 2 2 2 6. sin θw and proton decay in SU(6) grand unification (p → e π ) = A |αH | (1 + D + F ) 16π f 2 L π Using the RGEs and the relations among the coupling con- × |c(e, dc)|2 +|c(ec, d)|2 , (36) stants corresponding to different gauge groups one can express 2 sin θw (M Z ) in terms of the different scales associated with the where mp = 938.3MeVis the proton mass, fπ = 139 MeV is SU(6) grand unified√ theory. Noting√ that for SU(6) grand unification the pion decay constant, AL is the long distance renormaliza- = = we have nY 5/3 and nX 2/ 3, the relation between normal- tion factor; D, F and αH are parameters of the chiral Lagrangian. 3 ized couplings at the scales M Z and M X are given by For a rough estimate, taking αH (1 + D + F ) ∼ 0.012 GeV [38, 39]; A = A ASD ∼ 3, where ASD is the short distance renormal- − R L R R −1 −1 5 N 1 α (M Z ) = α (M Z ) − α (M Z ), (30) ization factor; the parameter depending on the mixing matrices 2L em 3 Y Fq(V ) ∼ 5, we obtain − 1 4 − N 1 = −1 + N 1 αY (M X ) α3L (M X ) αX (M X ). (31) −1 2 4 5 5 − + α MU 1(p → e π 0) ∼ 1036 yrs GUT . (37) Using Eq. (30) it is straightforward to obtain 35 1016 GeV 440 F.F. Deppisch et al. / Physics Letters B 762 (2016) 432–440

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2 In fact for a more careful estimation, one should also take into account the GUT threshold corrections which might improve on this limit, however, given the uncertainties in the hadronic parameters here we do not worry about such effects.