E-6 grand unified theory with three generations from heterotic string theory

E

grand unified theory with three generations from heterotic string theory

Motoharu Ito,1,*Shogo Kuwakino,1,†Nobuhiro Maekawa,1,‡Sanefumi Moriyama,2,3,xKeijiro Takahashi,4,k Kazuaki Takei,1,{Shunsuke Teraguchi,1,5,**and Toshifumi Yamashita1,††

1_{Department of Physics, Nagoya University, Nagoya 464-8602, Japan} 2_{Kobayashi Maskawa Institute, Nagoya University, Nagoya 464-8602, Japan} 3_{Graduate School of Mathematics, Nagoya University, Nagoya 464-8602, Japan} 4

Department of Electrophysics, National Chiao-Tung University, Hsinchu, Taiwan, Republic of China 5_{WPI Immunology Frontier Research Center, Osaka University, Osaka 565-0871, Japan}

(Received 13 December 2010; published 19 May 2011)

We construct two more supersymmetricE6 grand unified models with three generations within the framework ofZ12asymmetric orbifold compactification of the heterotic string theory. Such an asymmetric orbifold is missing in the classification in the literature, which concludes that only oneE6model is possible. In both of the new models, an adjoint Higgs field is obtained in virtue of the diagonal embedding method. This method mods out the three E6 factors of an even self-dual momentum-lattice by a permutation symmetry. In order to realize the ðE6Þ3even self-dual lattice, we utilize the lattice engineering technique. Among the eight possible orbifold actions in our setup, two lead to newE6models. Though these models still share the unsatisfactory issues with the known one, our discovery raises hopes that excellent models that solve all the problems in the supersymmetric grand unified models will be found in this framework.

DOI:10.1103/PhysRevD.83.091703 PACS numbers: 11.25.Wx, 11.25.Mj, 12.10.Dm

I. INTRODUCTION

Superstring theory is one of the most promising candi-dates that describe quantum gravity and unify all the four fundamental forces of nature. Usually, however, super-string theory is defined in a ten-dimensional (10D) space-time and its characteristic scale is taken to be the Planck scale, incredibly larger than that of the standard model (SM). If superstring theory truly describes our world, it must be an indispensable subject to find the way to the SM from superstring. Unfortunately, superstring has a lot of perturbative vacua, and so far, the way has not been estab-lished. Therefore, it is worthwhile to ask phenomenologi-cal studies for hints.

Independently of the developments on superstring, the supersymmetric (SUSY) grand unified theory (GUT) [1] is known as an interesting candidate for the model beyond the SM. It unifies the three gauge groupsSUð3Þ_C
SUð2Þ_L
Uð1ÞYin the SM into a single gauge group. This unification is quantitatively supported by experiments which have revealed that the three gauge couplings in the minimal supersymmetric SM meet with a very good accuracy at a very high scale (the GUT scale) close to the Planck scale. Moreover, it unifies one generation of quarks and leptons,

which is dispersed among five multiplets in the SM, into one or two multiplets. This matter unification is qualita-tively supported by the measurements of quark/lepton masses and mixings: The pattern of the various hierarchical structures of the masses and the mixings can be explained by a simple assumption that the hierarchies of the Yukawa couplings are induced mainly by the 10 multiplets of SUð5Þ [2–4].

Among the SUSY-GUTs, theE6GUT [5], which unifies all one generation quarks and leptons into a single 27 multiplet, has an advantage that the above assumption for the Yukawa hierarchies, which must be made by hand in the SUð5Þ unification, is naturally derived [3,4]. This ad-vantage is particularly important since it seems difficult to explain the hierarchical structure in the minimal symmetric SM-like models obtained directly from super-string [6,7]. Thus, apart from remaining issues, such as the so-called doublet-triplet (DT) splitting problem and the SUSY-flavor/CP problem, it is plausible that this E6 struc-ture is realized.

In addition, consistently with the E6 structure, it has been shown that the anomalous Uð1Þ_A gauge symmetry [8] and theSUð2Þ_H(orSUð3Þ_H) family symmetry [4] with a spontaneousCP violation [9] respectively serve solutions to the DT splitting problem and the SUSY-flavor/CP prob-lem. Interestingly, both of the above two additional sym-metries can be simultaneously adopted. Then, the resulting models are really promising, where almost all the phe-nomenological problems are solved, the realistic quark/ lepton mass matrices are obtained naturally and all the three generations are unified into two multiplets (or a single one forSUð3Þ_H).

*[email protected] †_{[email protected]} ‡_{[email protected]} x [email protected] k [email protected] { [email protected] **[email protected] ††_{[email protected]} PHYSICAL REVIEW D 83, 091703(R) (2011)

Thus, when we seek for the way to the SM from superstring, the above scenario must be valuable to be considered, though we should also keep in mind the pos-sibilities that not all of the above additional symmetries, such as the anomalous Uð1Þ_A symmetry and theSUð2Þ_H symmetry, are actually realized. Therefore, we assume only the four-dimensional (4D) SUSY E6 unification in this letter. Namely, our strategy is to construct phenome-nological string models with the following minimal re-quirements,

(i) E6unification group, (ii) 4DN ¼ 1 SUSY, (iii) adjoint Higgs field, (iv) three families,

anticipating to find models with all or some of the addi-tional symmetries mentioned above.

In the literature, despite decades of research, only one model with these requirements has been reported [10]. The authors of the reference claimed that they classified the models with the minimal requirements and a hidden non-Abelian gauge symmetry as well which may be useful to break the SUSY dynamically. Unfortunately, however, the above-mentioned additional symmetries are not real-ized in their model. Therefore, we would like to search for more 4D N ¼ 1 SUSY E6 models in string theory, by relaxing the requirement of the hidden non-Abelian gauge symmetry as it is possible to break the SUSY in other ways (for example, in a meta-stable vacuum [11,12]).

II. STRATEGY

Let us work on the heterotic string theory [13], where the E6unified group can be realized without much difficulty. In contrast to the F-theory [14] which is another related framework realizing the E6 group [15] and has attracted attention recently [16], the heterotic string theory has a microscopic description by a Lagrangian and thus any quantity is, at least in principle, calculable.

In the heterotic string, to obtain the 4D effective theory, twenty-two extradimensions in the left-moving bosonic string and six in the right-moving superstring should be compactified. Then, both the left-moving and the right-moving momenta are quantized and compose a lattice. For the consistency of string theory, the partition function, namely, the one-loop vacuum diagram of the closed string must be invariant under the modular transformations T :
!
þ 1 and S:
! 1=
, where
is the moduli of the worldsheet torus. This requires that the lattice is even and self-dual with a (22, 6)-Lorentzian signature. A space-time gauge symmetryG (in particular E6) is realized when this momentum-lattice contains the appropriately normal-ized Lie lattice ofG in the left-moving part.

To reduce the 4DN ¼ 4 SUSY to N ¼ 1, six com-ponents among eight of the massless 10D spinor mode in the right-moving superstring have to be projected out. This

is achieved by an orbifold compactification [17], which identifies the compactified space under an action of a point group that leaves the lattice unchanged. In particular, the compactified right-moving six dimensions should be fully rotated with three nontrivial rotating anglest_Rithat satisfy P₃

i¼1tRi¼ 0 (mod 2
2).

In general, when the heterotic string realizes a spacetime gauge symmetry, the currents of the corresponding world-sheet theory form a Kac-Moody algebra: ½ja_m; jb_n ¼ ifab

cjcmþnþ kmabmþn;0. Here,jamis the Laurent coeffi-cient of the world sheet current jaðzÞ ¼P_m2Zja_mzm1, fab

c is a structure constant and the integer k is a Kac-Moody level. The usual orbifold construction described above realizes the lowest Kac-Moody level (k ¼ 1), while it is known [18] that a higher level is necessary to obtain adjoint Higgs fields. A way to increase the level is the so-called diagonal embedding method [19], whereK-copies of the current ðjÞ_Iwith levelk ¼ 1 are permuted by an orbifold action so that only the diagonal part j_diag¼ PK_I¼1ðjÞ_I re-mains phaseless under the action. It is easy to see thatjdiag satisfies the Kac-Moody algebra with k ¼ K. The other eigenstates have nontrivial phases, and thus do not contrib-ute to the 4D gauge multiplets, while some of them may couple with chiral multiplets in the right-mover to cancel the phases, resulting in adjoint Higgs fields. It is also possible that adjoint Higgs fields appear in twisted sectors. Unfortunately, it is not easy to clarify the condition in string theory to construct models with three generations, while there is a conjecture that the number of the gener-ations is proportional to the Kac-Moody level [10]. Therefore, we start with the construction of 4D N ¼ 1 SUSYE6 models with an adjoint Higgs field, leaving the numbers of generations to be determined model-by-model. To summarize, we take the following strategy:

(1) we prepare a (22,6)-dimensional even self-dual lat-tice with equivalentK-copies of the left-moving E₆ lattice,

(2) we consider an orbifold identification that includes (a) a permutation among theE₆ factors,

(b) rotations of the right-moving six dimensions with three nonzero angles t_Ri satisfying P3

i¼1tRi¼ 0 (mod 2
2),

(3) we find out the number of the generations.

According to the conjecture, k ¼ 3 is needed for the three generations, and we take this choice hereafter. In this case, the left-moving ðE6Þ3 lattice occupies 18 dimensions and cannot be fitted in the 16 extradimensions (with respect to the 10D viewpoint). This means that the usual left-right symmetric treatment of the six extradimensions is not valid and we have to work in the asymmetric orbifold [20] with a Narain compactification [21]. In contrast to the symmetric orbifold, the general rules for consistent models are rather involved in the asymmetric orbifold [10], and thus, we calculate the one-loop partition functions explicitly to check the modular invariance (see Ref. [22] for the details).

III. SETUP

The lattice engineering technique [23] is helpful in constructing desired lattices. The essence of the technique is that a lattice (for example, the A₂ lattice) transforms oppositely as its complement lattice in the Euclidean even self-dual E₈ lattice (the E₆ lattice for the A₂ example) under the modular transformation. Thanks to this, we can always replace the left-moving A2 lattice by the right-moving
E6 lattice (denoted with a bar) and vice versa. Subsequently, since the
E6lattice is decomposed into three

A2lattices, we can construct a left-moving ðE6Þ3lattice out of the right-moving ð
A2Þ3 lattice using the same technique again. Thus, we can easily convert anA2 lattice into three equivalentE6 lattices.

With all these insights in mind, let us pick up the E6

E6 lattice as our starting point. After decomposing E6 into ðA2Þ3, we end up with ½ðA2Þ2
ðE6Þ3

E6 using the above technique. Though this lattice is the same as the one used in Ref. [10] constructed from the 10D SOð32Þ heterotic string by a compactification with Wilson lines, our method is more direct and hence many Narain lattices with ðE6Þ3 symmetry are accessible with their discrete symmetries manifest.

The orbifold action on the three left-movingE6factors is chosen to be a permutation among them. It turns out that a shift along the diagonal factor has to be introduced in addition, as a source of the asymmetry between the num-bers of the generations and the antigenerations in the twisted sectors.

A natural candidate for the action on the right-moving factor is the rotation by the Coxeter element of the right-moving
E6 lattice, which is an element of a point group Z12¼ Z3
Z4. Though it was claimed thatZ2 is the only

possible symmetry to add to theZ₃symmetry for the above permutation [10], we find no reasons to exclude this pos-sibility. Thus, we choose this rotation, which corresponds to the one with three anglest_R ¼ 2ð1; 4; 5Þ=12 classi-fied asZ₁₂-I [24], as part of our setup.

Then, the remaining options are actions on the two left-moving A2 factors. The allowed choices on each factor, labeled byi, are

(i) shifts_Li, with 12s_Li¼ 0 (mod roots), (ii) rotation with an anglet_Li¼ 2=3, (iii) Weyl reflection.

There are a lot of choices of s_Li but many of them are related to each other by transformations under the symme-try of the A2 lattice, leading to identical models. In addi-tion, the modular invariance does not allow arbitrary choices, but only certain combinations. Thus, there remain only a few possible actions:

fð2;0Þ;ð4;0Þ;“rot”gfð0;0Þ;ð6;0Þg; ð1;0Þð3;6Þ; ð1;6Þð3;0Þ;

where “rot” denotes the 1=3 rotation while ðn; mÞ repre-sents the shift defined by the vectors ¼ ðn1þ m2Þ=12, with_i being the simple roots ofA2. In the first line, we have three options for the action on one ofA₂ lattices, and two for the other. Therefore, there are, in total, 3
2 þ 1
1 þ 1
1 ¼ 8 consistent models possible in this setup. Note that the order is irrelevant since the two A₂ factors are equivalent.

TABLE I. The massless spectra of the models with three generations:U and T_denote the untwisted and various twisted sectors, respectively. The quantum numbers of left-handed chiral multiplets and the normalization of theUð1Þ charges are shown. The gravity and gauge multiplets are omitted.

Model 1 Model 2 Model 3

gauge symmetry E6
SUð2Þ
Uð1Þ3 E6
SUð2Þ
Uð1Þ3 E6
Uð1Þ4

U ð1; 1; þ6; 0; 0ÞL ð1; 1; þ6; 3; 0ÞL ð1; 6; 0; 0; 0ÞL ð78; 1; 0; 0; 0ÞL ð78; 1; 0; 0; 0ÞL ð1; þ3; 6; 0; 0ÞL ð78; 0; 0; 0; 0ÞL T1 ð27; 1; þ1; 0; 1ÞL    ð27; 1; 1; þ1; 0ÞL T2 ð27; 1; 1; 1; 0ÞL ð27; 1; þ2; 0; 2ÞL ð27; þ1; 0; 0; 1ÞL T3 2ð1; 1; 3; 0; 3ÞL ð1; 1; 3; 3; 3ÞL ð1; þ3; 3; þ3; 0ÞL ð1; þ3; þ3; 3; 0ÞL T4 ð27; 1; 2; 0; 0ÞL ð27; 1; 2; 1; 0ÞL ð27; þ2; 0; 0; 0ÞL ð27; 1; 2; 0; 0ÞL T5 ð27; 1; þ1; 0; 1ÞL ð27; 1; þ1; 1; þ1ÞL ð27; 1; þ1; 1; 0ÞL ð1; 3; 0; 0; 3ÞL T6 ð1; 2; 0; 0; 3ÞL ð1; 2; 0; 3; 0ÞL ð1; 0; þ6; 2; 0ÞL ð1; 1; þ3; 3; 0ÞL ð1; 1; 6; 0; þ6ÞL ð1; 0; 6; þ2; 0ÞL normalization ofUð1Þ ðpffiffi2 6 ; ffiffi 6 p 6 ; ffiffi 6 p 6Þ ð ffiffi 2 p 12; ffiffi 6 p 6 ; ffiffi 6 p 12Þ ð ffiffi 2 p 6 ; ffiffi 6 p 12; ffiffi 2 p 4; ffiffi 6 p 6Þ

IV. MODELS WITH THREE GENERATIONS Once fixing the orbifold action, one can calculate the partition function (see, for example, Refs. [10,22]), which contains information of the spectrum of the model. It turns out that, among the above eight models, three lead to vanishing net generation numbers, while other two and the remaining three, respectively, have nine and three net generations. Here, we concentrate on the last three,

ð2; 0Þ  ð6; 0Þ; ð1; 0Þ  ð3; 6Þ; ð1; 6Þ  ð3; 0Þ; and we call them Model 1, 2, 3, respectively.

Model 1 and Model 2 have the gauge group E₆
SUð2Þ
Uð1Þ3_{, and Model 3 hasE}

6
Uð1Þ4. Their mass-less spectra are listed in TableI. We find five generations and two antigenerations in Model 1 and Model 3, while four and one in Model 2. Thus, we obtain three models with three generations. Each model contains an E6 adjoint Higgs field in the untwisted sector.

Model 1 results in the same massless spectrum as theZ6 model in Ref. [10]. The other two, Model 2 and Model 3, are new. Model 3 does not contain any hidden non-Abelian gauge symmetry, which is one of the requirements of the classification in Ref. [10], while Model 1 and Model 2 do. We also find that these models have ðZ3Þ3symmetry which remains unbroken even after all the singlets develop non-vanishing vacuum expectation values and, unfortunately, do not possess the additional symmetries [4,8,9]. Thus, the traditional SUSY-GUT problems, such as the DT splitting problem and the SUSY-flavor/CP problem, are not re-solved in these models.

V. SUMMARY

In this letter, we construct 4D SUSY level-3E₆models. The k ¼ 3 E₆ gauge symmetry is realized from three copies ofk ¼ 1 E₆ symmetry via the diagonal embedding. We utilize the lattice engineering technique, instead of the

compactification of the usual 10D heterotic string models with Wilson lines, to construct Narain lattices containing three copies of theE6lattices. This technique allows us to construct new even self-dual lattices from a known one in a simple way, and thus, makes it easier to access new models. Though here we work only on the same lattice as the one studied in Ref. [10] where the lattice is obtained through Wilson lines, we show that Narain lattices with desired three copies of E6 can be immediately constructed from any lattice containingA₂.

Then, we examine all the possibleZ12actions which are missing in the classification in the literature [10], and we find three models with the minimal requirements. One of them has the same spectrum as the model [10] that has been the only one proposed so far. The other two are new. While one does not have any hidden non-Abelian gauge symme-try, the other does, and thus, should be added into the classification.

The two new models contain neither anSUð2Þ_H family symmetry nor an anomalousUð1Þ_Agauge symmetry which make the E6 models more attractive. Given that we have shown there are E6 models besides the unique one pro-posed so far, it is worthwhile to look for moreE6 models, especially the excellent models with the above additional symmetries. For this purpose, our systematic construction of theE6 models will be useful.

ACKNOWLEDGMENTS

We appreciate Y. Kawamura, K. Hosomichi, H. Kanno, T. Kobayashi, J. C. Lee, S. Mizoguchi, Y. Sugawara for valuable discussions. S. M. would like to thank Yukawa Institute for hospitality. This work was partially supported by the Grant-in-Aid for Nagoya University Global COE Program (G07), by MEXT of Japan [Grant No. 22011004 (N. M.), Grant No. 21740176 (S. M.)] and by JSPS (T. Y.).

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