Abstract
The discovery that the number of physically consistent string vacua is on the order of 10500 has prompted several statistical studies of string phenomenology. Focusing on the Weakly Coupled Free Fermionic String formalism, we present 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 gauge groups (and possibly matter content) are identical. This study is parallel to the extensions of the NAHE set itself and presents statistics related to similar model properties. Statistical coupling between specific gauge groups and spacetime supersymmetry is also examined. Finally, a model with completely mirrored gauge groups is discussed. It is found that the region of the landscape explored generates no physically realistic models due to a lack of three net chiral generations.
1. Introduction
The large number of string vacua [1, 2] has prompted both computational and analytical examinations of the landscape, for example, [3–11]. The Weakly Coupled Free Fermionic Heterotic String (WCFFHS) [12–15] approach to string model construction has produced some of the most phenomenologically realistic string models to date [16–57]. The present study focuses on the systematic extension of an NAHE Variation [50] thereby scanning a region the WCFFHS parameter space yet to be explored. The NAHE Variation is of particular interest because it is conducive to the generation of mirror models. Additionally, it was hoped that this regime would produce models with three net chiral generations which turns out not to be the case. Traditionally, the number of fermion families is linked to the topological structure of the compactification; however, in the language of the WCFFHS this connection is difficult to explore analytically because many such models do not have a well-defined geometric interpretation. This work parallels that presented in [58] regarding NAHE extension.
1.1. The NAHE Variation
While there have been many quasirealistic models constructed from the NAHE basis, other bases can be used to create different classes of realistic and quasirealistic heterotic string models. 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 means that hidden sector content matches the observable sector, making the dark matter and observable matter gauge charges identical. Several scenarios with mirrored dark matter have been presented as viable phenomenological descriptions of the universe [59–62].
The NAHE set does not have a tendency to produce mirrored models because the boundary conditions making up the gauge groups break the mirroring between the elements , , and . We can remedy this by ensuring that the worldsheet fermions and have the same boundary conditions as . In doing so, the NAHE variation basis vectors generate a model with gauge group . The basis vectors making up this set are presented in Table 1 with the resulting particle content of the NAHE variation model presented in Table 2.
The observable sector is generally regarded as being the ; however, contributions to the observable sector may come from the breaking of the . As compared to the NAHE set, the large number of s and non-Abelian singlets is less phenomenologically favorable; however, the quantities of both can be reduced drastically which are shown in the statistics for single-layer extensions.
In Section 2, layer , order () extensions of the NAHE variation are investigated, with a focus on statistics. In Section 3, extensions are similarly examined. In Section 4, the statistics of GUT and of spacetime supersymmetries of both orders are determined. Section 5 offers an example of a near mirrored model, and Section 6 reviews the findings of the prior sections.
2. Layer 1, Order 2 Extensions
There were quasi-unique models out of 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 were removed by hand. Approximately 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 5(a).
The most common gauge group in this data set is , while the most common non-Abelian gauge group is , though less than half of the models contain it. The other pertinent feature of these models is the presence of nonsimply laced gauge groups with high rank. The groups range from rank up to rank . Finally, about one third of the models retain their symmetry. The stability of the is in contrast to the more common breaking of , the observable sector, in NAHE-based models [58]. These models will be revisited later with the 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; see Figure 1(a). The distribution of the number of gauge group factors across the unique models peaks around , suggesting that, roughly, the most common effect of extension is the breaking of only one group factor. In a few models, some of the factors have enhancements, typically the groups. Additional adjoint content distributions are provided in Figure 1(c), with GUT model distributions presented in Table 4, but will not be discussed in detail here.
(a) Number of gauge group factors
(b) Number of spacetime SUSYs
| (c) Number of factors |
| (d) Number of non-Abelian singlets- extension |
| (e) Number of non-Abelian singlets- extension |
Regarding the matter content, the number of ST SUSYs is plotted in Figure 1(b), and the number of non-Abelian singlets is plotted in Figure 1(d). It is clear from the latter that the number of non-Abelian singlets can get quite high. While most models have between and , there can be up to non-Abelian singlets in a model. This implies that many models in this data set cannot be viable candidates for quasirealistic or realistic models.
3. Layer 1, Order 3 Extensions
As was the case with the NAHE extensions, there are more distinct NAHE variation extensions than extensions. Out of models built of them were unique. Based on the order-2 redundancies, the systematic uncertainty for this data set is estimated to be . Their gauge group content is tabulated in Table 5(b).
As was the case with the data set, is the most common gauge group. However, the percentage is significantly lower here, about as opposed to . This suggests that some of the added basis vectors are unifying the five s in the NAHE variation into larger gauge groups. Also of note is the number of models with gauge groups of rank higher than . In the data set, there were only three models of this type, about . In the data set, there were models with this property, about .
While it may seem from Table 5(b) that the order- models are more prone to enhancements, Figure 1(a) makes it clear that is not the case. The distribution of the number of gauge group factors for a model peaks between and factors, as opposed to the peak at factors for the order- models. However, there are several models with enhancements, even some models with as few as distinct gauge group factors in them, something not seen with the order- models. This implies there is a class of order- basis vectors that greatly enhances the gauge group symmetries, while most order- models break them.
The number of gauge groups per model is plotted in Figure 1(c). The distribution of peaks between and . More interestingly, a nontrivial number of models do not have symmetries at all. This implies, when combined with Figure 1(a), that in some models the are enhancing larger (but still small relative to and ) gauge groups. The mechanism producing this effect warrants further study, as it could be used to reduce the number of factors for order-layer combinations that tend to produce too many . The frequency of the GUT groups is presented in Table 4.
The number of ST SUSYs is presented in Figure 1(b). While there are a statistically significant number of enhanced ST SUSYs (expected from models with odd-ordered right movers), the majority of these models has ST SUSY.
The number of non-Abelian singlets is plotted in Figure 1(e). The distribution of non-Abelian singlets indicates that a large number of models do not have any non-Abelian singlets. It is possible that this is related to the number of models with no factors.
4. Models with GUT Groups
As a parallel to the NAHE extension study, the subsets of models containing the GUT groups , , , (Pati-Salam), (Left-Right Symmetric), and (MSSM) are examined (see Figure 2). 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 groups with different matter representations. Each one would be counted individually when examining the number of charged exotics and net chiral generations.
| (a) gauge group factors |
| (b) - factors |
| (c) gauge group factors |
| (d) - factors |
(e) Remaining GUTs-gauge group factors
| (f) Remaining GUTs- factors |
In order to calculate the net number of chiral fermion generations, we utilize the following expressions:
Upon analysis, it is found that the extensions yield and observable sectors with net chiral generations while no models with , , , nor have this property. This is a consequence of the fact that the latter groups only arise from extensions which are not conducive to production of net chiral generations. The distribution of net chiral generations, as well as charged exotic matter, by gauge group is provided in Figures 6 and 7. The distributions of number of non-Abelian singlets, by gauge group, can be found in Figures 8 and 9.
In addition to matter content, the hidden sector gauge content is tabulated for each of the aforementioned gauge groups: Tables 6, 7, and 8. We can see from Table 4 that the NAHE variation extensions favor and over the other groups. This is easily understood as is already present and the breaking to is rather straight forward. However, in order to produce the low-rank groups, either the s must be enhanced or there must be significant breaking of either the or . However, neither of these readily occur with a single layer or at low order.
4.1. ST SUSYs
The distributions of ST SUSYs for the entire data set can be found in Figure 1(b) with a breakdown by gauge group in Figure 3.
| (a) -number of ST SUSYs |
| (b) -number of ST SUSYs |
(c) Number of ST SUSYs
The models all have the same distributions regardless of which GUT is chosen. In these models, the gauge content does not statistically couple with the ST SUSY. For the models, however, some of the GUT groups do appear to have such a coupling. In particular, the occurrence of models ST SUSY is disproportionately high while , Left-Right Symmetric, and MSSM models with ST SUSY have a reduced occurrence. As all of the models containing these GUTs have at least a single , there could be a correlation between the number of s and the number of ST SUSYs. Further investigations of these findings show several statistical couplings for higher ST SUSY models containing certain gauge group factors. The methodology used to analyze these couplings was detailed in [58]. The observed significances are plotted in Figures 4 and 5 for the and NAHE variation extensions, respectively.
| (a) -chiral matter generations |
| (b) -chiral matter generations |
| (c) -charged exotics |
| (d) -charged exotics |
| (a) |
(b) Pati-Salam
(c) Left-Right Symmetric
(d) MSSM
| (a) -number of NA singlets () |
| (b) -number of NA singlets () |
| (c) -number of NA singlets () |
| (d) -number of NA singlets () |
| (a) |
(b) Pati-Salam
(c) Left-Right Symmetric
(d) MSSM
While there are no significant gauge groups in the extensions, several groups are significant with regard to enhanced ST SUSYs in the NAHE extensions. In particular, the three exceptional groups, as well as , , , , and , all have a significant statistical correlation with the average number of ST SUSYs. This is likely due to the additional basis vector adding a gravitino generating sector, which is common with odd-order extensions, and additional roots for the gauge groups. Further analysis will be needed to confirm the cause of this significance. It is also worth noting that one group, , has a negative impact on ST SUSYs. If this trend occurs for more odd-ordered extensions of the NAHE variation, it may affect the viability of realistic flipped- models derived from this variation.
5. Models with Mirroring
The larger sets of matching boundary conditions, seen in Table 1, are expected to lead to models with mirrored gauge groups and matter states. Only one model, generated by Table 9(a), in those discussed thus far exhibits full-gauge mirroring. However, the matter states are not mirrored. The particle content of that model is presented in Table 9(b).
The gauge groups are completely mirrored, and the matter representations are almost mirrored between one another. There is a state charged as a 16 under both groups and one charged as a 128 under one of the groups, but not the other. Thus, the matter is not mirrored. The potential for mirroring is clear from the basis vectors: and are mirrored with . There are also many models in which the observable and some of the hidden matter are mirrored, but include a shadow sector gauge group for which matter representations are not coupled.
These have been presented and discussed in [50].
6. 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 three-generation models were found. These conclusions are summarized in Table 3.
While there were more models with GUT gauge groups in the NAHE variation extensions, 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 distribution of ST SUSYs across the subsets of GUT models was also examined. It was concluded that, as was the case with the NAHE study, 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 factor with a GUT group had fewer models with 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 quasi-mirrored models has not yet been completed, it will be used in future work to examine models with this property.
Acknowledgment
This work was supported by funding from Baylor University.