A Z 2 × Z 2 standard model

We present a Z 2 × Z 2 orbifold compactiﬁcation of the E 8 × E 8 heterotic string which gives rise to the exact chiral MSSM spectrum. The GUT breaking SU ( 5 ) → SU ( 3 ) C × SU ( 2 ) L × U ( 1 ) Y is realized by modding out a freely acting symmetry. This ensures precision gauge coupling uniﬁcation. Further, it allows us to break the GUT group without switching on ﬂux in hypercharge direction, such that the standard model gauge bosons can remain massless when the orbifold singularities are blown up. The model has vacuum conﬁgurations with matter parity, a large top Yukawa coupling and other phenomenologically appealing features. of the 1 models They moduli in the an effective ﬁeld theory with physical like and couplings. interacting CFT, the limit, of the full string theory, limit,


Introduction
Heterotic model building [1][2][3][4][5] has received renewed increased attention over the past few years. Almost simultaneously, two constructions of models have been found that give rise to the exact (chiral) spectrum of the minimal supersymmetric extension of the standard model (SM), the MSSM. One of them is based on heterotic Z 6 -II orbifolds [6] and the other on smooth Calabi-Yau (CY) compactifications [7,8]. (See e.g. [9] for a review of recent progress in getting the MSSM from string theory.) Let us start by recalling some important properties of orbifold and CY compactifications. Orbifolds are exact string compactifications in which one directly goes from string theory in ten dimensions to an effective, four-dimensional (4D) field theory. This ensures that one has an ultra-violet complete framework in which couplings are accurately computable. The mini-landscape of Z 6 -II orbifolds [10][11][12] provides a large class of phenomenological appealing models of this kind: for example, the MSSM matter spectrum is reproduced, vector-like exotics can be decoupled and realistic features like a large top-Yukawa coupling, hierarchical couplings and non-trivial flavor mixing emerge.
Orbifolds correspond to very special points in the string landscape, where the worldsheet theory reduces to a combination of free conformal field theories (CFTs). Generically, such orbifold models contain unwanted gauge group factors and massless (vector-like) exotic states, which are charged but not part of the standard model. In order to obtain phenomenologically attractive vacua, non-trivial vacuum expectation values (VEVs) need to be switched on. Furthermore, often non-vanishing VEVs cannot be avoided due to the presence of a Fayet-Iliopoulos (FI) D-term for an anomalous U (1). This means that the orbifold point almost never constitutes a true and final vacuum configuration. Yet in all known examples "nearby vacua" can be found in which some fields attain VEVs such as to cancel the FI term (see [13]).
Like orbifolds, generic compactifications of the heterotic string that preserve N = 1 supersymmetry can give rise to models with chiral spectra. They possess a large moduli space, which may have special points like the large volume limit, the orbifold and conifold points, etc. Each point in the moduli space leads to an effective field theory with certain physical predictions, like masses and couplings. However, as the corresponding worldsheet theory involves a complicated interacting CFT, most commonly only the supergravity limit, i.e. the lowest order in α approximation of the full string theory, is considered. In this limit, generic compactifications have a clear geometrical interpretation in terms of CY manifolds, and the computation of the chiral spectra based on index theorems is well under control. On the other hand, the validity of the supergravity description requires moderately large radii, which is sometimes problematic as this easily leads to too small gauge couplings. Moreover, since the underlying CYs are complicated spaces, the calculation of couplings, needed to make detailed predictions for phenomenology, is still far from straightforward.
As is well known, orbifolds and CYs are not unrelated; rather, in many cases orbifold singularities can be resolved, thus reproducing compactifications based on smooth manifolds (see e.g. [14][15][16][17][18][19]). The transition from an orbifold to a smooth compactification is achieved by giving VEVs to twisted states, stringy degrees of freedom residing at the orbifold singularities. The reverse of the "blow-up" process, in which a compact hyper surface (i.e. exceptional divisor) shrinks down to zero size, is commonly referred to as a "blow-down". 1 A setting in which one has both an exact orbifold CFT picture as well as a smooth CY description, would be quite powerful, because one can combine the calculability of the orbifold with the generic features of CY compactifications.
So far no phenomenologically appealing model has been obtained that allows for an orbifold as well as a CY description. It is unknown whether a complete blow-down of the potentially realistic smooth compactifications, obtained so far [7,8,21], to an exact (free orbifold) CFT description exists. On the other hand, many phenomenologically attractive orbifold models [10][11][12] cannot be completely blown up without destroying the phenomenological viability of these settings, as the hypercharge or another part of the standard model gauge group gets broken in the complete blow-up [19].
Our aim is to describe in detail how to construct (phenomenologically attractive) orbifolds which allow for complete blow-ups without breaking the standard model gauge group We base our discussion on an explicit model that can be seen as a specific realization of a proposal made by Witten [22] in that the GUT breaking SU(5) → G SM is achieved by dividing out a freely acting symmetry. (The idea of associating a Wilson line with an involution of the underlying CY manifold in concrete model building has been employed already for some time [23].) This ensures that there is no flux in hypercharge direction, such that U(1) Y remains unbroken in the smooth limit. In addition, such settings allow us to avoid GUT scale threshold corrections to the gauge couplings and therefore fit particularly well to the paradigm of MSSM precision gauge coupling unification [24,25]. This Letter is organized as follows: Section 2 is devoted to the construction of a concrete orbifold model that allows for a freely acting symmetry. The next section demonstrates that by switching on appropriate VEVs quasi-realistic vacuum configurations can be obtained from our construction. Section 4 contains a tentative discussion of how to relate the model to a smooth compactification in which all orbifold singularities have been resolved. In Section 5 we present our conclusions and an outlook. Finally, Appendices A, B contain the full spectrum of our model and the string selection rules for allowed couplings.

A Z Z Z 2 × Z Z Z 2 orbifold model with a Z Z Z 2 involution
We construct our model in two steps described in the following two subsections. We start with a Z 2 × Z 2 orbifold based on the product of three two-tori (for a detailed description of such orbifolds see [26] and [27] for the free fermionic formulation) that leads to an SU(5) GUT with six generations and vector-like exotics.
In the second step we mod out a freely acting symmetry of order two. The resulting geometry was first discussed in [28] and corresponds to model  in the classification by Donagi and Wendland [29]. Since this Z 2 acts freely, the 48 fixed tori of the Z 2 × Z 2 orbifold are mapped to each other pairwise resulting in 24 fixed tori. Hence, the number of chiral generations is reduced to three. The final crucial ingredient of our model is a non-standard gauge embedding that accompanies this involution and breaks SU(5) to G SM . In the final two subsections we discuss some general phenomenological properties of this model, like the massless spectrum and gauge coupling unification.

Underlying
The orbifold model is defined by a torus lattice that is spanned by six orthogonal vectors e α , α = 1, . . . , 6, the Z 2 × Z 2 twist vectors v 1 = (0, 1/2, −1/2) and v 2 = (−1/2, 0, 1/2), the associated shifts and six discrete Wilson lines  1 Since the VEV and the corresponding volume are schematically related by VEV ∼ exp(volume), the naive definition of the volume has to go to −∞ in order to arrive at the orbifold point [19,20]. Hence, the blow-down in the supergravity sense does not describe the orbifold point.
x 6 (1, 1; 1, 2, 1) (0,0) z corresponding to the six torus directions e α . These shifts and Wilson lines satisfy the modular invariance conditions and, furthermore, fulfill the consistency requirements of Ref. [30]. Eq. (3) is obtained by noticing that the theta-functions inside the corresponding partition function are periodic under the change of the modular parameter (ρ → ρ + 2 for an order two element) up to some phase factors that needs to be cancelled.
The states in the spectrum originate from different sectors: the untwisted sectors U i (with i = 1, 2, 3 corresponding to the ith plane, spanned by e 2i−1 and e 2i ) and the twisted sectors T (k, ) (corresponding to the orbifold twist kv 1 + v 2 ). In total, the spectrum contains 6 × 10 + 15 × 5 + 9 × 5 of SU (5), 52 non-Abelian singlets and some representations with respect to a hidden sector gauge group SU(4) 2 . Three of the nine vector-like pairs of 5/5-plets are part of the untwisted sectors U i , i = 1, 2, 3, originating from the 10D bulk; the remainder of the SU(5)-charged spectrum resides in the various twisted sectors. In particular, the six generations of SU(5) are all twisted states.

Modding out a freely acting Z 2 involution
Next, we divide out the Z 2 symmetry corresponding to with a gauge embedding denoted by W . Since τ acts freely, i.e. it does not produce fixed points, we refer to W as freely acting Wilson line. This is a slight abuse of terminology, since (field-theoretic) Wilson lines are always non-local. However, in the context of orbifold model building discrete Wilson lines usually denote the differences between local shifts, i.e. they are Wilson lines on the underlying torus but not on the orbifold (see e.g. [10,31]). By contrast, W is a Wilson line also on the orbifold.
The strict identification of W 2 , W 4 and W 6 in Eq. (2e) allows us to mod out τ . Further, from its definition (4) it follows that W is an element of order four, as W 2 is of order two.
Modular invariance of the resulting partition function for this order four element amends the conditions (3) by In particular, we have chosen the Wilson line W 2 in equation (2b) such that W satisfies all the conditions (6) and breaks the SU(5) GUT group down to G SM . By contrast, the Wilson line associated with an involution employed on smooth CY are taken to be perpendicular to the gauge bundle [7,8]. This is not the case in our construction; precisely for that reason our Wilson line W is an order 4 element instead of order 2.

Massless spectrum
After modding out τ , the 4D gauge group is G SM times eight U(1) factors and a non-Abelian hidden sector SU(3) × SU(2) × SU(2). One combination of the U(1) factors with generator t anom denotes the anomalous U(1). Furthermore, the standard hypercharge generator t Y from SU(5) can be identified and turns out to be orthogonal to the anomalous direction, The model has a local SU(5) GUT structure (for the discussion of the concept of local GUTs see [32] and cf. the related earlier discussion in [26,33]).
Dividing out the freely acting symmetry τ reduces the number of fixed points from 48 to 24 and breaks the symmetry from SU(5) to G SM . It further splits the untwisted 5-and 5-plets in the (e 1 , e 2 )-plane to a pair of Higgs candidates, denoted by h 1 and h 1 , removing the triplets. In the other two planes it removes the doublets, leaving two pairs of triplets/anti-triplets δ i / δ i (i = 1, 2) massless. A compact summary of the spectrum is given in Table 1; more complete details have been listed in Table 2 in Appendix A.
To understand the family structure note that, due to the absence of the Wilson line in the e 1 direction (W 1 = 0), states in the T (0, 1) and T (1,1) sectors form doublets under a discrete group D 4 , which is unaffected by modding out the freely acting symmetry τ . As two families reside in the T (1,1) sector, the two light families transform as a doublet under this D 4 flavor symmetry. The third family comes from T (1,0) sector and hence is a D 4 singlet. Such a D 4 symmetry is known to be phenomenologically attractive as it can ameliorate supersymmetric flavor problems [34]. In this respect the structure of the model is very similar to the Z 6 -II models discussed in [10,12,35].

Gauge coupling unification
As explained in Section 2.2, the GUT symmetry breaking is accomplished by the action of a freely acting symmetry τ , leading to a completely non-local breaking. This mechanism was introduced originally in the context of smooth manifold compactifications [22]. Later it was considered as an alternative to the standard (localized) breaking in orbifold constructions [36].
In order to discuss the virtues of non-local breaking, let us briefly recall the usual obstructions in embedding the beautiful picture of MSSM gauge coupling unification in the heterotic string. There are three main issues: 1. huge, "power-like" threshold corrections around the string scale; 2. the MSSM unification scale, M GUT = few · 10 16 GeV, is by an order 10 factor below the heterotic string scale; 3. the appearance of split multiplets at the high scale generically leads to logarithmic thresholds.
The first problem is absent in the scheme of 'local grand unification' as the bulk gauge group in extra dimensions contains G SM such that power-like corrections are universal. The second issue may be overcome by considering anisotropic compactifications [37,24]. As described in detail in [24], by using a discrete (rather than continuous) Wilson line associated with the involution to break the GUT symmetry, the breaking scale is related to the length of the corresponding Wilson line cycle. This length can be of order of M −1 GUT with the volume of compact space being so small that a description in terms of the perturbative heterotic string is still justifiable. This mechanism also ameliorates the third problem. In fact, the remaining logarithmic corrections may even mitigate the discrepancy between string and GUT scales [25]. In this respect our model is "cleaner" than the MSSMs based on Z 6 -II, where various logarithmic corrections to gauge unification from localized states and vector-like exotics coming in incomplete GUT multiplets are expected (cf. the discussion in [38]). Hence, the mechanism of non-local GUT breaking provides us with one of the most compelling realizations of precision gauge coupling unification. The implications of precision gauge unification for the MSSM superpartner spectrum have been discussed very recently in [39].

Semi-realistic VEV configuration
In order to obtain the MSSM, the unwanted U(1) gauge group factors have to be broken. This can be accomplished by switching on VEVs of standard model singlet fields consistently with vanishing F -and D-terms (cf. the discussion in [10,12]). In addition, these VEVs give rise to effective Yukawa couplings for quarks and leptons, they serve as effective mass terms decoupling the exotics and may generate dangerous proton decay operators. In order to avoid the latter ones (at least of dimension four), we identify vacuum configurations with a matter parity, using methods described in [40]. Like in the heterotic benchmark model in [12], this matter parity emerges as a Z 2 subgroup of a U(1) B−L gauge symmetry generated by t B−L = − and is given by e 2π i 3 2 q B−L = ±1. This matter parity will be referred to as Z R 2 . The configurations with preserved Z R 2 are such that we are left with the exact MSSM gauge group, three chiral generations, no R parity violating couplings, and are able to discriminate between lepton and Higgs doublets as well as between SM fields and exotics (see Table 1).
Let us now discuss a configuration in which all G SM × Z R 2 singlets φ (i) are assumed to attain VEVs Configurations in which all these 44 fields are non-trivial lead to 44 F -term equations for 44 fields, which in general have solutions (cf. the corresponding discussion in [10]). Furthermore, we have explicitly verified that these fields enter gauge invariant monomials, such as to ensure vanishing D-terms including a cancellation of the FI term of the anomalous U(1). Due to our ignorance of the coefficients of couplings we were not able to prove that the simultaneous solutions to F = D = 0 occur for small singlet expectation values; in the following we make this assumption.
Assigning VEVs to all the φ (i) fields breaks all extra U(1) factors and leads to effective mass terms for the non-chiral remnants w.r.t. the symmetry G SM × Z R 2 . This can be seen in detail by generating all couplings allowed by the relevant string selection rules, and compute the corresponding mass matrices M ij . As the freely acting symmetry slightly modifies the usual selection rules, we specify them explicitly in Appendix B. For the exotic δ i -δ j pairs we obtain the structure (10) where here and in the following φ n denotes a sum of known monomials in the VEVs of the fields of (9) with n being its lowest degree.
Obviously, due to the Z R 2 matter parity, there is no mixing between d quarks and the quark-like exotics δ. Switching on the VEVs of the untwisted states s 1 and s 2 is sufficient to decouple the untwisted triplets δ i , δ i , i = 1, 2. Even more, as can be seen from (10), all triplets decouple at linear order in the φ (i) fields. Similar features have been reported in the context of free fermionic model building (see e.g. [27]).
There are four Higgs pair candidates with mass matrix, defined by the superpotential terms Generically, they depend on the VEVs of all four Higgs-pairs and, due to Z R 2 , there is no mixing between the lepton-doublets and the Higgses h. The top-quark couples to h 1 already at order φ 0 , hence this coupling is not suppressed compared to the first and second generations. Moreover, as the Higgs h 1 is part of the untwisted sector, i.e. an internal part of the 10D gauge field, it couples with a strength proportional to the gauge coupling, realizing a gauge-top unification [41]. In general, couplings between localized states exhibit SU(5) relations, as they are not subject to the non-local symmetry breakdown due to W . Furthermore, as a consequence of our choice of U(1) B−L , the three generations of quarks and leptons originate only from the twisted sectors, hence their couplings originate from SU(5). This explains why the charged lepton mass matrix M e and the d-type mass matrix M d are identical in Eq. (12), a feature that is actually only desirable for the third generation.

Interpretation as a complete blow-up
The second objective of our work is to show that our orbifold model may be related to a smooth Calabi-Yau compactification. The Z 2 × Z 2 orbifold has 48 Z 2 fixed tori that constitute codimension four singularities, which are identified pairwise by the freely acting symmetry. To obtain a smooth space all these singularities have to be polished out. Below we will explain why the configuration discussed in the previous section defines a complete blow-up within the effective 4D theory.
From the perspective of the orbifold model smoothing out singularities corresponds to non-vanishing VEVs of twisted states. To smooth out all singularities at least one twisted state per fixed torus needs to acquire a VEV. Such a VEV can either lead to a blow-up or to a deformation of these singularities [42]. In the former case the cycle hidden inside the singularity, called the exceptional divisor, acquires a finite volume w.r.t. the Kähler form of the geometry. When the singularity is deformed, i.e. the complex structure is modified, its volume remains zero, and is in this sense still singular. As all twisted states of the Z 2 × Z 2 orbifold are six dimensional, they form hyper multiplets of N = 1 in 6D. Which of the two chiral multiplets within these hyper multiplets takes a VEV decides whether one has a blow-up or a deformation.
The blow-up described within the effective 4D theory takes into account only those twisted states as blow-up-modes that are massless in 4D. Due to the presence of Wilson lines it may happen that some orbifold singularities do not provide 4D massless states. In fact, a quick glance over Table 2 in Appendix A reveals that three fixed tori of the orbifold model defined in Section 2 do not support 4D zero modes. So they might remain singular in a complete blow-up within the effective 4D theory. On the other hand, each fixed torus supports 6D massless twisted states, which may develop non-trivial profiles over the internal tori that might remove the singularities. However, a detailed discussion of these issues is beyond the scope of the present Letter.
In general, VEV configurations correspond to complicated gauge bundles on the Calabi-Yau space that need to fulfill the integrated Bianchi identities S tr R 2 − tr F 2 = 0 (13) for all four-cycles S. These consistency equations can be viewed as the smooth analog of the modular invariance conditions, equation (3), for the shifts V 1 , V 2 and the Wilson lines W α . However, there seems to be no condition(s) on the Wilson line W associated with the involution τ for our blow-up or for other smooth CY constructions. By contrast, on the orbifold we encounter the additional requirements (6). Their derivation necessarily involves winding modes. The fact that in the supergravity approximation they are usually ignored, might explain why the modular invariance conditions of the freely acting Wilson line W do not have a smooth counterpart. Nevertheless, such conditions might be essential to ensure that a given smooth CY compactification of supergravity has a full string lift.
The VEV configuration (9) has been chosen such that the standard model group and matter parity (in particular also the hypercharge) remain unbroken. Moreover, according to Table 2 in Appendix A all non-empty fixed tori support twisted states to the 4D theory with VEVs switched on. Therefore, this corresponds to a complete blow-up in the effective 4D theory. Hence, it shows that the obstructions to a full blow-up encountered in the Z 6 -II mini-landscape models can be overcome in settings with non-local GUT breaking.
To summarize, we have shown that the Z 2 × Z 2 orbifold model with a freely acting Z 2 involution allows for VEV configurations where 4D zero modes originating from all non-empty fixed tori are switched on without breaking the standard model gauge group. The construction and interpretation of such configuration from the point of view of smooth compactifications will be discussed elsewhere [43].

Conclusions
We have presented a Z 2 × Z 2 orbifold compactification of the heterotic string exhibiting the exact chiral MSSM spectrum and gauge group as well as matter parity. The starting point of this model is the Z 2 × Z 2 orbifold with SU(5) gauge group. The SU(5) GUT symmetry is non-locally broken by modding out a freely acting symmetry. This ensures that there is no flux in hypercharge direction such that there is no obstruction to a complete blow-up. Further, Wilson line breaking is known to avoid large thresholds to the gauge coupling such that our construction complies with the beautiful picture of MSSM gauge coupling unification.
Accompanying an involution of the geometry with a Wilson line has been considered previously in smooth compactifications leading to the MSSM [7,8]. However, in our approach we encounter novel modular invariance conditions on this freely acting Wilson line that seem to have no analog in smooth CY compactifications in the supergravity approximation. This might suggest that some of the smooth CYs with involutions dressed with Wilson lines may exist only as supergravity models, but do not have a lift to consistent string theory constructions.
The model has the chiral MSSM spectrum and many other phenomenologically appealing features, like non-trivial Yukawa couplings and admits vacua with matter parity. On the other hand, we cannot claim that the configuration presented here is fully realistic. In more detail, due to the presence of a D-term for an anomalous U(1) some states need to acquire VEVs. We identified and discussed a specific VEV configuration with an exact matter parity (hence proton decay is avoided at the dimension four level), where all unwanted U(1) gauge group factors are broken, and all exotics decouple. Unfortunately, also the Higgs fields generically attain large masses. This unpleasant feature is shared with smooth CY MSSM models [7,21], where generically the μ-term is of the order of the fundamental scale [21,44] (while in orbifold models there are symmetries that allow us to relate the size of μ to the scale of supersymmetry breakdown [12,45,46]). Our model also avoids the problem of an additional U(1) B−L symmetry that cannot be broken without breaking supersymmetry. In summary, we have presented an explicit orbifold compactification satisfying all stringy consistency conditions. We identified vacua which correspond to resolutions of the orbifold fixed points, have properties very similar to those of the most promising smooth heterotic compactifications known so far, and are, in addition, endowed with an exact matter parity.

Outlook
The main achievement in this Letter was to show how to construct a concrete orbifold compactification of the heterotic string in which the breaking SU(5) → G SM is non-local, i.e. due to a Wilson line. We have argued that this may allow us to obtain a potentially realistic model with an explicit orbifold limit and a clear interpretation in terms of smooth geometry. Our analysis is incomplete in three main respects. First, the phenomenological viability of the model has to be studied in more detail. The configuration discussed in this Letter suffers from the problem that the Higgses generically get ultra-heavy. Possible solutions to the μ-problem will be discussed in a forthcoming publication [47].
Secondly, the configuration with VEVs discussed in this work seems to indicate that a complete blow-up within the effective 4D theory is possible. However to really show that this configuration corresponds to a smooth compactification, one has to construct the gauge bundle on the resolution of the compact orbifold explicitly and check that it fulfills all Bianchi identities for consistency. Work in this direction is in progress [43].
Finally, we have seen that some twisted sectors are empty in 4D. This seems to indicate that the corresponding orbifold singularity remain unresolved. Therefore, they correspond to partial (rather than full) blow-ups of the geometry. A geometric interpretation of such settings still needs to be obtained. (1, 1, 1, 1, 1) −1, 0, 0