application/xmlPrecursors from S(1)/Z2×Z2′ orbifold GUTsFilipe Paccetti CorreiaMichael SchmidtZurab TavartkiladzePhysics Letters B 567 (2003) 93-99. doi:10.1016/j.physletb.2003.06.037journalPhysics Letters BCopyright © unknown. Published by Elsevier B.V.Elsevier B.V.0370-26935671-27 August 20032003-08-0793-99939910.1016/j.physletb.2003.06.037http://dx.doi.org/10.1016/j.physletb.2003.06.037doi:10.1016/j.physletb.2003.06.037http://vtw.elsevier.com/data/voc/oa/OpenAccessStatus#Full2014-01-01T00:14:32ZSCOAP3 - Sponsoring Consortium for Open Access Publishing in Particle Physicshttp://vtw.elsevier.com/data/voc/oa/SponsorType#FundingBodyhttp://creativecommons.org/licenses/by/3.0/JournalsS300.2PLB20014S0370-2693(03)00934-110.1016/j.physletb.2003.06.037PhenomenologyTable 1Extended 5D SUSY SU(5) with light precursors. For all cases α3(MZ)=0.119CaseN, N′NENLMG/MEMG/MLMG/GeVμ0(I)1013213×10133×10132×10161 TeV(II)1010426.4×1092.1×10111.8×1013912 GeV(III)1010436.4×1093.6×1071.8×1013912 GeVTable 2Extended 5D non-SUSY SU(5) with light precursors. For all cases α3(MZ)=0.119CaseN, N′NENLMG/MEMG/MLMG/GeVμ0(I)5×1011842.5×10131.2×10103.7×101637 TeV(II)5×1091157.4×10103.3×10103.4×10133.4 TeV(III)1081479.5×1084.6×1082.3×10111.1 TeVPrecursors from S(1)/Z2×Z2′ orbifold GUTsFilipePaccetti CorreiaaMichaelSchmidtaZurabTavartkiladzeabz.tavartkiladze@thphys.uni-heidelberg.deaInstitut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16, D-69120 Heidelberg, GermanybInstitute of Physics, Georgian Academy of Sciences, Tbilisi 380077, GeorgiaEditor: G.F. GiudiceAbstractThe possibility of appearance of GUT precursors near the TeV scale (suggested by Dienes–Dudas–Gherghetta) is addressed within 5D GUTs compactified on an S(1)/Z2×Z2′ orbifold. For a low compactification scale (large radius), there is a significant non universal logarithmic contribution in the relative running of gauge couplings. This within 5D SU(5), with the minimal field content, gives wrong prediction for α3(MZ) unless one goes beyond the minimal setting. The realization of the light precursors' idea thus requires some specific extensions. As a scenario alternative to SU(5) we also consider an SU(6) orbifold GUT, whose minimal non-SUSY version gives natural unification. In all the presented unification scenarios with light precursors, various GUT scales are realized. This allows the model to be naturally embedded in either heterotic or Type I string theories.1IntroductionOne of the phenomenological motivations for SUSY GUTs is the successful unification of the gauge couplings near the scale MG≃2×1016 GeV. Besides the nice unification picture SUSY provides a natural understanding of the gauge hierarchy problem, but there are puzzles which need to be understood. Namely, one should find natural ways for baryon number conservation and the resolution of the doublet–triplet (DT) splitting problem. The minimal SU(5) and SO(10) GUTs also suffer from the problem of wrong asymptotic mass relations: Me0=Md0 and Me0=Md0∝Mu0 for SU(5) and SO(10), respectively. Higher-dimensional orbifold constructions suggest rather economical ways for simultaneously resolving these problems [2–6]. Therefore, they give new insights for GUT model building.11Proton stability and DT splitting in superstring derived models were discussed in [7]. There are scenarios (not orbifold GUT constructions), which can give unification near the TeV scale [8]. However, let us note, that within orbifold GUTs only in some particular cases power law low scale unifications is possible [6]. While usually, within 5D S(1)/Z2×Z2′ GUTs, the unification is logarithmic (like in 4D) and MG is still in the 1016 GeV range [4,6], with all GUT states lying far above the electro-weak scale (∼100 GeV). This makes a test of most GUT models impossible in present and future high energy colliders.In a recent paper [1] by Dienes, Dudas and Gherghetta the possibility of an orbifold construction was suggested where the compactification scale μ0=1/R (R is the radius of the compact extra dimension(s)) lies in the TeV range, while the GUT scale is still near 1016 GeV. If GUT symmetry breaking occurs by boundary conditions upon compactification, then above μ0 there appear signatures of the GUT model: the KK modes of gauge bosons, which correspond to broken generators, will have masses ≳μ0. In [1], these states were called GUT precursors, since their appearance indeed would be a characteristic feature of the GUT model.In this Letter we present a detailed study of the possibility of light precursors coming from 5D GUTs compactified on an S(1)/Z2×Z2′ orbifold. We show that even if there is no power law relative running of gauge couplings, there is a non-universal logarithmic contribution which for low μ0 becomes significantly large and does not allow unification with minimal field content. For unification, some extension should be done. We present an extension of 5D SUSY SU(5), with additional matter introduced on a brane, which can have unification in a range 1010–1016 GeV, depending on the selection of extended matter. We analyze also the 5D non-SUSY SU(5) model and it turns out that also this one requires extensions, which seem to be rather complicated. As an alternative GUT model, we study the extended GUT gauge group SU(6). As it turns out, the non-SUSY version of the 5D SU(6) GUT can naturally give unification with light precursors. In all considered scenarios with successful unification, the GUT scale can lie between scales 1010 GEV and 1016 GeV, while the precursors have ∼TeV masses. These values of MG allow one to embed the GUT scenario either in heterotic or in Type I string theory (depending on which value for MG is realized).2Renormalization for S(1)/Z2×Z2′ orbifold 5D GUTsIn this section we present expressions, which will be useful for studying gauge coupling unification for 5D GUTs.As it was shown in [2], a realistic 5D SU(5) GUT can be built if compactification occurs on an S(1)/Z2×Z2′ orbifold. For this case, on one of the fixed points we have the SU(3)c×SU(2)L×U(1)Y≡G321 gauge group and minimal field content. If instead of SU(5) some extended gauge group is considered, compactification on an S(1)/Z2×Z2′ orbifold still turns out to be an economical possibility for realistic model building.It is assumed that the fifth (space-like) dimension y parameterizes a compact S(1) circle with radius R. All states, introduced at 5D level, should have definite Z2×Z2′ parities (P,P′), where Z2:y→−y, Z2′:y′→−y′ (y′=y+πR/2). Therefore, there are only the following options for parity prescription: (2.1)(P,P′)=(+,+),(+,−),(−,+),(−,−), and the corresponding KK states have masses (2.2)2nμ0,(2n+1)μ0,(2n+1)μ0,(2n+2)μ0, respectively, were μ0=1/R is the compactification scale and n is the quantum number in the KK mode expansion. In a GUT scenario, KK states become relevant for gauge coupling running if μ0 lies below the GUT scale MG. The solution of one loop RGE has the form (2.3)αa−1(MG)=αa−1(MZ)−ba2πlnMGMZ+Δa, with (2.4)Δa=Δa0+ΔaKK, where ba corresponds to the contribution of SM or MSSM states (depending which case we consider) and (2.5)Δa0=−(baMI)α2πlnMG(MI)α includes contributions from all additional brane and zero-mode states α with mass (MI)α. ΔaKK comes from the contributions of the KK states (except zero-modes) (2.6)ΔaKK=−γa2πS1−δa2πS2, where S1 and S2 include contributions from KK states with masses (2n+2)μ0 and (2n+1)μ0, respectively: (2.7)S1=∑n=0NlnMG(2n+2)μ0,S2=∑n=0N′lnMG(2n+1)μ0. In (2.7), N and N′ are the maximal numbers of appropriate KK states which lie below MG, e.g., (2.8)(2N+2)μ0≲MG,(2N′+1)μ0≲MG. KK states with masses larger than MG are irrelevant. For a given MG/μ0 the N and N′ can be calculated from (2.8). Imposing the condition of gauge coupling unification (2.9)α1(MG)=α2(MG)=α3(MG)≡αG, from (2.3), eliminating αG and lnMG/MZ, we find for the strong coupling at the MZ scale (2.10)α3−1=b1−b3b1−b2α2−1−b2−b3b1−b2α1−1+b1−b3b1−b2Δ2−b2−b3b1−b2Δ1−Δ3, where αa in (2.10) stands for αa(MZ). Also, from (2.3) one can obtain (2.11)lnMGMZ=2πb1−b2α1−1−α2−1+2πb1−b2(Δ1−Δ2), and finally the value of the unified gauge coupling (2.12)αG−1=α2−1−b22πlnMGMZ+Δ2.It is straightforward to present some expressions, which will also be useful for further estimates. For N,N′⪢1 the S1 and S2 can be approximated by using Stirling's formula (2.13)S1≃(N+1)lnMGμ0−N+32ln(N+1)+(1−ln2)(N+1)−ln2π−112(N+1)+⋯,S2≃(N′+1)lnMGμ0−2N′+32ln(2N′+1)+N′+12lnN′+(1+ln2)N′+1+⋯. The combination, which will matter for the analysis below, is (2.14)S=S2−S1, which according to (2.13), for N=N′⪢1 reduces to the simple form (2.15)S≃lnπN+712N+O1N2.35D SU(5) GUT on S(1)/Z2×Z2′ orbifoldWe start our studies of orbifold GUT models with a 5D SU(5) theory. The fifth dimension is compact and is considered to be an S(1)/Z2×Z2′ orbifold. In terms of the G321, the adjoint of SU(5) reads (3.1)24=C(8,1)0+W(1,3)0+S(1,1)0+X(3,2̄)5+Y(3̄,2)−5, where subscripts are the hypercharge Y=160Diag(2,2,2,−3,−3) in the 1/60 units. In the 5D gauge multiplet the 4D gauge field A(24) is accompanied by an adjoint scalar Φ(24). In order to achieve SU(5)→G321 breaking, the gauge fragments AX, AY (see decomposition in (3.1)) must have negative orbifold parity. This will insure the masses of their KK modes to be proportional to μ0. Since the fragments ΦX, ΦY should become the longitudinal modes of AX, AY, the former should carry opposite orbifold parity. Having only one Z2, the ΦX, ΦY would contain massless zero modes, which are phenomenologically unacceptable. That is why the projection on an S(1)/Z2×Z2′ orbifold should be considered. Below we study SUSY and non-SUSY cases separately.3.1SUSY caseThe 5D N=1 SUSY gauge multiplet in 4D notation constitutes a N=2 SUSY supermultiplet VN=2=(V,Σ), where V is 4D N=1 gauge supermultiplet and Σ is a chiral superfield. Ascribing to the fragments (in decomposition (3.1)) of V(24) and Σ(24) the following Z2×Z2′ parities (3.2)(VC,VW,VS)∼(+,+),(VX,VY)∼(−,+),(ΣC,ΣW,ΣS)∼(−,−),(ΣX,ΣY)∼(+,−), at the y=0 fixed point we will have a 4D N=1 SUSY G321 gauge theory. We introduce three families of quark–lepton superfields and two MSSM Higgs doublets at the y=0 fixed point (this 3-brane is identified with our 4D world). We are looking for cases in which μ0 lies much below MG; in order to have perturbativity up to the GUT scale, we then always assume that matter and higgses are introduced on the brane. This means that they do not have KK excitations. Therefore, the zero modes at the y=0 brane are just the MSSM content. Other states, including the X, Y gauge bosons, are projected out. Taking all this into account, and also (3.2), (2.1), (2.2), for ba and γa, δa (defined in (2.6)) we have (b1,b2,b3)=(335,1,−3), (3.3)(γ1,γ2,γ3)=(0,−4,−6),(δ1,δ2,δ3)=(−10,−6,−4). From this and (2.10)–(2.12), taking also into account (2.4)–(2.7), we get for ‘minimal’ 5D SUSY SU(5) the following relations: (3.4)α3−1=127α2−1−57α1−1−37πS,(3.5)lnMGMZ=5π14α1−1−α2−1+57S,(3.6)αG−1=α2−1−12πlnMGMZ+2πS1+3πS2. Here, (3.4) and (3.5) show that α3 and MG are determined by the function S, which is defined in (2.14). For simplicity let's take N=N′, which for large values allows to use approximation (2.13). With this, from (3.4) we see that already N=N′=106 gives an unacceptably large α3(MZ) (≃0.13), while (3.5) gives an increased value of the GUT scale (≃4×1018 GeV). The situation becomes even worse for larger N, N′, because the function S grows logarithmically. Therefore, we conclude that, in this minimal setting, the compactification scale μ0≃MG/(2N) cannot be lowered below 4×1012 GeV. The first two terms in (3.4) would give a nice value for α3 (≃0.116), which coincides with the one loop 4D SU(5) prediction. To maintain this, one should take S≃0, which means μ0≃MG. This excludes light precursors in the framework of minimal 5D SUSY SU(5).Light precursors from extended 5D SUSY SU(5)With specific extension of minimal 5D SUSY SU(5), it is possible to get successful unification with low μ0. Let's at the y=0 brane introduce the following vector like states (3.7)NE×(Ec+Ec),NL×(L+L), where under G321, Ec and L have precisely the same transformation properties as a right handed charged lepton and a left-handed lepton doublet, respectively. Ec, L have conjugate quantum numbers and NE, NL denote the numbers of corresponding vector like pairs. Assuming that they have 4D masses ME, ML, above these scales these states will contribute to the b-factors as (3.8)(b1,b2,b3)E=65,0,0NE,(b1,b2,b3)L=35,1,0NL. In this case, the RGEs are modified and have the form (3.9)α3−1=127α2−1−57α1−1+3NE7πlnMGME−9NL14πlnMGML−37πS,(3.10)lnMGMZ=5π14α1−1−α2−1−3NE14πlnMGME+NL14πlnMGML+57S,(3.11)αG−1=α2−1−12πlnMGMZ−NL2πlnMGML+2πS1+3πS2. From (3.9) and (3.10) we see that by suitable selections of NE, NL and of the mass scales, it is possible to cancel the large logarithmic contribution coming from S and obtain a reasonable value for α3(MZ). At the same time, it is possible to get various values of MG. The different cases of unification, estimated through (3.9), (3.10), are presented in Table 1. As we see, for relatively large NE and NL, it is also possible to have a lower unification scale. For the considered cases the μ0 lies in the TeV range.3.2Non-SUSY caseIn this subsection, we study the possibility of light precursors for the non-SUSY case. For the non supersymmetric model, the higher-dimensional extension is more straightforward than for the SUSY one. As was pointed out at the beginning of Section 3, in a 5D extension the 4-dimensional gauge field should be accompanied by a real scalar. As far as the matter and Higgs fields are concerned, we introduce them on the brane. The S(1)/Z2×Z2′ orbifold parities for bulk states still are chosen in such a way as to break SU(5) down to the G321. It is easy to check that also for minimal 5D non-SUSY SU(5), the light precursors are incompatible with unification due to large logarithmic corrections (caused by S of (2.14)) coming from bulk states.Light precursors from extended 5D non-SUSY SU(5)Also here we extend the model by introducing (at the y=0 brane) vector like states with the transformation properties of (3.7), but now instead of superfields, under Ec, Ec, L, L we assume fermionic states. Their contribution to the b-factors are (3.12)(b1,b2,b3)E=45,0,0NE,(b1,b2,b3)L=25,23,0NL. With this setting the one loop solutions of RGEs have the forms (3.13)α3−1=333218α2−1−115218α1−1+23NE109πlnMGME−44NL109πlnMGML−21218πS,(3.14)lnMGMZ=30π109α1−1−α2−1−12NE109πlnMGME+4NL109πlnMGML+105109S,(3.15)αG−1=α2−1+1912πlnMGMZ−NL3πlnMGML+72πS1+214πS2. From them it follows that one can get successful unification with light precursors. Cases with different mass scales and number of vector-like states are presented in Table 2. As we see, the NE, NL should be large. This indicates that an extension is complicated.45D SU(6) GUT on S(1)/Z2×Z2′ orbifoldIn the previous section we have seen that in order to have light precursors within 5D SU(5) GUTs, one has to extend the matter sector. We will now discuss the extension of the gauge group. The smallest unitary group (in rank), which includes SU(5) is an SU(6) and we will consider here its 5D gauge version. The symmetry breaking of SU(6) should occur in two stages. Through compactification, by proper selection of the S(1)/Z2×Z2′ orbifold parities, the SU(6) can be reduced to one of its subgroups H. For the latter, there are three possibilities: H=H331=SU(3)c×SU(3)L×U(1), H=H421=SU(4)c×SU(2)L×U(1) and H=H51=SU(5)×U(1). The H gauge symmetry is realized at the y=0 fixed point and its further breaking to the G321 must occur spontaneously, by the VEV of an appropriate scalar field. As it turns out, the cases of H=H421 or H51 do not allow for light precursors, while the non-SUSY case of H=H331 gives an interesting possibility. The 5D SUSY SU(6) with minimal field content does not lead to unification with light precursors. Of course, some extensions of SUSY SU(6) versions in the matter sector can give the desirable result, but since we are looking for an SU(6) model with minimal matter/Higgs content, we will not pursue this possibility and concentrate on the non-SUSY version.Light precursors from 5D non-SUSY SU(6)The adjoint representation 35 of the SU(6), in terms of H331=SU(3)c×SU(3)L×U(1) decomposes as (4.1)35=C(8,1)0+L(1,8)0+S(1,1)0+CL(3,3̄)2+CL(3̄,3)−2, where the subscripts denote U(1) hypercharges (4.2)YU(1)=112(1,1,1,−1,−1,−1), in 1/12 units (the SU(6) normalization). Choosing the Z2×Z2′ parities of the fragments of Aμ(35) and Φ(35) (together they form a 5D gauge field A=(A,Φ)) as (4.3)(AC,AL,AS)∼(+,+),(ACL,ACL)∼(−,+),(ΦC,ΦL,ΦS)∼(−,−),(ΦCL,ΦCL)∼(+,−), at the y=0 fixed point we will have H331 gauge symmetry. For its breaking down to G321, we introduce a Higgs field H at the y=0 brane transforming under H331 as (1,3)−1. Its third component's non-zero VEV 〈H〉≡v induces the breaking SU(3)L×U(1)→vSU(2)L×U(1)Y, where (4.4)Y=−15YU(1)′+25YU(1), and YU(1)′ is the SU(3)L generator (4.5)YU(1)′=112Diag(1,1,−2). At the y=0 fixed point we also introduce three families of matter (4.6)uc(3̄,1)2,Q(3,3)0,Ec(1,3̄)−2,dc1,2(3̄,1)−1,L1,2(1,3̄)1, where we marked the transformation properties under H331. Under G321(4.7)Q=(q,Dc),Ec=(ec,L),L=(l,ξ), where ξ is a SM singlet. It is easy to verify that (4.6) effectively constitute anomaly free 15+6̄1,2 chiral multiplets of the SU(6) gauge group. Together with these, at the y=0 brane, we introduce an h(1,3)−1 Higgs, containing the SM Higgs doublet. With the brane couplings Qdc1,2H+, EcL1,2H+, after substituting the H's VEV v, the extra Dc and L states form massive states with one superposition of dc1,2 and l1,2, respectively, and therefore decouple. So, below the scale v, we have the SM with its minimal content. Taking all this into account, above the scale v, the b-factors are (4.8)(b1,b3L,b3)=376,−1,−5, while taking into account (4.3) the γ and δ-factors read (4.9)(γ1,γ3L,γ3)=0,−212,−212,(δ1,δ3L,δ3)=−21,−212,−212. From (4.4), it follows that (4.10)γY=−2110,δY=−18910,bY=7115. Using (4.8)–(4.10) we derive (4.11)α3−1=333218α2−1−115218α1−1−319654πlnMGv−483218πS,(4.12)lnvMZ=30π109α1−1−α2−1−215218πlnMGv−63109S. From (4.11) we already see, that with help of the last two terms, the wrong prediction of minimal non-SUSY SU(5) ((α3−1)minSU(5)=333218α2−1−115218α1−1≃1/0.07) can be improved. Namely, for N=N′=3×106, MG/v=1.27 we obtain successful unification with α3(MZ)=0.119, v≃7×1010 GeV. Therefore, unification occurs at MG≃8.8×1010 GeV and the compactification scale is μ0=14.7 TeV, i.e., we have light precursors.5Discussions and conclusionsIn this Letter we have presented a detailed study of the possibility of light precursors within concrete 5D GUTs. We have shown that compactification on a S(1)/Z2×Z2′ orbifold introduces corrections, which significantly affect gauge coupling unification. In order to compensate these corrections, specific extensions are needed. Achieving successful unification with light precursors, the prediction for α3(MZ) will not be affected by possible brane localized gauge kinetic terms, since the bulk is sufficiently large. Although the relative running of gauge couplings is logarithmic, the coupling itself has power law running. According to (2.13), in the ultraviolet limit (N≃N′→∞) we have S1≃S2→(1−ln2)N and the dominant contribution to the renormalization (2.3) comes from KK states: αa−1→−γa+δa2π(1−ln2)N. For a given GUT, γa+δa=const≡b̃ and the effective coupling is [1]αaeff=(1−ln2)Nαa=−2πb̃. Indeed, α5−1∼(αeff)−1Λ, where Λ is the ultraviolet cut off and α5 is the 5D gauge coupling. The αeff remains perturbative if b̃ is large and negative: −2πb̃⪡4π. It is easy to check that, in all models considered above, this condition is well satisfied. Therefore, at the ultraviolet limit αeff approaches a perturbatively fixed value.As far as the fundamental scale M∗ (5D Planck scale) is concerned, with the simplest setting the four- and five-dimensional Planck scales are related as [9]MPl2∼M∗3R and for 1/R∼ TeV we have M∗∼1014 GeV. To have a self-consistent picture of unification, we need the unification scale MG to lie below the M∗. Attempting to embed the presented scenarios into string theory, one also should make the values of scales self-consistent.22For detailed discussions see [1]. In perturbative heterotic string theory Mstring∼M∗ and there is a similar relation (MPl2∼M∗3R). Therefore, MG still should lie below the 1014 GeV. However, within the context of Type I (non-perturbative) strings, M∗ can be close to 1016 GeV, which allows to have unification in this region. Within the scenarios considered in this Letter, it is possible to have MG in the range of (1011–1016) GeV. This allows the models to be embedded either in heterotic or in Type I string theory.Note addedAfter the submission of this Letter into the arXive, we were informed by authors of [1], that the presence of additional logarithmic contributions into the relative gauge coupling runnings was reported in the Pascos'03 meeting by E. Dudas [10].AcknowledgementsResearch of F.P.C. is supported by Fundação para a Ciência e a Tecnologia (grant SFRH/BD/4973/2001).References[1]K.DienesE.DudasT.Gherghettahep-th/0210294[2]Y.KawamuraProg. Theor. Phys.1052001999Y.KawamuraProg. Theor. Phys.1052001691[3]G.AltarelliF.FeruglioPhys. Lett. B5112001257A.KobakhidzePhys. Lett. B5142001131M.KakizakiM.YamaguchiProg. Theor. Phys.1072002433[4]L.HallY.NomuraPhys. Rev. D642001055003A.HebeckerJ.March-RusselNucl. Phys. B61320013C.CsakiG.KribsJ.TerningPhys. Rev. D652002015004H.ChengK.MatchevJ.WangPhys. Lett. B5212001308A.HebeckerA.Westphalhep-ph/0212175[5]R.BarbieriL.HallY.NomuraNucl. Phys. B62463200263T.AsakaW.BuchmüllerL.CoviPhys. Lett. B5232001199T.LiNucl. Phys. B619200175Q.ShafiZ.Tavartkiladzehep-ph/0108247hep-ph/0210181F.Paccetti CorreiaM.G.SchmidtZ.TavartkiladzePhys. Lett. B5452002153I.GogoladzeY.MimuraS.Nandihep-ph/0210320hep-ph/0301014B.KyaeQ.Shafihep-ph/0212331[6]F.Paccetti CorreiaM.G.SchmidtZ.TavartkiladzeNucl. Phys. B649200339[7]R.ArnowittP.NathPhys. Rev. Lett.6219892225A.FontL.IbanezF.QuevedoPhys. Lett. B228198979J.EllisJ.LopezD.NanopoulosPhys. Lett. B252199053A.FaraggiNucl. Phys. B4281994111A.Faraggihep-ph/0107094[8]I.AntoniadisPhys. Lett. B2461990377K.DinesE.DudasT.GherghettaPhys. Lett. B436199855Z.KakushadzeNucl. Phys. B5481999205C.Caronehep-ph/9902407A.DelgadoM.Quiroshep-ph/9903400P.FramptonA.Rasinhep-ph/9903479A.Perez-LorenzanaR.N.Mohapatrahep-ph/9904504T.KobayashiNucl. Phys. B550199999Z.BerezhianiI.GogoladzeA.KobakhidzePhys. Lett. B5222001107[9]N.Arkani-HamedS.DimopoulosG.DvaliPhys. Lett. B4291998263N.Arkani-HamedS.DimopoulosG.DvaliPhys. Rev. D591999086004I.AntoniadisN.Arkani-HamedS.DimopoulosG.DvaliPhys. Lett. B4361998257[10]Talk presented by E. Dudas in Bombay, at PASCOS'03 Meeting, http://theory.theory.tifr.res.in/pascos/