Towards the minimal renormalizable supersymmetric $E_6$ model

We find an explicit renormalizable supersymmetric $E_6$ model with all the ingredients for being realistic. It consists of the Higgs sector $351'+\overline{351'}+27+\overline{27}$, which breaks $E_6$ directly to the Standard Model gauge group. Three copies of $27$ dimensional representations then describe the matter sector, while an extra $27+\overline{27}$ pair is needed to successfully split the Standard Model Higgs doublet from the heavy Higgs triplet. We perform the analysis of the vacuum structure and the Yukawa sector of this model, as well as compute contributions to proton decay. Also, we show why some other simpler $E_6$ models fail to be realistic at the renormalizable level.


Introduction
In spite of the E 6 group being introduced [1] quite soon after the first attempt of grand unification [2], to our knowledge there exists no complete model in the literature so far. With complete we mean not only the Yukawa sector, which has been studied occasionally [3,4,5,6,7,8,9,10,11,12,13], but also the whole Higgs sector and symmetry breaking (exceptions being the simplest renormalizable [14] or non-renormalizable [15] supersymmetric case or the renormalizable non-supersymmetric example in [16,7]), as well as the determination of the Standard Model (SM) Higgs doublet, i.e. the doublet-triplet splitting. This state of affairs is probably due to two reasons. The first one is the complexity of the E 6 group, which is a bit less familiar than the SU(N) [2] or SO(N) [17] structures that can be easily generalized from simpler low dimensional cases. The second one is the lack of a serious motivation. The fact that the quantum numbers of the fundamental 27-dimensional representation can accommodate both the matter 16 and the Higgs 10 of SO(10) is nice in principle, but not easy to make it useful and realistic. Not only that, but at least part of what one gains coming from SU (5) to SO (10), for example automatic R-parity conservation in SO (10) [18,19,20] with 126 breaking rank [21,22,23], is lost when the same SO (10) is embedded in E 6 . Also, the simplest supersymmetric model with the lowest dimensional Higgs representations 27+27+78 can break at the renormalizable level only to SO(10) [14]. It is the purpose of this paper to fill this gap and present a fully realistic E 6 grand unified theory (GUT). In doing this we simplify our analysis by assuming renormalizability and supersymmetry. Contrary to the approach in [24,25], we will not consider the orthogonal problem of supersymmetry breaking in our model. The simplest realistic model we were able to find is made out of 351 ′ + 351 ′ + 27 + 27, needed to break E 6 →SU(3)×SU(2)×U (1), an extra 27 + 27 to achieve the doublet-triplet splitting and so determine explicitly where the SM Higgs doublets live, and three copies of the fundamental 27 for the matter sector. Although we will not present a complete proof that this is really the minimal or simplest renormalizable version, we will give various arguments for why some other representations like 78, 351 or extra 27's in many cases cannot do the same job.
This paper is an extended and detailed version of [26]. We will start first in section 2 with some general description of the E 6 representations, maximal subgroups, and how to construct invariants. In section 3 we will show why some examples cannot work, while in section 4 we will present the minimal Higgs sector, and show the explicit solution which spontaneously breaks the gauge group to the SM one. Section 5 will be devoted to the doublet-triplet splitting and the need for an extra fundamental-antifundamental pair. Section 6 will be devoted to the Yukawa sector, and its peculiarities: the matter fields consist in SU(5) language of 3 generations of 10 +5, 3 vector-like 5 +5 pairs, which, after being integrated out, lead to the needed flavor structure, and 3 pairs of SU(5) singlets. Section 7 is an analysis of the contributions to D = 5 proton decay in our model. We will conclude in section 8 with a list of open problems and possible future projects. Three appendices will give computational details on the issues of state identification, checks on the vacuum solution and the doublet-triplet splitting.
Note on convention: for greater clarity, we color code the vacuum expectation values according to their mass scale: red signifies a GUT mass scale, while blue signifies an Electroweak scale.
2 All we need to know about E 6 E 6 , similarly to the SU(N) groups, has two type of tensor indices: the fundamental or upper index, and the anti-fundamental or lower index. They both run from 1 to 27, which is the dimensionality of the fundamental and anti-fundamental representations. Tensors are constructed with these indices, and extra constraints like simmetricity or antisymmetricity can be further imposed to get irreducible representations. Finally, similarly to the case of the completely antisymmetric SU(N) invariant Levi-Civita tensor ǫ α 1 ...α N or ǫ α 1 ...α N , we have in E 6 the 3-index completely symmetric invariant tensors d µνλ and d µνλ with µ, ν, λ = 1 . . . 27.
The lowest dimensional (< 1000) nontrivial representations are [27] with t A the generators of the E 6 algebra, with the adjoint indices A = 1 . . . 78 and the fundamental indices µ, ν = 1 . . . 27. Note that our convention for labeling representations exchanges 351 and 351, as well as 351 ′ and 351 ′ compared to [27]; in our convention, the representations without bars contain fundamental (upper) indices, while the barred representations contain antifundamental (lower) indices.
To get an explicit form for the invariant d-tensors, we can follow [28]. We organize the fields in 27 according to their quantum numbers: we introduce three 3 × 3 matrices L, M, N, which contain all the fields in 27 Note that the first and third terms on the right have a minus sign compared to [28] due to the different embedding of the SU(3) L and SU(3) R parts of the maximal subgroup SU(3) C × SU(3) L × SU(3) R . Our embedding conforms to the one in [27], which is more useful from the physics point of view. Another difference from [28] is the factor 1 6 = 1 3! in front of d µνλ on the left, which is needed to ensure the normalization claimed in [28]. The tensor d µνλ with all upper indices is taken to have the same numerical values as the tensor d µνλ with lower indices.
With the above definition and normalization, the only nonzero values of the d-tensor are either 1 or −1. Another important property of the d-tensor can be deduced from equation (12): although d is symmetric in its indices, it gives zero as soon as two of the three indices take the same value, similar to the completely antisymmetric tensors ε α 1 ...α N .
With all this we can now see some explicit examples of models.

Some simple unsuccessful examples
We assign the SM matter fields to three copies of the 27-dimensional representation. In order to avoid possible issues with R-parity breaking nonzero vacuum expectation values in the 16 of SO(10), we will enforce a Z 2 parity, under which the matter 27's are odd, while all other "Higgs" representations are even. In this and in the next section, we will consider various Higgs sectors and assess, whether they enable a direct breaking of E 6 to the SM group SU 3.1 n 27 × 27 + n 27 × 27 + n 78 × 78 Trying to build realistic renormalizable models just from the fundamental, antifundamental and adjoint representations 27, 27 and 78, possibly in multiple copies, proves to be impossible due to group-theoretic reasons alone. We shall describe these reasons below where we label the number of copies of the representations 27, 27 and 78 in the model by n 27 , n 27 and n 78 respectively. In order to break E 6 to the SM gauge group, only SM singlets can acquire a nonzero vacuum expectation value (VEV). First note the SO(10) decompositions of the representations under consideration: Only the representations 1, 16, 16 and 45 of SO(10) contain SM singlets. The SM singlets of 1, 16 and 16 are also SU(5) singlets, while the 45 contains both a 1 and a 24 of SU (5). In total, the representation 27 contains 2 SM singlets, both of which are also SU(5) singlets, while the 78 contains 5 singlets, 4 of which are SU(5) singlets and one is in a 24 of SU(5).
Taking n 78 = 0, we are left only with SU(5) singlets in the 27's and 27's, so SU(5) remains unbroken, regardless of the number of 27 and 27 copies. To break the SU(5) part to the SM, we need a model with at least one 78, which acquires a nonzero VEV in the 24 of SU(5). We will show below that no 24 in a 78 can be nonzero, no matter how many 27, 27 and 78 copies in the renormalizable model, and thus SU(5) remains unbroken.
Consider first all types of invariants, which can be formed from the representations 27 i , 27 j and 78 k . Besides the mass terms, we have the following cubic invariants: The only relevant invariants are the last two, since they alone contain a 24 ⊂ 78. We analyze which combinations of VEVs can form terms of these invariants.
The invariant (19) does not contain a term with the 24 of 78, since the 27 contains only SU(5) singlets and 1 × 24 × 1 does not contain a singlet in the SU(5) language, so this term cannot be present in the invariant.
The cubic invariant (20) is antisymmetric in the 78-factors, so we need at least three different 78 copies in the model in order for this invariant to be nonzero. Assuming n 78 ≥ 3, the 24's can enter into the invariant in two possible factor combinations in the SU(5) language: 1 × 24 × 24 and 24 × 24 × 24. But only symmetric products of the 24 form an SU(5) invariant, while our case is antisymmetric due to the cubic 78 3 being antisymmetric. Again, no terms containing any 24 's are present in the invariant.
Since no cubic invariants contain the 24 's, these VEVs only have their mass terms, which force us to 24 = 0. Only SU(5) singlets can therefore acquire VEVs and SU(5) remains unbroken. Note that the 24 as a part of 78 in E 6 behaves differently from the 24 in SU(5). In SU(5), the 24 can acquire a VEV due to the existence of both the quadratic and cubic invariants Tr 24 2 and Tr 24 3 . In E 6 , it is the cubic term which is missing due to the antisymmetry of 78 3 : Tr 78 3 = 0.
This conclusion also applies to the special case n 27 = n 27 = n 78 = 1 in the literature [14]. This renormalizable model breaks E 6 to SO (10), which contains the invariant SU(5). Our result shows that adding extra copies of representations to this model, while still at the renormalizable level, will never allow breaking to the SM.
Note that due to the antisymmetry of 351 and 351, these invariants are trivially zero if we have just a single copy of 27 and 27.
Assume we have more than a single copy of 27 and 27 so that the invariants (22)

351 ′ + 351 ′
The representation 351 ′ is a two index symmetric representation, with a cubic invariant 351 ′3 symmetric in its factors.
In contrast with the previous models, we cannot discount this model by simple grouptheoretic arguments alone. An explicit computation shows this model breaks E 6 to the Pati-Salam (PS) group SU(2) L × SU(2) R × SU(4) C . Since this result is a special case of our proposed model, we postpone the discussion of this model until section 4.5.
The most general renormalizable superpotential of the model can be written as Explicit computation yields the following expressions for the superpotential invariants: Note that the invariants 27 3 and 27 3 are not trivially zero, they just do not contain any terms with only SM singlets, which are relevant for the equations of motion. The zero result can be easily understood under the decomposition SO(10) × U(1) ′′ ⊂ E 6 : by noting that the U(1) ′′ quantum numbers of both singlets in 27 have the same sign, no triple product of the two VEV singlets will yield a net U(1) ′′ charge to be zero, which is a requirement for the term to be present in the invariant. Alternatively, from the point of view of the invariant d-tensor, since there are only 2 SM singlets in the 27, only components with two same-value indices of the d-tensor contribute, but these are zero, as already mentioned in section 2. Analogous considerations apply to the invariant 27 3 .

Equations of motion
Our model is supersymmetric, so the equation of motion consist of both F -terms and D-terms. The F -terms are determined by the equations with i = 1, 2 and j = 1, . . . , 5. The F terms give in total 14 holomorphic equations containing 14 holomorphic variables. Since we have already written the superpotential W explicitly in equations (29)-(37), the equations of motion can be trivially derived by the reader.
The D-terms on the other hand, take the form with the representations 27, 27, 351 ′ and 351 ′ containing the 14 SM singlet VEVs. The index A = 1, . . . , 78 is the E 6 adjoint index and thet A is the action of the A-th algebra generator on the states in a given representation. As per the usual tensor methods in group theory, the actions of the A-th generator on different representations is where the symbol (t A ) µ ν denotes the components of the A-th generator as a 27 × 27 matrix, and * denotes complex conjugation.
Out of possible 78 D-terms, only 5 are non-trivial and they correspond exactly to the 5 SM singlets in the 78. The singlet generators are the following generators of the maximal We therefore label the nonzero D-terms accordingly: R of the non-trivial D-terms is trivially zero, since this combinations of generators corresponds to the SM hypercharge generator t Y : The independent equations for the D-terms can be further simplified by taking their linear combinations. The 4 independent D-term real constraints can be written as The term D III is complex, so it represents 2 real equations.

Symmetries and the general solving strategy
Before proceeding to solve the equations of motion, it is instructive to note two types of symmetry they possess. These symmetries will have implications on the general strategy, how to solve these equations.
1. Conjugation symmetry: the Higgs sector contains representations in complex conjugate pairs. This symmetry exchanges between the representation and its conjugate, e.g. 27 ↔ 27 and 351 ′ ↔ 351 ′ . But since the superpotential contains asymmetric invariants with respect to this symmetry (the cubic invariants for example), we also have to exchange the parameters in front of the invariants. Explicitly, conjugation symmetry can be written as Under this symmetry operations, the superpotential W is invariant, so the F -terms do not change, while the D-terms change according to D I → −D I , D II → −D II , D III → −D III * , so we get an equivalent set of equations of motion.
The exchange of parameters λ in front of invariants under conjugation of representations will have major consequences on our strategy of solving the equations of motion. In the absence of this feature, we could start with an ansatz 351 ′ = 351 ′ and 27 = 27 (more specifically c i = d i and e i = f i ), which automatically solves the D-terms, and then only the F -terms would remain. But due to the exchange in λ's, this ansatz leads to a consistent set of F -terms only if the VEVs vanish or we make an exact fine-tuning λ 1 = λ 2 and λ 3 = λ 4 . Since we would like to avoid relations among parameters altogether, at least for symmetry breaking, we abandon this route. The D-terms will have to be solved in a non-trivial way, which is not a priori obvious, so we find the best strategy to first solve the F -terms, and only then proceed to the D-terms.
2. Alignment symmetry: by examining the equations of motion, it is also possible to see a symmetry under the exchanges The superpotential W remains unchanged under these exchanges, since every single invariant remains unchanged. Furthermore, the D-terms are exchanged according to D I ↔ D II and D III ↔ D III * . All equations of motion thus remain unchanged.
When performing the alignment symmetry operation, we are in fact exchanging the two5's of SU(5) in the representation 27. By doing this, we are also changing the way SO(10) and its subgroups (such as Pati-Salam) are embedded in E 6 , while still containing the same SM group. To elucidate this argument further, consider the E 6 subgroup SU(2) ′ R defined by the generators t 6 R , t 7 R and t 3 R − √ 3t 8 R (these are all SM singlets). This SU(2) ′ R is a subgroup of SU(3) R in E 6 , which rotates the second and third component in the 3 of SU(3) R . The Standard Model generators, and even SU(5) generators, commute with SU(2) ′ R rotations, therefore ensuring that the SU(2) ′ R rotations do not change the embedding of either SU(5) or the SM into E 6 . But SU(2) ′ R rotations do not commute with the standard SU(2) R in SU(3) R , so the embedding of SU(2) R is changed. Since both Pati-Salam and SO(10) contain the SU(2) R , the embedding of these two groups also changes.

The main branch of solutions
In accordance with the discussion on conjugation symmetry, we first start by solving the F -terms obtained from the superpotential in equation (29). Since the superpotential is renormalizable and its highest order terms are cubic, we get a holomorphic system of 14 quadratic polynomial equations containing 14 variables. The general strategy of solving consists of finding equations with a variable only in a linear term, so that we can express this variable from the equation in a unique way. There are two main branches of solutions, which partly overlap. The first branch conforms to the assumptions c 1 , d 1 , e 5 , f 5 = 0, while the second branch assumes c 2 , d 2 , e 4 , f 4 = 0.
The two branches are main branches in the sense that the assumptions of nonzero VEVs are general, while zero VEVs would be considered a special kind of ansatz. The assumptions of the two main branches are exchanged under alignment symmetry; this symmetry also brings one main branch into the other, so we will limit our discussion to the first branch.
First, it is possible to express e 1 , f 1 , e 3 , f 3 from the terms F f 1 , F e 1 , F f 3 and F e 3 , respectively. To proceed along the first branch, we then express e 2 and f 2 from F d 2 and F c 2 respectively, where the assumptions c 1 , d 1 = 0 are needed, since c 1 and d 1 come into the denominators of expressions. Next, we determine e 4 and f 4 from equations F e 2 , F f 2 , respectively, where the assumption e 5 , f 5 = 0 is needed for the same reason. With this procedure, solving the remaining F -terms also for d 1 and f 5 , we get the analytic ansatz of the first branch, which solves all the F -terms: The VEVs c 1 , c 2 , d 2 and e 5 remain undetermined in the expressions of the ansatz and we use them as variables for the remaining VEVs. They can be determined by solving the D-term equations (45)-(47) with the above ansatz plugged-in. Obtaining all the possible solutions in the branch would involve solving a very complicated system of non-holomorphic polynomials; but a simple solution does exist, if we assume the ansatz c 2 = d 2 = 0: equation D III is then solved trivially, while equation D II determines e 5 . We get a specific solution The term D I becomes a polynomial condition for |c 1 | 2 : Note that the positive constant and negative coefficient in front of the highest power of |c 1 | ensure, that this polynomial always has a positive solution for |c 1 |. The explicit form of c 1 will not be needed. We get the other main branch of solutions, if we perform the alignment symmetry operation on the ansatz for the first main branch. We also get a specific solution to the D-terms from the specific solution of the first branch by alignment symmetry; this is equivalent to a 90 • real rotation by SU(2) ′ R , which brings the second entry of the 3 of SU(3) R to the third entry. There also exists a specific solution, which corresponds to a 45 • SU(2) ′ R rotation: we get it by the alignment symmetric ansatz This alignment symmetric solution has all VEVs nonzero and is in the overlap of the two main branches.
The solutions in the main branches are in a sense equivalent, since choosing one or the other essentially means choosing, which combination of the 5's in the 27 is the Standard Model 5.
To show that the specific solution really breaks to the SM, with no flat directions, we explicitly compute the masses of the gauge bosons and of the SM singlets. Everything checks out OK, with further details provided in Appendix B.

Discussion of alternative solutions
Beside the main branches, there are numerous other possible solutions to the equations of motion, which we get by carefully avoiding the assumptions of the two main branches. But it turns out all other solutions of the equations of motion do not break to the SM group, so the above branches are the only solutions for a direct E 6 breaking.
In fact, all but one of the alternative solutions leave SU(5) unbroken. The exception is the solution with the ansazt 27 = 27 = 0: this case corresponds to the model with breaking sector 351 ′ +351 ′ in section 3.3. Due to the presence of the cubic invariants, this is again an example of a model where the ansatz 351 ′ = 351 ′ is not valid and the D-terms need to be solved nontrivially.
Solving the F -terms in the case 27 = 27 = 0 is a simple matter: By explicit computation we discover 21 massless gauge bosons before even solving for the D-terms. This scenario thus breaks to a Pati-Salam group, which is in general embedded into E 6 in a (possibly) non-canonical way. The canonical embedding is recovered by the ansatz e 5 = f 5 = 0, which properly aligns the invariant Pati-Salam. In fact the only nonzero VEVs are then e 3 , f 3 , e 4 and f 4 , exactly the ones which are singlets under the canonical Pati-Salam (see Table 1).
It was therefore the addition of the VEVs from 27 + 27, which enabled E 6 to be broken all the way to the SM.

Where are the MSSM Higgses?
The MSSM Higgses would be expected to reside in the breaking sector 27 + 27 + 351 ′ + 351 ′ . More specifically, they would need to reside at least partly in representations, which couple to the fermionic pair 27 i F 27 j F , so they should be present in 27 and 351 ′ . They cannot reside in 27 F , since there is no cubic term 27 3 F due to matter parity. The usual procedure would be to compute the mass matrices of the doublets (1, 2, + 1 2 ) and antidoublets (1, 2, − 1 2 ) and perform a fine-tuning of Lagrangian parameters to get one doublet mode massless; the soft supersymmetry-breaking terms would then enable this mode to get an electroweak (EW) scale VEV. At the same time, we need to make sure the fine-tuning still keeps the triplets (3, 1, − 1 3 ) and antitriplets (3, 1, + 1 3 ) heavy (of the order of the GUT scale), since they mediate proton decay. This separation of scales is called the doublet-triplet (DT) splitting problem. Although fine-tuning is not considered to be aesthetically pleasing, it usually does the job.
But curiously, in our case, DT splitting cannot be performed: all vacua, which break to the SM, have this inability, since making the doublet massless automatically does the same to the triplet. Further computational details on the this DT splitting attempt in the breaking sector are provided in Appendix C, as well as a list of possible reasons for failure. The inability to perform DT splitting is disappointing, since it is only this usually trivial hurdle which prevents the 27 + 27 + 351 ′ + 351 ′ model to be realistic.
The above DT problem can be cured by a reasonably simple action: we add an extra pair 27 + 27 to the model. To simplify the analysis we will assume that these extra 27 + 27 couple only quadratically with the Higgs fields with large VEVs. In this way we are making automatic the solution of the old equations of motion for vanishing VEVs of these new tilde fields. In addition, the mass matrices of the doublets and triplets in the tilde fields decouple from the DT matrices of the breaking sector.
We thus get the following tilde-field superpotential needed for the doublet-triplet splitting: The tilde sector contains 3 doublet/antidoublet and triplet/antitriplet pairs, so the mass matrices will be 3 × 3. For our labeling convention and list of the doublets and triplets in the tilde representations, see Table 2. The mass matrices can be written as where the mass terms are written as doublet,triplet ⊂ SU(5) ⊂ SO(10) ⊂ E 6 doublet VEV A fine tuning among the new κ parameters will ensure DT splitting. If we plug the vacuum solution into the mass matrices M doublets and M triplets , we get the following DT splitting conditions: These two conditions insure a massless doublet mode, but keep all triplet modes heavy. Both can be simultaneously satisfied by a fine-tuning The above fine-tuning of κ 1 gives the following modes of doublets and antidoublets to be massless: Notice that the massless modes have components of all doublets and antidoublets present: in particular, the Higgs is present in the components D 1 , D 2 and D 3 of 27, so the corresponding EW VEVs v 1 ,v 2 andv 3 all become nonzero. The presence of these VEVs will be important in the analysis of the Yukawa sector.

The Yukawa sector
Assuming a Z 2 "matter parity", which avoids potentially dangerous R-parity violating terms, with and with all other fields even, we can write down the most general Yukawa sector: Excluding the tilde part, our E 6 Yukawa sector is completely analogous to the Yukawa sector in the minimal renormalizable SO(10) model [29,30]: Notice, however, that the flavor-mixing mechanism in the two cases is completely different. In the renormalizable SO(10) we have the usual case of GUTs where the EW Higgs is present in two representations: 10 and 126. Since the two generic matrices Y 10 and Y 126 cannot be diagonalized simultaneously, we get flavor mixing.
Flavor mixing in our model is more subtle. The MSSM Higgs doublets are present only in 27 (and 27, which is not present in the Yukawa sector), but the representations 27 and 351 ′ of the breaking sector acquire GUT scale VEVs. Through these large SU(5) breaking VEVs, they contribute to mix the5 in 16 with the5 in 10 of 27. Flavor mixing is therefore not due to the presence of Higgs in two different representations at the EW scale, but due to the different mixing of vector-like heavy pairs at the GUT scale. This situation is analogous to [8], which we further elaborate on below.
The triplets get nonzero VEVs as usual: the superpotential terms are Integrating out the heavy triplets yields To get the light fermion mass matrices explicitly, we will integrate out the heavy vectorlike pairs following for example [31,32]. Our matrices are of block form where with relations we get Integrating out the heavy states in the lower right part, we are left in leading order of m W /M GU T with the matrix for light states For our solution with the ansatz and defining the light fermion masses become Notice the combined contributions of both type I [33,34,35,36,37] (proportional to v 2 1 ) and type II [38,39,40,41] (proportional to ∆ 1,2,3 ) seesaw in (109). As always, the seesaw mechanism gives neutrino masses of the scale O(m 2 W /M GU T ), which can be seen from the factors  This means we recover the usual low-energy degrees of freedom from the MSSM. Also, the 16 and 10 of SO(10) in the 27 F 's mix, i.e. the light states do not live just in the 16.
The explicit fitting of these mass matrices to the experimental values of the masses and mixings is complicated by the nonlinear way the various matrices enter into the equations. This is typical for contributions from vector-like families. Although the full analysis is beyond the scope of this paper, we note here that there are 3 Yukawa matrices involved. The number of free parameters seems more than likely large enough to allow a successful fit. We leave the full analysis for a future publication.

Proton decay
The D = 5 proton decay [43,44,45,46] The relevant couplings for proton decay can be written in terms of SM irreducible representations generically as where i, j are generation indices and A, B = 1, . . . , 15 are indices over all the color triplets/antitriplets, with sums over repeated indices; we define T 12+A := T A and T 12+A := T A (with A = 1, 2, 3). We suppress the SU(3) C and SU(2) L indices in our notation; the indices in the fields are contracted with the epsilon tensors in the order the fields are written, with ε 123 = ε 12 = 1. The triplet mass matrix M T has contributions from the following terms in equation (110): the mass terms m 27 , m 351 ′ and m 27 , the λ-terms and the κ-terms. The tilde and non-tilde fields do not mix in the mass terms because the tilde fields have vanishing VEVs, so M T has the block form The two matrices are These matrices can also be found in equations (143)  The coefficients C are computed to be Notice the different coefficients in front of δ A 12 , a consequence of different Clebsch-Gordan coefficients in Table 5.
Integrating out the triplets T A and antitriplets T A from the relevant terms, we obtain (to lowest order in the operators) Note that we have written the inverse matrixM −1 T with a hat. A triplet mode and an antitriplet mode are massless, which means the block M triplets cannot be inverted. The massless modes correspond to the would-be Goldstone bosons in the Higgs mechanism; these unphysical degrees of freedom can be rotated out of the Yukawa terms by a gauge transformation, which is equivalent to plugging a zero for their field value. This is formally equivalent to introducing a mass term for these modes, integrating them out, and then pushing the introduced mass to infinity, so they decouple from the theory. A basis independent ansatz for the computation of the inverse of the physical degrees of freedom, while automatically decoupling the would-be Goldstone bosons, iŝ where e A and f A are the components of right and left null eigenvectors of M, respectively. They need not be normalized, since the normalization factors can be absorbed into M. In our basis, we can take Although formally elegant, this method is hard to implement since it requires to first explicitly invert a large matrix and only then take the limit M → ∞. An equivalent but more explicit procedure would be to rotate the (N + 1)-dimensional system of triplets into the N-dimensional part orthogonal to the Nambu-Goldstone zero mode. Let the normalized right and left Nambu-Goldstone eigenstates respectively be with the columns α = α a andᾱ =ᾱ a , a = 1, . . . , N. The unitary (N + 1) × (N + 1) matrix then transforms the old basis where now T B = (T 0 , T a ) with T 0 the would-be Nambu-Goldstone triplet. The T A are analogously transformed by U(ᾱ). The choice of U represents just one simple possibility of choosing the transformations matrix; it is not unique since we could have composed it with an arbitrary rotation in the orthogonal complement of the zero mode (space of T a 's). Dropping the zero modes T 0 , T 0 , equation (111) can now be written as Defining now a N × N invertible matrix we arrive to (121) with the inverse of the M triplets block given by Finally, we have to project onto the light matter superfields. Remember that the 27 F 's contain a vector-like pair of quarks and leptons in the 10 of SO (10). This 10 mixes with the 16, so d c mixes with d ′c and L mixes with L ′ . This is rotated to the basis of light and heavy states with the help of the matrix U in equation (96). Generic particles q and q ′ can be decomposed into light and heavy states q l and q H , respectively, with This implies the following projections to the light statesd c andL: where X 0 is defined in equation (105). Writing in terms of only light states (those at the scale m W ) and only for the lepton and baryon number violating operators, we get the following low-energy effective operators for D = 5 proton decay: In spite of the fact that the final expression is rather complicated, we can draw some general conclusions and leave the numerical analysis in combination with the study of the Yukawa part for a future publication.
• Since we are in E 6 with several possible heavy thresholds, there is no necessary light color triplet as in the minimal renormalizable SU(5) with low-scale supersymmetry [47].
• Only some elements of the inverse matrix need to be small, similar to the SO(10) case, which also has multiple triplet contributions. Notice that the triplets in 351 ′ , 27 and 27 do not couple to the matter fields, and neither do some triplets in 351 ′ (seen from the C-coefficients).
• The final expressions are functions of a number of parameters: the masses, the λ and κ parameters, as well as three Yukawa matrices. Since the constraints on these parameters come from the fit to a smaller number of parameters of the SM Yukawas, there will likely be some residual freedom in parameter space, which would allow for proton decay supression.
All these reasons make nucleon decay amplitude suppressions probable. Finally, if all this fails, we can still use some version of a (moderately) split supersymmetric spectrum.

Conclusions
This paper represents a first attempt to write down and solve a realistic E 6 model. We concentrated on a supersymmetric and renormalizable case and found strong evidence that such a model includes the fields 351 ′ + 351 ′ + 27 + 27 to spontaneously break E 6 into the SM, another pair of 27 + 27 for the MSSM Higgs fields, and three copies of matter 27's. To simplify the analysis we made two assumptions: an extra Z 2 symmetry which automatically preserves R-parity, and some vanishing couplings of the superfields that contain the MSSM Higgses. Although the first assumption is probably unavoidable, the second may not be needed.
We noticed some interesting features which are not usually encountered in theories with lower groups: • The existence of asymmetric solutions: although the D-terms are satisfied by the natural solution |φ i | = |φ i | with i going over all complex Higgs representations, the F-terms are not, unless the same VEVs vanish or there are some fine-tuned relations among the superpotential parameter. We avoided such an assumption, and found an asymmetric solution |φ i | = |φ i |.
• The minimal sector that breaks into the SM could not describe the MSSM Higgses in spite of the fact that it contains fields with the right quantum numbers. The reason is the impossibility of performing a realistic DT splitting.
• The automatic presence of 3 vector-like families, which makes the analysis of the Yukawa sector nonlinear.
An obvious problem with this type of models is the Landau pole which occurs at a scale Λ less than one order of magnitude above the GUT scale (the gauge beta function is -153 compared to -109 of the minimal SO(10) [21,22,23]). So even if we believe that for some reason gravity will not produce higher dimensional operators suppressed by inverse powers of M P lanck , a consistent perturbative treatment of our type of models should assume the absence of operators suppressed by inverse powers of Λ as well. There is nothing we can say in defense of this, except that well studied SO(10) models have similar problems.
There are several open questions, which are beyond the scope of this paper. First, it would be interesting to check what happens if other terms in the light Higgs superpotential are introduced, i.e. if the terms linear or cubic in "tilde" fields appear. Second, since the neutrino masses need a slightly lower see-saw scale than the GUT scale, the usual approximate one-step supersymmetric unification [48,49,50,51] may be at risk due to large representations involved, and the knowledge of the mass spectrum may turn out to be necessary. Third, more elegant solutions to the doublet-triplet splitting problem can be looked for: both the missing VEV [52,53] and the missing partner [54,55,56] however seem to need the 650: a full minimization with this field should thus be performed. Fourth, a thorough study of the E 6 could be studied at the non-renormalizable level: in this case we could consider an asymptotically free theory with 78+27+27 only in the Higgs sector. Fifth, non-supersymmetric theories could be considered, where the Higgs sectors are typically

A Detailed identification of various states
It is possible to refer to states in various E 6 representations by using familiar labels of particles. The fundamental 27 representation is 16 + 10 + 1 in SO(10) language, so we denote the various SM irreducible representations in the following way: • The 16 contains Q ∼ (3, 2, • The SO(10) singlet is labeled by s ∼ (1, 1, 0).
The conjugate representation of 27 contains exactly the conjugate SM representations of the listed ones. We denote them by bars, so the particle content of 27 is labeled byQ, L,d c ,ū c ,ē c ,ν c ,L ′ ,L ′c ,d ′c ,d ′ ands.
Similarly, we can use this particle notation also for other irreducible representations. The representation 351 ′ is contained in the symmetric product of 27 × 27, so we label the states with two successive labels of the 27, while suppressing manifest symmetricity in our notation; to get the 351 ′ in the 27 × 27 matrix, we also have to project out the 27 with the d-tensor, but that does not change the rules of notation. Due to simplicity, we also suppress any color or weak indices from our notation, but summation over them is sometimes implicit.
As an example, consider the antitriplet (3, 1, + 1 3 ) contained in the product of two (3, 2, + 1 6 ): we label it simply by QQ, but this written out explicitly would be ε abc Q bi Q cj ε ij , where a, b, c = 1, 2, 3 are color indices, i, j = 1, 2 are weak indices and ε are the antisymmetric tensors; note that the only remaining free index is the lower a, which indicates the state is a 3 under SU(3) C . Another example is the antidoublet (1, 2, − 1 2 ) in Qu c : explicitly, we actually mean Q ai (u c ) a + (u c ) a Q ai (note also the symmetricity), with the same convention for the color and weak indices as before. Notice that we do not write (overall) normalization factors in particle notation; we use this notation only to identify the relevant states (which is non-trivial in the case of 351 ′ , since it is necessary to project out the 27), while in explicit computations we always use properly normalized states, such that for the doublet and triplets (see Table 5), we have We list the labels and particle identifications for the relevant fields in various tables below. The SM singlets are listed in Table 4, while the weak doublets and triplets relevant for DT splitting are listed in Table 5. We write them out explicitly in particle notation only in the unbarred representations, since the corresponding states in the conjugate representation would have the same form, but with ordinary letters substituted by barred letters. Also, the list of weak triplets contributing to type II seesaw has already been presented in Table 3.
Note that the states are part of distinct representations, so they need to be orthogonal. This can be easily checked by the reader from particle notation (i.e. the different two-label states are orthogonal basis vectors), but one should not forget about our convention of suppression of indices; what looks like a single term at first glance may indeed be a sum of multiple terms. Table 4: Singlet labels and their identification in particle notation. Table 5: Doublet and triplet labels and identification in particle notation. label E 6 ⊇ SO(10) ⊇ SU(5) label E 6 ⊇ SO(10) ⊇ SU(5) doublet in p.n. triplet in p.n.

B Details of the vacuum
We present here some details of the SM (supersymmetric) vacuum, obtained by the particular solution in equations (66)-(72).

B.1 Gauge boson masses
The masses of the gauge bosons A µ a can be computed explicitly. The gauge boson mass terms can be written as where g is the E 6 gauge coupling constant, and the the mass-square matrix M ab is defined as The symbolt a denotes the action of the a-th generator on the representation. Knowing the explicit form of the generators t a in the fundamental representation, and using E 6 tensor methods, we can compute the matrix M ab explicitly, and determine the squares of the gauge boson masses by diagonalization. The results for the specific solution with c 2 = d 2 = e 2 = f 2 = e 4 = f 4 = 0 are given in Table 7. The nonzero VEVs of the solution are not plugged in. Notice that the SU (5) (3, 1, + 2 3 ) 45 10 (1, 1, +1) (1, 1, −1) 10 (3, 1, + 2 3 ) 16 10 (1, 1, +1) (1, 1, 0) (1, 1, 0) They mix:

B.2 No flat directions check
In order to check that our solution is an isolated point and that there are no flat directions in the F -terms, we check the mass matrix of the SM VEV-acquiring singlets in our model. The relevant singlets live in the representations 27, 27, 351 ′ and 351 ′ of the Higgs sector. We label the singlets by s x , where x is the label of the VEV; our singlets are therefore s c i , s d i , s e j and s f j , where i = 1, 2 and j = 1, . . . , 5.
The mass term can be written as where M singlets is the matrix  Plugging in the specific solution from equations (66)-(72), the matrix gets 4 massless modes.
Since the E 6 → SM breaking also breaks 4 of the 5 SM singlet generators in the 78, we expect that any solution of the equations of motion breaking to the SM will automatically produce 4 massless singlet modes of scalars due to the Higgs mechanism. Any additional massless modes would correspond to a flat direction of the superpotential around the specific vacuum. Since there are no additional massless singlet modes, there are no flat directions in our solution.

C The doublet-triplet splitting
Suppose we have a Higgs sector consisting of fields 351 ′ + 351 ′ + 27 + 27 and we want the SM Higgs to live in both the 27 and 351 ′ , similar to how the Higgs lives both in the 10 and 126 in the SO(10) model. The mass terms connecting doublets (1, 2, + 1 2 ) to antidoublets (1, 2, − 1 2 ) and the triplets (3, 1, − 1 3 ) to antitriplets (3, 1, + 1 3 ) will come from the breaking part of the superpotential in equation (29). The mass matrices of the doublets and triplets in the fermionic 27 i F are distinct and do not mix with the mass matrices in the breaking sector due to Z 2 matter parity.
The list of all the doublets and triplets, along with our label conventions, is already compiled in Table 5. There are 11 doublet/antidoublet pairs and 12 triplet/antitriplet pairs in the breaking sector, so the doublet and triplet mass matrices are 11 × 11 and 12 × 12, respectively.
Writing the doublet and triplet mass terms as − √ 3 for triplets. This is due to the fact that the β entries couple the doublets/triplets in the 45 and 5 (or equivalently 45 and 5) of SU(5); the − √ 3 is due to the normalization of the doublets and triplets in the representations 45 and 45.
Performing doublet-triplet splitting would involve a fine-tuning of parameters m 351 ′ , m 27 and λ i , such that we get a massless doublet mode, while keeping all the triplets heavy. Solving the equations of motion and plugging their solution into the mass matrices (i.e. using equations (56)-(65)), we automatically get a massless doublet/antidoublet and triplet/antitriplet mode. This should come as no surprise, since the breaking E 6 → SM involves breaking the generators transforming as 16 + 16 of SO (10), which contain exactly one doublet/antidoublet pair and one triplet/antitriplet pair: the doublet and triplet massless modes of scalars are eaten up by the corresponding gauge bosons due to the Higgs mechanism. We thus have the extra complication of already having a massless doublet and triplet mode, so DT splitting involves making a second doublet mode massless.
In principle, the massless modes can be straightforwardly extracted from the scalar squared-mass matrices M † doublets M doublets and M † triplets M triplets . In our case, however, squaring a matrix would unnecessarily complicate the calculation, so we use methods which work on the matrices M doublets and M triplets themselves. Already We see that due to the simplicity of the conditions, which are just a product of the Lagrangian parameters, we cannot perform a fine-tuning on the doublets independently from the triplets: an extra massless doublet necessarily implies an extra massless triplet. The usual procedure of DT splitting via fine-tuning is therefore not possible in this case. We cure this problem of the model by adding an extra 27 + 27 pair.
Although explicit calculation shows DT splitting is not possible without the tilde fields, we are unable to find a clear-cut reason, which would explain -without calculationwhy the usual method of fine-tuning fails. Note that the mass matrices by themselves do not have this feature: the impossibility of DT splitting shows itself only after solving the F -term equations of motion and plugging in the solutions. In the following, we list peculiar details and possible reasons, a combination of which might contribute to this inability: • It seems DT splitting is a problem only for solutions breaking to the SM. It is possible to fine-tune in the alternative solution 27 = 27 = 0, which breaks to Pati-Salam.
• Without the tilde fields, the SM Higgs would live in representations already involved in the E 6 breaking: these representations would thus acquire both GUT and EW scale VEVs.
• We already have a massless doublet and triplet mode present in the mass matrices due to the Higgs mechanism.
• There are only a few specific places in the mass matrices (in the M 3×3 block), where there is a sum of the type A 1 + B 24 , which enables DT splitting in the simplest SU(5) case. When plugging in a specific solution with the ansatz c 2 = d 2 = e 4 = f 4 = 0, one of the two terms always disappears. Also, the mass matrices contain a lot of zero entries.
As a final point, we allude to the missing partner mechanism (MP mechanism) [54,55,56] for DT splitting as it pertains to our model. At first glance, the MP mechanism would seem promising for our model, since we already have one more triplet than a doublet due to the presence of 50 and 50 of SU(5) in the representations 351 ′ and 351 ′ . The above explicit computation already shows that the mechanism is not at work in our case, which is perhaps expected: the MP mechanism involves a specific setup of fields and form of the mass matrix, so it would be too optimistic to expect it for free in a general setup. To implement the MP mechanism, it is necessary to have the 50 of SU(5) with the triplet but not the doublet, but also the 75, which couples the triplet in the 50 with the antitriplet in the 5. The lowest dimensional E 6 representation, which contains the 75 of SU(5), is the 650. In our case, simplicity therefore seems to dictate to forego the MP mechanism, and just add the extra 27 + 27 pair.