All the realistic free fermionic models utilize a set of basis vectors, the
NAHE set, that correspond to $Z_2\times Z_2$ orbifold compactification with
nontrivial background fields. I argue that the realistic features of free
fermionic models, like the number of generations and the fermion mass spectrum
are due to the underlying $Z_2\times Z_2$ orbifold compactification.Comment: 4 pages, latex, To appear in proceedings of the Beyond the Standard
  Model IV Conference, Lake Tahoe, CA, Dec 13-18, 1994. IASSNS preprint number
  is corrected

Faraggi, Alon

English

arXiv

All the realistic free fermionic models utilize a set of basis vectors, the
NAHE set, that correspond to $Z_2\times Z_2$ orbifold compactification with
nontrivial background fields. I argue that the realistic features of free
fermionic models, like the number of generations and the fermion mass spectrum
are due to the underlying $Z_2\times Z_2$ orbifold compactification

University of Liverpool Repository

arXiv:hep-ph/9501288v2  16 Jan 1995IASSNS-HEP-94/114hep-ph/9501288Z2 × Z2 orbifold compactification –the origin ofrealistic free fermionic modelsAlon E. FaraggiSchool of Natural Sciences, Institute for Advanced StudyOlden Lane, Princeton, NJ 08540E-mail:faraggi@sns.ias.eduABSTRACTAll the realistic free fermionic models utilize a set of basis vectors, the NAHE set,that correspond to Z2×Z2 orbifold compactification with nontrivial backgroundfields. I argue that the realistic features of free fermionic models, like the numberof generations and the fermion mass spectrum are due to the underlying Z2×Z2orbifold compactification.As a unified theory of gravity and the gauge interactions, heterotic string theory1should reproduce the matter and symmetry content of the Standard Model and de-termine the fermion mass spectrum. Presently we do not know what is the dynamicalmechanism that selects the unique string vacuum, and, a priori, there is a large num-ber of potentially viable superstring models.The notion, however, that there is a huge number of string models is somewhatmisleading. By just imposing one or two phenomenological criteria, like three gen-erations and a gauge group that can be reduced at low energies to the standardmodel gauge group, already one finds that the number of possibilities is substantiallyreduced. Imposing further phenomenological constraints may indeed single out aunique superstring model. If such a model is constructed, it will certainly be of usein trying to learn about the dynamical mechanism that chooses the string vacuum.The task of constructing phenomenologically viable string models seems hopeless.While in ten dimensions the string vacuum is more or less unique, in four dimensionsthere is a huge number of equivalent candidates. The string consistency constraintsimpose a number of degrees of freedom and those degrees of freedom produce asymmetry that is larger than the observed symmetry at low energies. Furthermorethe number of chiral generations is also determined in the four dimensional vacuumand is correlated with the gauge degrees of freedom. A bottom–up approach, in whichdifferent blocks of the standard model are assembled together piece by piece, is notadequate. Rather, what is required is a top–bottom approach in which the featuresof the standard model are carved out of the more symmetric string vacua.To appear in Proceedings of the Beyond the Standard Model IV Conference, Lake Tahoe, California,December 13–18, 1994.Is there a guiding principle that may distinguish among the equivalent stringvacuum ? String vacua exhibits a new kind of symmetry, usually referred to as target–space duality2, which is a generalization of the R → 1/R duality in the case of S1. Atthe self–dual point, Rj = 1/Rj , space–time symmetries are enhanced. For appropriatechoices of the background fields the space–time symmetries are maximally enhanced3.At the maximally symmetric point the internal degrees of freedom that are needed tocancel the conformal anomaly may be represented in terms of internal free fermionspropagating on the string world–sheet. It is plausible that if string theory is realizedin nature then the true string vacuum is in the vicinity of the highly symmetric self–dual point. It may turn out that near that point the free fermionic formulation4provides a good approximation to the true string vacuum. However, the number ofconsistent free fermionic models is still enormous.As is well known in (2,2) string models that admit a geometrical interpretationthe number of chiral generations is half the Euler number of the six dimensionalcompactified manifold. Following LEP data it is plausible to assume that only threecomplete generations exist in nature. How can three generations arise from a sixdimensional compactified space. The answer may be simple. The six dimensionalcompactified space is divided into three factors of two. In the orbifold language5,divide the six dimensional space, which is compactified on a flat torus, by a Z2 × Z2discrete symmetry. In that case the Z2 × Z2 orbifold model produces exactly threetwisted sectors. In the Z2×Z2 orbifold on a six dimensional space, the number threeis deeply rooted in the structure of the models. Thus, the Z2×Z2 can very naturallylead to models with three generations. Namely, each light generation comes from adifferent twisted sector of the Z2 × Z2 orbifold model.It appears that Z2×Z2 orbifold on the flat torus of the six dimensional compactifiedspace, can very naturally lead to three generations. However, in general, Z2 × Z2orbifold on generic lattices do not lead to three generation models. For example theZ2×Z2 orbifold on SO(4)3 lattice did not yield three generation models. In contrast,the Z2 × Z2 models at the free fermionic point in toroidal compactification space,realized by the NAHE set6,7, do produce three generation models. The difference isseen by examining the number of fixed points in the two compactifications with (2,2)world–sheet supersymmetry. On the SO(4)3 lattice the Z2 × Z2 produces sixteengenerations, from each twisted sector. On the SO(12) lattice, which corresponds tothe free fermionic point in the toroidal compactification space, it produces eight chiralgenerations, from each twisted sector. In the fermionic three generation constructionseach one of the complex planes of the Z2 × Z2 orbifold is modded out by additionalZ32symmetries, thus reducing the number of generations to one generation from eachtwisted sector.In the (2,2) fermionic constructions one starts from a set of boundary conditionvectors that produces an N = 4 supersymmetric model with SO(12) × E8 × E8gauge group3. One then adds two boundary condition vectors that correspond to theZ2×Z2 twisting. The resulting gauge group is SO(4)3×E6×U(1)2×E8 with N = 1space–time supersymmetry. In this model there are twenty four chiral generationsfrom the boundary condition vectors that correspond to twisted sectors and threepairs of chiral and anti chiral generations from the untwisted sector. The number oftwisted and untwisted moduli is equal to the number of generations. In addition theuntwisted and twisted sectors produce E6 × E8 singlets that are obtained by actingon the vacuum with oscillators that arise from the fermionic degrees of freedom thatcorrespond to the six internal compactified dimensions.In the orbifold formulation5 the same model is obtained by applying a Z2 × Z2twist to a torodialy compactified SO(12) lattice and E8 × E8 gauge symmetry. The36 free parameters of the six dimensional metric and the antisymmetric tensor fieldparameterize the six dimensional compactified space. For generic values of theseparameters the gauge symmetry that arises from the six dimensional compactifiedtorus is U(1)6. For specific choices of the background parameters the U(1)6 of thecompactified torus is enlarged. To reproduce the SO(12) × E8 × E8 gauge groupof the free fermionic model, the metric Gij is the Cartan matrix of SO(12) and theantisymmetric tensor field is given by, Bij = Gij for i > j; Bij = 0 for i = j andBij = −Gij for i < j. For RI =√2 and with the chosen background fields, theright–moving momenta produce the root vectors of SO(12), thus reproducing thesame gauge group as in the free fermionic model. The orbifold model is obtained bymoding out the six dimensional torus by a discrete symmetry group. The masslessspectrum contains states from the untwisted and twisted sectors. In the case of“standard embedding” the number of chiral families is given by one half of the Eulercharacteristic. To translate the fermionic boundary conditions to twists and shifts inthe bosonic formulation the real fermionic degrees of freedom that correspond to thecompactified dimensions are bosonized. The fermionic boundary condition vectors, b1and b2, then translate to Z2 × Z2 twist on the compactified coordinates and to shiftson the gauge degrees of freedom. It is then seen that symmetries and spectrum ofthe orbifold model coincide with those of the corresponding fermionic model3.The realistic free fermionic models correspond to models with (2,0), rather than(2,2), world–sheet supersymmetry. The transition from the (2,2) models to the (2,0)models can be regarded as choosing a GSO phase between the two boundary conditionvectors that produce the spinorial of SO(16). The GSO projection projects out themassless states from these sectors and the resulting gauge group is SO(12)×SO(16)×SO(16), with N = 4 space–time supersymmetry. Alternatively, one of the spinorialvectors may be enlarged with additional four periodic complex fermions in the hiddensector. The E8 × E8 gauge group is modified to SO(16) × SO(16), as in the firstconstruction. The analysis with respect to the number of fixed points is identical tothe case with (2,2) world–sheet supersymmetry. However, in this case the observablegauge group after applying the Z2 × Z2 is SO(10) × U(1) rather than E6, and theU(1) is “anomalous”. The twisted sectors produce spinorial and vectorial sixteen ofthe observable SO(10) and hidden SO(16) gauge groups, respectively.The structure of the Z2 × Z2 orbifold with (2,0) world–sheet supersymmetry andstandard embedding, is common to all the realistic free fermionic models. Threegeneration models are obtained by adding three additional boundary condition basisvectors, beyond the NAHE set. The additional boundary condition vectors mod outeach of the three complex planes of the Z2×Z2 orbifold by a Z32symmetry and breakthe observable SO(10) symmetry to one of its maximal subgroups SU(5) × U(1),SO(6)× SO(4) or SU(3)× SU(2)× U(1)2.The fermion mass spectrum is also seen to originate from the Z2 × Z2 orbifoldstructure, realized by the NAHE set. The untwisted sector produces three pairsof Higgs doublets and a combination of the vectors that break the SO(10) symme-try produces one or two additional pairs. Due to the horizontal symmetries in theZ2×Z2 orbifold models, each pair of Higgs doublets couples only to states from one ofthe twisted sectors, producing couplings 16j16j10j j = 1, 2, 3. The cancelation of theanomalous U(1) D–term equation by singlet VEVs, gives Planck scale mass to severalHiggs doublets. As a result, there exist models in which only one mass term, namelythe top quark mass term, exist at the cubic level of the superpotential. The massterms for the lighter quarks and leptons are obtained from nonrenormalizable terms.The nonrenormalizable terms contain SO(10) singlets with nonvanishing VEVs, thatare required to cancel the anomalous U(1) D–term equation. Thus, the nonrenormal-izable terms become effective renormalizable terms that are suppressed relative to theleading cubic level terms. Due to the horizontal symmetries and the singlet VEVsone generation is necessarily light8. Similarly, the mixing terms arise generically fromnonrenormalizable terms of the form 16i16j1016i16jφn, where the first two 16 are inthe spinorial representation of the observable SO(10), the 10 is in the vector represen-tation of the observable SO(10), the last two 16 are in the vector representation of thehidden SO(16) and φn is a combination of SO(10)× SO(16) scalar singlets9,3. TheZ2×Z2 orbifold structure gives rise to the horizontal symmetries that may be neededto understand the matter mass spectrum. Requiring adequate generation mixing andthe form of the mixing terms necessitates that we give nonvanishing VEVs to some ofthe hidden sector 16 representations. In Ref. [10] it was shown that this is possiblythe source of supersymmetry breaking in these models.AcknowledgementsThis work is supported by DOE grant DE-FG02-90ER40542.References1. D.J. Gross et. al., Nucl.Phys. B256 (1985) 253.2. A. Giveon et. al., Phys.Rep. 244 (1994) 77, and references therein.3. A.E. Faraggi, Phys.Lett. B326 (1994) 62, hep-ph/9311312..4. I. Antoniadis, C. Bachas, and C. Kounnas, Nucl.Phys. B289 (1987) 87; H. Kawai,D.C. Lewellen, and S.H.-H. Tye, Nucl.Phys. B288 (1987) 1.5. L. Dixon, J.A. Harvey, C. Vafa and E. Witten, Nucl.Phys. B274 (1986) 285.6. I. Antoniadis et. al., Phys.Lett. B231 (1989) 65.7. A.E. Faraggi, Nucl.Phys. B387 (1992) 239, hep-th/9208024.8. A.E. Faraggi, Nucl.Phys. B403 (1993) 101, hep-th/9208023; Nucl.Phys. B407(1993) 57, hep-ph/9210256.9. A.E. Faraggi and E.Halyo, Phys.Lett. B307 (1993) 305, hep-ph/9301261;Nucl.Phys. B416 (1994) 63, hep-ph/9306235.10. A.E. Faraggi and E.Halyo, IASSNS–HEP–94/17, hep-ph/9405223.

$Z_2\times Z_2$ orbifold compactification -- the origin of realistic
  free fermionic models

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$Z_2\times Z_2$ orbifold compactification -- the origin of realistic free fermionic models

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Z2×Z2Z_2\times Z_2Z2​×Z2​ orbifold compactification -- the origin of realistic free fermionic models

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$Z_2\times Z_2$ orbifold compactification -- the origin of realistic free fermionic models