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Physics Letters B 645 (2007) 88-94

www.elsevier.com/locate/physletb

A mini-landscape of exact MSSM spectra in heterotic orbifolds

Oleg Lebedeva b, Hans Peter Nillesa, Stuart Rabyc, Saúl Ramos-Sáncheza, Michael Ratzd *,

Patrick K.S. Vaudrevangea, Akin Wingerterc

a Physikalisches Institut der Universität Bonn, Nussallee 12, 53115 Bonn, Germany b CERN, Theory Division, CH-1211 Geneva 23, Switzerland c Department of Physics, The Ohio State University, 191 W. Woodruff Ave., Columbus, OH 43210, USA d Physik Department T30, Technische Universität München, James-Franck-Strasse, 85748 Garching, Germany

Received 9 November 2006; accepted 6 December 2006

Available online 15 December 2006

Editor: L. Alvarez-Gaumé

Abstract

We explore a "fertile patch" of the heterotic landscape based on a Zg-II orbifold with S0(10) and E6 local GUT structures. We search for models allowing for the exact MSSM spectrum. Our result is that of order 100 out of a total 3 x 104 inequivalent models satisfy this requirement. © 2006 Elsevier B.V. All rights reserved.

1. Introduction

Although there are only a few consistent 10D string theories, there is a huge number of 4D string compactifications [1,2]. This leads to the picture that string theory has a vast landscape of vacua [3]. The (supersymmetric) standard model (SM) corresponds to one or more possible vacua which a priori might not be better than others. To obtain predictions from string theory one can employ the following strategy: First seek vacua that are consistent with observations and then study their properties. Optimistically, one might hope to identify certain features common to all realistic vacua, which would lead to predictions. Even if this is not the case, one might still be able to assign probabilities to certain features, allowing one to exclude certain patches of the landscape on a statistical basis. However, realistic vacua are very rare. For instance, in the context of orientifolds of Gepner models, the fraction of models with the chiral matter content of the standard model is about 10-14 [4,5]. The probability of getting something close to the MSSM in the context of intersecting D-branes in an orientifold background is 10-9 , even if one allows for chiral exotics. In this study, we show that certain patches of the heterotic landscape are more "fertile" in the sense that the analogous probabilities are at the percent level.

We base our model scan on the heterotic E8 x E8 string [8,9] compactified on an orbifold [10-16]. Our study is motivated by recent work on an orbifold GUT interpretation of heterotic string models [17-19]. We focus on the Z6-II orbifold, which is described in detail in [17,19,20]. The search strategy is based on the concept of "local GUTs" [20-23] which inherits certain features of standard grand unification [24-26]. Local GUTs are specific to certain points in the compact space, while the 4D gauge symmetry is that of the SM. If matter fields are localized at such points, they form a complete GUT representation. This applies, in particular, to a 16-plet of a local S0(10), which comprises one generation of the SM matter plus a right-handed neutrino [26,27],

16 = (3,2)1/6 + (3, 1)—2/3 + (3, 1)1/3 + (1,2)_1/2 + (1,1)1 + (1,1)0, (1)

where representations with respect to SU(3)C x SU(2)L are shown in parentheses and the subscript denotes hypercharge. On the other hand, bulk fields are partially projected out and form incomplete GUT multiplets. This offers an intuitive explanation for the

* Corresponding author. E-mail address: mratz@ph.tum.de (M. Ratz).

0370-2693/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2006.12.012

observed multiplet structure of the SM [20-23]. This framework is consistent with MSSM gauge coupling unification as long as the SM gauge group is embedded in a simple local GUT Glocal 2 SU(5), which leads to the standard hypercharge normalization.

We find that the above search strategy, as opposed to a random scan, is successful and a considerable fraction of the models with S0(10) and E6 local GUT structures pass our criteria. Out of about 3 x 104 inequivalent models which involve 2 Wilson lines, 0(100) are phenomenologically attractive and can serve as an ultraviolet completion of the MSSM.

2. MSSM search strategy: Local GUTs

It is well known that with a suitable choice of Wilson lines it is not difficult to obtain the SM gauge group up to U(1) factors. The real challenge is to get the correct matter spectrum and the GUT hypercharge normalization. To this end, we base our strategy on the concept of local GUTs. An orbifold model is defined by the orbifold twist, the torus lattice and the gauge embedding of the orbifold action, i.e. the gauge shift V and the Wilson lines Wn .We consider only the gauge shifts V which allow for a local S0(10) or E6 structure. That is, V are such that the left-moving momenta p (we use the standard notation, for details see e.g. [18-20]) satisfying

p • V = 0 mod 1, p2 = 2 (2)

are roots of S0(10) or E6 (up to extra group factors). Furthermore, the massless states of the first twisted sector T\ are required to contain 16-plets of S0(10) at the fixed points with S0(10) symmetry or 27-plets of E6 at the fixed points with E6 symmetry.

Since these massless states from T\ are automatically invariant under the orbifold action, they all survive in 4D and appear as complete GUT multiplets. In the case of S0(10), that gives one complete SM generation, while in the case of E6 we have 27 = 16 + 10 + 1 under S0(10). It is thus necessary to decouple all (or part) of the 10-plets from the low energy theory.

The Wilson lines are chosen such that the standard model gauge group is embedded into the local GUT as

Gsm C SU(5) c S0(10) or E6, (3)

such that the hypercharge is that of standard GUTs and thus consistent with gauge coupling unification. The spectrum has certain features of traditional 4D GUTs, e.g. localized matter fields form complete GUT representations, yet there are important differences. In particular, interactions generally break GUT relations since different local GUTs are supported at different fixed points. Also, gauge coupling unification is due to the fact that the 10D (not 4D) theory is described by a single coupling.

0ur model search is carried out in the Z6-II orbifold compactification of the heterotic E8 x E8 string, which is described in detail in [19,20]. In this construction, there are 2 gauge shifts leading to a local S0(10) GUT [28],

VS0(10),1 = (1, 1, 1, 0, 0,0,0, 0) (1, 0, 0,0,0, 0, 0,0),

VS0(10),2 = (1, 1, 1, 0, 0,0,0, 0) (1,1, 0,0,0, 0, 0,0), (4)

and 2 shifts leading to a local E6 GUT,

VE6,1 = (1, 3,1, 0,0,0, 0,0) (0, 0, 0,0, 0, 0,0, 0),

VE6,2 = (3, 3, 3, 0,0,0, 0,0) (1, 1, 0,0, 0, 0,0, 0). (5)

We will focus on these shifts and scan over possible Wilson lines to get the SM gauge group. The Z6-II orbifold allows for up to two Wilson lines of order 2 and for one Wilson line of order 3 (cf. [19,20,29]).

The next question is how to get 3 matter generations. The simplest possibility is to use 3 equivalent fixed points with 16-plets [21] which appear in models with 2 Wilson lines of order 2. If the extra states are vectorlike and can be given large masses, the low energy spectrum will contain 3 matter families. However, this strategy fails since all such models contain chiral exotic states [20]. In the case of E6, it does not work either since one cannot obtain GSM C SU(5) C S0(10) C E6 with 2 Wilson lines of order 2.

The next-to-simplest possibility is to use 2 equivalent fixed points which give rise to 2 matter generations. The third generation would then have to come from other twisted or untwisted sectors. The appearance of the third family can be linked to the SM anomaly cancellation. Indeed, the untwisted sector contains part of a 16-plet. Then the simplest options consistent with the SM anomaly cancellation are that the remaining matter either completes the 16-plet or provides vector-like partners of the untwisted sector. In more complicated cases, additional 16- or 16-plets can appear. The localized 16- and 27-plets are true GUT multiplets, whereas the third or "bulk" generation only has the SM quantum numbers of an additional 16-plet. We find that the above strategy is successful and one often gets net 3 families. The other massless states are often vector-like with respect to the SM gauge group and can be given large masses consistent with string selection rules.

In our MSSM search, we focus on models of this type (although we include all models with 2 Wilson lines in the statistics). These are realized when 1 Wilson line of order 3 and 1 Wilson line of order 2 are present. We require that the spectrum contain 3 matter families plus vector-like states. Furthermore, we discard models in which the SU(5) hypercharge is anomalous. Although a non-anomalous hypercharge could be defined, typically it would not have the GUT normalization and thus would not be consistent with gauge unification.

Table 1

Statistics of Z6-II orbifolds based on the shifts VS0(10),1, VS0(10),2, vEg,1, vE6,2

with two Wilson lines

Criterion

V S0(10),1

VSO(10),2

© inequivalent models with 2 Wilson lines ® SM gauge group c SU(5) c S0(10) (or E6) © 3 net (3, 2)

® non-anomalous U(1)y c SU(5) © spectrum = 3 generations + vector-like

22000 3563 1170 528 128

7 800 1163 492 234 90

27 3 3 3

63 32 22 2

3. Results

Let us now present our results for models with the S0(10) local structure. For each of the S0(10) shifts of Eq. (4), we follow the steps:

© Generate Wilson lines W3 and W2.

© Identify "inequivalent" models.

© Select models with Gsm C SU(5) c S0(10).

© Select models with three net (3,2).

© Select models with non-anomalous U(1)y C SU(5).

© Select models with net 3 SM families + Higgses + vector-like.

Our results are presented in Table 1. The models with the chiral MSSM matter content are listed in [30].

Before continuing further, we make a few comments. In order to obtain the models listed under points ©-©, we generate all possible Wilson lines along the lines of Refs. [31] and [32]. However, due to the rapid growth in computing time, generating all inequivalent models is not possible using these tools. Thus the inequivalent models under point © have been generated by exploiting symmetries of the gauge lattice along the lines discussed in [33]. Two models are considered "equivalent" if they have identical spectra with respect to non-Abelian gauge groups and have the same number of non-Abelian singlets. Thus, models differing only in U(1) charges are treated as equivalent. Further ambiguities arise in certain cases when U(1)y can be defined in different ways. In addition, some models differ only by the localization of states on the different fixed points. We know that these ambiguities occur and it is possible that in some cases Yukawa couplings are affected. Hence our criterion may underestimate the number of truly inequivalent models.

In the E6 case, we consider the SM embedding

Again, models with 2 Wilson lines of order 2 fail and the analysis proceeds similarly to the S0(10) case. These results are also presented in Table 1.1

It is instructive to compare our model scan to others. In certain types of intersecting D-brane models, it was found that the probability of obtaining the SM gauge group and three generations of quarks and leptons, while allowing for chiral exotics, is 10-9 [6,7]. The criterion which comes closest to the requirements imposed in [6,7] is ©. We find that within our sample the corresponding probability is 5%.

In [4,5], orientifolds of Gepner models were scanned for chiral MSSM matter spectra, and it was found that the fraction of such models is 10-14. In our set of models, the corresponding probability, i.e. the fraction of models passing criterion ©, is of order 1%. Note also that, in all of our models, hypercharge is normalized as in standard GUTs and thus consistent with gauge coupling unification.

This comparison shows that our sample of heterotic orbifolds is unusually "fertile" compared to other constructions. The probability of finding something close to the MSSM is much higher than that in other patches of the landscape analyzed so far. It would be interesting to extend these results to other regions of the landscape where promising models exist [35-38] (see also [39]).

4. Towards realistic string models

The next step on the path towards realistic models is the decoupling of the vector-like extra matter (x;}. The mass terms for such states are provided by the superpotential

W = x;Xj (sasb •••}, (7)

1 In the analysis of [34] looking at non-supersymmetric heterotic string vacua, about 10% of the models scanned contained the SM gauge group. Our result (step

Gsm C SU(5) c S0(10) c E6

3) is comparable.

Table 2

A subset of the MSSM candidates

Criterion

V SQ(10),1

V S0(10),2

V E6-1

V E6,2

® heavy top

® exotics decouple at order 8

where sa,sb,... are SM singlets. Some singlets are required to get large (close to Mstr) VEVs in order to cancel the Fayet-Iliopoulos (FI) term of an anomalous U(1). The supersymmetric field configurations are quite complicated and generally there are vacua in which all or most of the SM singlets get large VEVs. This breaks many of the gauge group factors, such that the low energy gauge group can be GSM up to a hidden sector,

where the SM matter is neutral under Ghidden. Furthermore, if the relevant Yukawa couplings are allowed by string selection rules, this makes the vector-like matter heavy; thus it decouples from the low energy theory. We note that there are in general several pairs of Higgs doublets with a matrix of ^.-like mass terms, for which we require only one small eigenvalue.2

Clearly, one cannot switch on the singlet VEVs at will. Instead one has to ensure that they are consistent with supersymmetry. Supersymmetry requires vanishing of the F- and O-terms. The number of the F-term equations equals the number of complex fields sa, therefore there are in general non-trivial singlet configurations with vanishing F-potential. The O-terms can be made zero by complexified gauge transformations [42] if each field enters a gauge invariant monomial [43]. Thus, to ensure that the decoupling of exotics is consistent with supersymmetry, one has to show that all SM singlets appearing in the mass matrices for the exotics enter gauge invariant monomials involving only SM singlets and carrying anomalous charge. In this Letter, we assume that the relevant singlets develop large supersymmetric VEVs.

In the process of decoupling, the vector-like states can mix with the localized 16- and 27-plets (if it is allowed by the SM quantum numbers) such that the physical states at low energies are neither localized nor "true" GUT multiplets. Nevertheless, it is clear that whatever the mixing, in the end exactly 3 SM families will be left, if the mass matrices have maximal rank.

To show that the decoupling of exotics is consistent with string selection rules is a technically involved and time consuming issue. In order to simplify the task and to reduce the number of models, we first impose an additional condition. We require that the models possess a renormalizable top-Yukawa coupling as motivated by phenomenology. Then we consider only superpotential couplings up to order 8. Thus, the next two steps in our selection procedure are:

® Select models with a heavy top.

® Select models in which the exotics decouple at order 8.

First, we require a renormalizable O(1) Yukawa coupling (3,2)1/6(3, 1)_2/3(1,2)1/2, i.e. one of the following types

UUU, UTT, TTT, (9)

where U and T denote generic untwisted and twisted fields, respectively. The UUU coupling is given by the gauge coupling, UTT is a local coupling and thus is unsuppressed, while the TTT coupling is significant only when the twisted fields are localized at the same fixed point. We discard models in which the above couplings are absent or suppressed.

In the next step ®, we select models in which the mass matrices for the exotics (cf. Eq. (7)) have a maximal rank such that no exotic states appear at low energies. Here, we consider only superpotential couplings up to order 8 and for this analysis we assume that all relevant singlets can obtain supersymmetric vevs.3 We find that a significant fraction of our models passes requirements ® and ® (see Table 2 and for further details [30]). In particular, we identify 93 models that can serve as an ultraviolet completion of the MSSM in string theory.

To verify whether an MSSM candidate is consistent with phenomenology requires addressing several questions. The most important issues include

• realistic flavour structures,

• absence of fast proton decay,

• hierarchically small supersymmetry breaking.

2 To get a pair of massless Higgs doublets usually requires fine-tuning in the VEVs of the SM singlets such that the mass matrix for the (1, 2)_1/2, (1, 2)1/2 states gets a zero eigenvalue. This is the notorious supersymmetric ^-problem. The fine-tuning can be ameliorated if the vacuum respects certain (approximate) symmetries [40,41].

3 We also address to some extent the question of O-flatness. In many of the models, we find that all SM singlets enter gauge invariant monomials. A full analysis of this issue is deferred to a subsequent publication.

Gsm X Ghidden,

Table 3

Massless spectrum. The quantum numbers are shown with respect to SU(3)c x SU(2)l x S0(8) x SU(2), the hypercharge is given by the subscript

# Irrep Label # Anti-irrep Label # Irrep Label

3 (3' 2 1'1)1/6 qt 4 (1' 2; 1' 1)0 mt

8 (1' 2 1' 1)-1/2 ft 5 (1' 2; 1' 1)1/2 ft 2 (1' 2; 1' 2)0 m' t

3 (1' 1 1' 1)1 et 47 (1' 1; 1' 1)0 st

3 (3' 1 1' 1) -2/3 ñt 26 (1' 1; 1' 2)0 hi

7 (3' 1 1'1)1/3 di 4 (3' 1; 1' 1)-1/3 di 9 (1' 1; 8' 1)0 wt

4 (3' 1 1'1)1/6 vt 4 (3' 1; 1' 1)-1/6 vi

20 (1' 1 1' 1)1/2 s+ 20 (1' 1; 1' 1)-1/2 s. t

2 (1' 1 1' 2)1/2 s+ I 2 (1' 1; 1' 2)-1/2 s-t

Table 4

Renormalizable interactions involving the SM fields

Coupling qtfjñk ñt dj dk qtfjdk etfjfk

# 1 0 4 4

A model that passes all of our criteria ®~® and comes very close to the supersymmetric standard model has been presented in [20,22]. In our scan, we obtain many comparable models. In what follows, we substantiate this statement by studying a specific example, leaving a complete survey for future work.

Example

The model is based on the gauge shift

VS0(10), 1 = (3, -2, -2, 0 , 0 , 0 , 0 , 0) (1, -1, -1, -2, -1, -2, -1,2), (10)

where we have added an E8 x E8 lattice vector to simplify computations. The Wilson lines are chosen as

W2- (1 -1 -1 -1 -111 1) Í1 -1 - 5 - 3 -1 - 5 _ 3 3 )

W2 — W' 4' 4' 4' 4' 4' 4' 4/ V1' 2' 2' 2' 2' 2'2/'

W3 — ( -2' -1' 1' 6' 6' 1' 6' 6) (10' 0' - 6 ' - 3' -3' - 5 ' - 3 ' 3). (11)

The standard SU(5) hypercharge generator is given by

t7 = (0 ,0 , 0 , -2, -2,1,1,|) (0 ,0 , 0 ,0 ,0 ,0 , 0 ,0). (12) The gauge group after compactification is

Gsm x S0(8) x SU(2) x U(1)7, (13)

while the massless spectrum is given in Table 3.

Renormalizable Yukawa couplings involving the SM fields are shown in Table 4. The top Yukawa coupling comes from the UUU interaction qiljuk, which allows us to identify the right-handed top, the up-type Higgs doublet and the quark doublet of the third generation. (Here we denote the leptons and Higgses collectively by l;, l;.) 0ther renormalizable interactions q;ljdk and eiljlk can produce the down-type quark and lepton masses as well as lepton number violating interactions. What happens precisely depends on the form of the matrix of ^-like mass terms for the vector-like states and, thus, on the vacuum configuration. We note that, due to the absence of the U idjdk operator, the proton is stable at this level.

The model has three generations of SM matter plus vector-like exotics. 0nce the SM singlets s; get VEVs, the gauge group reduces to

GSM x Ghidden (14)

where Ghidden = S0(8) x SU(2). At the same time, the vector-like states get large masses. We have checked that the rank of all the mass matrices is maximal, such that the exotics do decouple (assuming all singlets acquire supersymmetric vevs). Below we present most of them. An entry sn indicates that the coupling appears first when n singlets are involved. Each entry usually contains

many terms and involves different singlets as well as coupling strengths

s6 s6 s3 s6 s6

s6 s6 s3 s6 s6

s3 0 0 s3 0

s6 s3 0 s6 s3

A „1

J^m.'m.' —

Mvv —

s5\ s5

s3 s1 s1 s1 s1

s1 s3 s3 s3 s3

s1 s3 s3 s3 s3

s1 s3 s3 s3 s3

s1 s3 s3 s3 s3 ,

s1 s3 s6 s6 s3

s4 s2 s6 s2 s2

Vs4 s2 s6 s2 s2

0 s5 s6 s6

^^mm — s5 s6 0 s6 s6 0 s6 s5

s6 s6 s5 0

Similarly, the mass matrices for s± and s± have a maximal rank. The dd mass matrix is 4 x 7 such that there are 3 massless d states. The 11 mass matrix is 8 x 5, so there are 3 lepton doublets. By choosing a special vacuum configuration one can reduce the rank of the 11 mass matrix to 4 such that there is a pair of massless Higgs doublets. (This is just the supersymmetric "^.-problem"). Thus we end up with the exact MSSM spectrum.

We have checked that the required vacuum configuration is D-flat. That is, one can assign large VEVs to the singlets without inducing the D-terms. Since the number of the F-term equations equals the number of the field variables, there are generally non-trivial solutions to F = 0. Then, using complexified gauge transformations, one can make the F- and D-terms vanish simultaneously. Such supersymmetric vacua would correspond to isolated solutions in field space. Although we expect such solutions to exist, their explicit form remains undetermined and will be studied elsewhere.

Finally, the model allows us to define a suitable B-L generator which leads to the standard charges for the SM matter,

tB-L — (1, 1, 0, 0, 0, -2, -2, -2) (2x - 1, 2, 0,x,x, 0, 0, 0)

with arbitrary x. An interesting feature is that the spectrum contains a pair of fields which have B-L charges ±2. If B-L gauge symmetry is broken by VEVs of these fields, the matter parity (or family reflection symmetry [44,45]) (—1)3(B-L) is conserved and proton decay is suppressed.

5. Conclusion

We have analyzed the heterotic E8 x E8 string compactified on a Z6-II orbifold, allowing for up to two discrete Wilson lines. Employing a search strategy based on the concept of local GUTs, we have obtained about 3 x 104 inequivalent models. Almost 1% of these models have the gauge group and the chiral matter content of the MSSM. This result shows that orbifold compactifications of the heterotic string considered here correspond to a particularly fertile region in the landscape and the probability of getting something close to the MSSM is significantly higher than that in other constructions.

The most important outcome of our scan is the construction of 0(100) models consistent with the MSSM gauge group and matter content, amended by a hidden sector. A detailed phenomenological analysis of these models is in progress.

Acknowledgements

We would like to thank A.N. Schellekens and W. Taylor for correspondence. This work was partially supported by the European Union 6th Framework Program MRTN-CT-2004-503369 "Quest for Unification" and MRTN-CT-2004-005104 "ForcesUniverse". S. Raby and A. Wingerter received partial support from DOE grant D0E/ER/01545-870, and would like to thank the KITP for their hospitality during the completion of this work. This research was supported in part by the National Science Foundation under Grant No. PHY99-07949.

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