Initial Systematic Investigations of the Landscape of Low Layer NAHE Variation Extensions

The discovery that the number of physically consistent string vacua is on the order of 10^500 has prompted several statistical studies of string phenomenology. Contained here is one such study that focuses on the Weakly Coupled Free Fermionic String (WCFFHS) formalism. Presented here are systematic extensions of a variation on the NAHE (Nanopoulos, Antoniadis, Hagelin, Ellis) set of basis vectors. This variation is more conducive to the production of"mirrored"models, in which the observable and hidden sector particles are identical. This study is a parallel to the extensions of the NAHE set itself (referenced herein), and presents statistics related to similar model properties. Statistical coupling between specific gauge groups and ST SUSYs is also examined. Finally, a model with completely mirrored gauge groups is discussed.


I. INTRODUCTION
Physically consistent string/M-theory derived models number at least approximately 10 500 [1,2]. The large number of vacua has prompted both computational and analytical examinations of the the "landscape" of possible string vacua, e.g. [3][4][5][6][7][8][9][10][11]. The Weakly Coupled Free Fermionic Heterotic String (WCFFHS) [12][13][14][15] approach to string model construction has produced some of the most phenomenologically realistic string models to date . The present study focuses on systematic extensions of a NAHE Variation, first presented in [50]. It is a parallel study to the work presented in [58]. Background on WCFFHS model building as well as systematic searches is presented there.

A. The NAHE Variation
While there have been many quasi-realistic models constructed from the NAHE basis, other bases can be used to create different classes of realistic and quasi-realistic heterotic string models. In this paper, one such basis will be discussed, called the NAHE variation [50]. Like the NAHE set, the NAHE variation is a collection of five order-2 basis vectors.
However, the sets of matching boundary conditions are larger than those of the NAHE set.
This allows for a new class of models with "mirrored" groups -that is, with gauge groups that occur in even factors. Some also have mirrored matter representations that do not interact with one another. This mirroring means that the hidden sector content matches the observable sector content, making the dark matter identical to the observable matter in terms of gauge charges. Several scenarios with mirrored dark matter have been presented as viable phenomenological descriptions of the universe [59][60][61][62].
The NAHE set does not have a tendency to produce mirrored models because the bound- providing for the possibility of mirrored gauge groups and matter representations.
The particle content of the NAHE variation model is presented in Table II. The observable sector is generally regarded as being the E 6 , as it is a GUT group. However, additional observable sectors may come out of the SO(22) as well. The large number of U (1) groups and non-Abelian singlets (when compared to the NAHE set) is less phenomenologically favorable. However, the quantities of both can reduce drastically, as will be seen shortly when the statistics for single layer extensions are presented.
In section II, layer-1, order 2 extensions of the NAHE variation are investigated, with a focus on statistics. In section III, layer-1, order-3 extensions are similarly examined. In section IV, the statistics of GUT and of ST SUSY of both orders are determined. Section V offers examples of mirror models. Section VI reviews the findings of the prior sections.

II. ORDER 2, LAYER 1 EXTENSIONS
There were 309 quasi-unique models out of 1,315,328 total consistent models built given the input parameters. A redundancy related to the rotation of the gauge groups, discussed in detail in [58], is also present. Duplicate models within the set of 309 were removed by hand. Approximately 2% of the models in the data set without rank cuts were duplicates, while none of the models with rank cuts had duplicates. The gauge group content of those models is presented in Table III. The most common gauge group in this data set is U (1), while the most common non-Abelian gauge group is SU (2). However, less than half of the models in the data set contain SU (2). The other pertinent feature of these gauge groups is the presence of non-simply laced gauge groups with high rank. The SO(2n+1) groups range from rank 2 up to rank 10. This is a feature unique to the NAHE variation extensions (at least within this study). Finally, about one third of the models retain their E 6 symmetry.
The stability of E 6 is in contrast to the more common breaking of SO(10) in NAHE-based models [58]. These models will be revisited later with the E 6 treated as an observable sector gauge group, and the number of chiral matter generations they have will be statistically examined.
Also of interest regarding the gauge group content of this data set is the number of gauge group factors present in each model. Those are plotted in Figure 1. The distribution of the number of gauge group factors across the unique models in this data set have a peak around 8. This is close to the "initial" value (from the NAHE variation alone) of 7. There are a few models in which some of the factors have enhancements. These are likely the result of enhancements to the U (1) groups in most cases.
Also related to the gauge group content is the distribution of the number of U (1)'s in this data set. That is plotted in Figure 2. The distribution in Figure 2 has a clear peak at 5, implying that many of these order-2 extensions do not alter the number of U (1)'s.
The frequencies of the GUT groups in this data set are tabulated in Table IV. Regarding  the matter content, the number of ST SUSYs is plotted in Figure 3, and the number of non-Abelian singlets is plotted in Figure 4. It is clear from the latter that the number of non-Abelian singlets can get quite high. While most models have between 50 and 80, there can be up to 250 non-Abelian singlets in a model. This implies that many models in this data set cannot be viable candidates for quasi-realistic or realistic models.

III. ORDER 3, LAYER 1 EXTENSIONS
As was the case with the NAHE set order-3 extensions, there are more distinct models in the NAHE variation + O3L1 data set. Out of 442,272 models built 1,166 of them were unique. Based on the order-2 redundancies, the systematic uncertainty for this data set is estimated to be 2%. Their gauge group content is tabulated in Table V. As was the case with the O2L1 data set, U (1) is the most common gauge group. However, the percentage is significantly lower here, about 86% as opposed to 98%. This suggests that some of the added basis vectors are unifying the five U (1)'s in the NAHE variation into larger gauge groups. Also of note is the number of models with gauge groups of higher rank than 11.
In the O2L1 data set, there were only three models of this type, about 1%. In the O3L1 data set, there were 28 models with this property, about 2.4%. While it may seem from Tables III and V that the order-3 models are more prone to enhancements, Figure 5 makes it clear that is not the case. The distribution of the number of gauge group factors for a model peaks around 9-11 factors, as opposed to the peak around 8 factors for the order-2 models. However, there are several models with enhancements, even some models with as few as 2 distinct gauge group factors in them, something not seen with the order-2 models.
This implies there is a class of order-3 basis vectors that greatly enhances the gauge group symmetries, while most order-3 models break them.
The number of U (1) gauge groups per model is plotted in Figure 6. The distribution of U (1) peaks between 5 and 7. More interestingly, a nontrivial number of models do not have U (1) symmetries at all. This implies, when combined with Figure 5, that in some models the U (1) are enhancing to larger (but still small relative to SO(22) and E 6 ) gauge groups. The mechanism producing this effect warrants further study, as it could be used to reduce the number of U (1) factors for order-layer combinations that tend to produce too many U (1)'s. The frequency of the GUT groups is presented in Table VI. The number of ST SUSYs is presented in Figure 7. While there are a statistically significant number of

IV. MODELS WITH GUT GROUPS
As a parallel to the NAHE set extension study, the subsets of models containing the GUT groups E 6 , SO(10), SU Like the NAHE study, the usual statistics will be reported along with the number of net chiral generations for models containing the GUT groups in question. If there is more than one way to configure an observable sector, each configuration will be counted when tallying the charged exotics and net chiral generations. For example, a model may have two E 6 groups with different matter representations. Each one would be counted individually when examining the number of charged exotics and net chiral generations.

A. E 6
The equation for calculating the number of net chiral generations is given in equation 1.
There are 101 unique models containing E 6 in the O2L1 data set, while the O3L1 data set has only 68 unique models. The hidden sector gauge group content for the O2L1 models is tabulated in Table VII. The hidden sector gauge groups for the O3L1 models is presented in   Figure 10 for the O2L1 data set and Figure 11 for the O3L1 data set.     The number of net chiral fermion generations for the SO(10) GUT group is given by equation (2).
There were 125 models with SO(10) in the O2L1 data set and 271 models with SO(10) in the O3L1 data set. The hidden sector gauge content is presented in Table IX for the O2L1 models and Table X for the O3L1 models. Note that in both data sets all of the models with SO (10) also come with a U (1) gauge group, implying the mechanism for reducing SO(22) (or more likely E 6 ) is not unifying the U (1)'s into a larger group. The number of net chiral fermion generations in the O2L1 data set with and without hidden sector duplicates is presented in Figure 12. No models in the O3L1 data set have any net chiral matter generations, but both data sets contained observable sector charged exotics. Those are also plotted in Figure   12. Since most models in the O2L1 data set and all of the models in the O3L1 data set have zero net chiral matter generations, it is clear that more complicated basis vector sets -either higher order or more layers -will be needed to produce quasi-realistic SO (10) GUT models. It is also clear that the number of observable sector charged exotics is peaked at zero. This would be advantageous if more models had a net number of chiral fermion generations. It is likely that the models without observable sector charged exotics do not have any net chiral matter generations, thus limiting the phenomenological advantages of having few exotics. The remaining statistics for the SO(10) models are presented in Figure   13 for the O2L1 models and Figure 14 for the O3L1 models.      The number of net chiral matter generations in a flipped SU (5) model is presented in equation (3).
As was the case with the NAHE set, there are not NAHE variation based O2L1 extensions producing an SU (5)⊗U (1) gauge group. The O3L1 extensions, however, produce 165 models containing SU (5) ⊗ U (1). The hidden sector gauge content of those models is presented in Table XI. Most of the models have the SU (5) GUT group accompanied by another SU (n+1) gauge group, a result of the models being built from an extension with an odd ordered RM.
The ranks of these SU (n + 1) groups can get quite large. This is likely due to the additional The number of observable sector charged exotics is presented in Figure 15. Other statistics related to the SU (5) ⊗ U (1) models in the NAHE variation + O3L1 data set are presented in Figure 16.
As was the case with the NAHE investigations, the generations of matter and anti-matter follow the same statistics, since all of the possible permutations of observable sectors are counted. There are no models in the O2L1 data set containing the Pati-Salam gauge group, but the O3L1 data set had 125 such models. The hidden sector gauge content of those models is presented in Table XII. The hidden sector gauge group appearing in the most models is U (1), which is expected. Additional U (1) groups are common with basis vectors of fractional phases, such as the ones in this data set. All of the models in this data set have zero net chiral matter generations, suggesting more complex basis vectors should be used to construct quasi-realistic Pati-Salam models originating with this NAHE variation.
The number of observable sector charged exotics is presented in Figure 17, while remaining statistics for these models is presented in Figure 18.

E. Left-Right Symmetric
The Left-Right Symmetric GUT group is a derivative of the Pati-Salam GUT, reducing the SU (4) that governed lepton and quark generations to an SU (3), which directly represents the QCD color force. As was the case with the NAHE extensions, only the quark generations will be statistically examined in this study. The number of net chiral quark generations is given by equation (5).
There are no models containing this gauge groups in the O2L1 data set (none contained SU (3)), but there are 61 models in the O3L1 data set with this GUT group. The hidden sector gauge group content is presented in Table XIII. Note that all of the models in this subset have U (1) factors. This means each of them also contains an MSSM gauge group as well. Statistics on MSSM models will be presented in the next section.
As was the case with the Pati-Salam models, all of the models in this data set have zero net quark generations. The number of observable sector charged exotics is presented in

F. MSSM-like Models
The MSSM 1 gauge group is SU (3) ⊗ SU (2) ⊗ U (1). As with the NAHE investigation, only the quark generations will be statistically examined. Thus, the term chiral matter generation here refers only to quark generations. The equation for the number of net chiral matter generations is given by equation (6), while the number of net chiral anti-generations is given by equation (7).
Due a lack of SU (3) groups, no models in the NAHE variation + O2L1 data set contain the MSSM gauge groups. The O3L1 data set has 63 models with the MSSM group. The hidden sector gauge content of these models is presented in Table XIV. A significant number of these models contain higher rank SU (n + 1) gauge groups, while not many contain higher ranking  Figure 21, while other statistics related to these models are presented in Figure 22.

G. ST SUSYs
The ST SUSY distributions across the GUT group subsets was examined in the NAHE set investigations. It will be examined here as well. The distributions of ST SUSYs for the full NAHE variation + O2L1 data set, the O2L1 E 6 models, and the O2L1 SO(10) models are plotted in Figure 23. The distributions of ST SUSYs for the full NAHE variation + O3L1 data set, the E 6 models, the SO(10) models, the SU (5) ⊗ U (1) models, and the Pati-Salam models in the O3L1 data set are plotted in Figure 24. The distributions of ST SUSYs for the full NAHE variation + O3L1 data set, the Left-Right Symmetric models, and the MSSM models in the O3L1 data set are plotted in Figure 25. The O2L1 models all have the same distributions regardless of which GUT is chosen. In these models, the gauge content   Further investigations of these findings show several statistical couplings for higher ST SUSY models containing certain gauge group factors. The statistical test to be used invokes the Central Limit Theorem, which is applicable to populations which are well behaved, such as the number of ST SUSY distributions discussed. We will average the number of ST SUSYs per model for models containing each of the gauge groups present. Random sample averages will be close to the average of the population; therefore any sample drawn based on gauge groups which has an average close to the population average indicates that the average number of ST SUSYs is not coupled to the gauge group. If the average number of ST SUSYs per model for a particular gauge group (for example, E 6 ) is higher than the population (and if the sample is large enough to be more significant), then we can conclude that the gauge group content has an effect on the number of ST SUSYs for a significant percentage of models. These significances are plotted in Figure 26 for the NAHE Variation + O2L1 data set, and Figure 27 for the NAHE Variation + O3L1 data set.
While there are no significant gauge groups in the NAHE Variation + O2L1 data set, several groups are significant with regard to enhanced ST SUSYs in the NAHE Variation + O3L1 data set. In particular, the three exceptional groups, as well as SO (12)

V. MODELS WITH MIRRORING
The larger sets of matching boundary conditions, seen in Table I, are expected to lead to models with mirrored gauge groups and matter states. Only one model in those discussed thus far exhibits full gauge mirroring, and the matter states are not mirrored. The particle content of that model is presented in Table XV.
The gauge groups are completely mirrored, and the matter representations are almost mirrored between one another. There is a state charged as a 16-dimensional representation under both SO(16) groups, however, so the matter mirroring is not complete. Additionally, there is a 128 dimensional representation under one of the SO(16) groups, but not the other.
The basis vectors for the model are presented in Table XVI. The mirroring is clear from the basis vectors:ψ 1,,5 andη 1,2,3 are mirrored withφ 1,..., 8 . There are also many models in which the observable matter is mirrored by hidden matter, but in addition there is a shadow sector gauge group whose matter representations are not coupled. These have been presented and discussed in [50].

VI. CONCLUSIONS
Though there were many models containing GUTs in the data sets explored in this study, a vast majority of them do not contain any net chiral fermion generations. No threegeneration models were found. These conculsions are summarized in Table XVII. While there were more models with GUT gauge groups in the NAHE variation + O3L1 data set, none of them had any net chiral matter generations, implying that the added basis vector produces the barred and unbarred generations in even pairs, if at all. More complicated basis vector sets will need to be studied to determine if any NAHE variation based quasi-realistic models can be constructed.
The distributions of ST SUSYs across the subsets of GUT models was also examined. It was concluded that, as was the case with the NAHE study, E 6 has a statistical coupling to enhanced ST SUSYs for order-3 models. Additionally, data sets in which all of the models contained at least one U (1) factor with a GUT group had fewer models with N = 1 ST

SUSY.
Models with partial gauge group mirroring were also discussed, with a model presented that has complete gauge group mirroring. While a statistical search algorithm for finding