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98/1561 CLNS TeV Scale Superstring and Extra Dimensions Gary Shiu and S.-H. Henry Tye Newman Laboratory of Nuclear Studies, Cornell University, Ithaca, NY 14853 (May 24, 1998) Abstract Utilizing the idea of extra large dimensions, it has been suggested that the gauge and gravity couplings uni cation can happen at a scale as low as 1 TeV. In this paper, we explore this phenomenological possibility within string theory. In particular, we discuss how the proton decay bound can be satis ed in Type I string theory. The string picture also suggests di erent scenarios of gauge and gravitational couplings uni cation. The various scenarios are explicitly illustrated with a speci c 4-dimensional N = 1 supersymmetric chiral Type I string model with Pati-Salam-like gauge symmetry. We point out certain features that should be generic in other Type I strings. 11.25.-w Typeset using REVTEX E-mail: E-mail: shiu@mail.lns.cornell.edu tye@mail.lns.cornell.edu 1 I. INTRODUCTION Probably the most important problem in elementary particle theory today is to nd out how superstring theory describes our universe. In the standard scenario, the Planck scale MP (i.e., 1019 GeV) de nes the string scale to be around 1017 GeV, while the uni cation of the gauge couplings happens at the grand uni ed scale MGU T (around 1016 GeV) [1]. Although the string scale and MGU T are quite close, the discrepancy between them may still be of some concern. However, a more practical problem with this scenario is the di culty in calculating physical observables. Since the natural string scale of this scenario is between MGU T and MP , while most of the physical observables are at the electroweak scale MEW , a typical comparison between theory and experiment requires a detailed analysis of a speci c string model. Unfortunately, our understanding of the string dynamics is still quite primitive, making such precise calculations essentially impossible. So the connection between string theory and our observable universe is rather tenuous in this scenario at this moment. It is therefore exciting that an alternative scenario has recently emerged. The idea of extra dimensions have been well studied in Kaluza-Klein theories and string theories. It was suggested by Antoniadis et. al. [2] that, beyond the usual 4 space-time dimensions that we live in, there are large extra dimensions that may be probed by upcoming experiments. It is by now well-known that, in some string models, the gravity lives in the bulk, while the gauge and charged matter elds live on the branes (which may be understood as special types of solitons and can have lower dimensions than the bulk [3]); our 4-dimensional universe may actually be inside the branes [4]. In particular, the extra dimensions that gravity feels can be as large as 1 mm, as recently pointed out by ArkaniHamed, Dimopoulos and Dvali [5], while the extra dimensions that the gauge and other 1 matter couplings feel can be as big as MEW , as recently pointed out by Dienes, Dudas and Gherghetta [6]. In this scenario, the Planck scale MP is traded for the size of the extra dimensions felt by gravity [5,7]. Likewise, gauge coupling uni cation can be preserved and remain perturbative, but now occurs at scales as low as a TeV [6]. One can therefore now have gravity and gauge coupling uni cation as low as a few hundred GeV to 1 TeV [5]. In string theory, this means that the string scale ms can be as low as a TeV [7]. Such a scenario has been suggested previously by Lykken [8]. One advantage of this TeV scale string scenario is obvious. Not only that near future experiments can probe the string scale and the large extra dimensions, it may even help us unravel the string dynamics and pinpoint the string vacuum we live in. In addition to the advantage of being experimentally testable, this new scenario may o er a simple qualitative explanation to the fermion mass hierarchy problem, as pointed out in Ref [6]. To be speci c, let us suppose the string scale ms = 1 TeV. Gravity, but not the standard model gauge and matter elds, lives in n large compacti ed dimensions, with radii ri . The radii Rj of the remaining compacti ed dimensions in which both gravity and 1 1 1 gauge elds live are somewhere between m 1 and MEW , so ms > Rj > MEW >> ri . s In this scenario, the e ective couplings at the ms scale are all irrelevant operators and so the dimensionless gauge couplings i and Yukawa couplings yf run as powers of the energy scale. If the di erent Yukawa couplings are comparable at the ms scale, they can easily di er by orders of magnitude at the electroweak scale due to this power-law behavior. The gauge couplings di er by only one order of magnitude because they are uni ed at the string 2 1 scale. Below the Rj scales, the dimensionless gravitational coupling runs a function of the 1 energy scale like E 2+n to the scales ri and then runs like E 2 , yielding a huge MP [4]. So the presence of the extra dimensions provide a qualitative explanation of the origin of the orders of magnitude di erences among the couplings. As pointed out in Ref [6], ms >> 1 TeV is perfectly acceptable. However, we have to treat ms and MEW as two di erent scales in this situation. In this paper, we study a number of issues in the 4-dimensional N = 1 chiral Type I string theory, which is the appropriate framework for the TeV scale string scenario. A typical model will have 9-branes, which ll the 10-dimensional spacetime, and 5-branes, which ll 6-dimensional spacetime. Both branes have a at 4-dimensional uncompacti ed spacetime. The string picture has been discussed previously in Ref [8,7]. Here, we review and extend the analysis. Among other observations in this paper, we note that: It is well-known that proton decay can be suppressed by symmetry. Here, we see that proton decay is suppressed by the presence of a custodial U(1) gauge symmetry. The presence of such a U(1) gauge symmetry is generic in Type I strings. As an alternative scenario, the standard model gauge symmetries can come from different types of branes, e.g., QCD SU(3) comes from one type of branes (say, 9-branes) while the weak SU(2) comes from another type (say, 5-branes). Since the 9-brane couplings are in general di erent from the 5-brane couplings, the standard model gauge couplings do not need to meet at the string scale. Rather, an appropriate choice of the sizes of the compacti ed dimensions is needed for the couplings to agree with experiment. Cavendish type experiments have tested Newton s Law to a scale of millimeters [9], providing an upper bound on the large radius. The strong and electroweak scatterings have 1 tested the small extra dimensions to a radius of MEW , providing an upper bound on the size of the small extra dimensions. Taking ms Rj 1, the relation between the large radii ri and MP is given by n 2 MP 32 2 g 4 m2 s (ms ri ) i=1 (1) where n is the number of large compacti ed dimensions, g is the gauge coupling [7]. The numerical factor follows from string unitarity and duality. Clearly ms must be bigger than MEW . Assume the gauge coupling g 2 1. For n = 2, r is about 10 4 meter for ms = 1 TeV. As pointed out in Ref [5], both the 1 mm scale and the 1 TeV scale can be tested by experiments in the near future. We point out that the n = 2 choice seems natural in a number of string scenarios. For example, in the speci c model that we consider in this paper, only the n = 2 choice gives rise to 3 chiral families. String theory has no global symmetry. However, some gauge couplings are proportional to r 1 . For very large r, they become so weak that the respective gauge symmetries may appear like global symmetries. In some situations, the corresponding matter elds with vanishingly small gauge couplings are suitable candidates for dark matter. To make the discussion concrete, we construct an explicit model to illustrate the TeV scale string scenario. Our analysis of the model is quite sketchy and cavalier. Our purpose is to draw attention to the model s features that are generic to other Type I string models. The model is a D = 4, N = 1 supersymmetric, chiral Type I string model, with 9-branes and 5-branes. Their gauge groups G9 and G5 (with gauge couplings g9 and g5 respectively) 3 are identical: G9 =G5 = U(4) U(2) U(2) . The massless open string spectrum is given in Table I. The U(1) s associated with the SU(2)s are anomalous, not unusual in string theories. The U(1) associated with the SU(4) provides the custodial symmetry to suppress proton decay. We shall use this model to discuss the following three scenarios: One may identify the SU(4) SU(2) SU(2) from the 9-brane sector as the PatiSalam group. Spontaneous symmetry breaking reduces it to SU(3) SU(2)L U(1). At the string scale, both the QCD coupling g3 and the weak coupling g2 are equal to the 9-brane coupling g9 , and sin2 W = 3/8. There is a U(1)B gauge symmetry associated with the baryon number, so the proton decay is suppressed. However, it seems that breaking this U(1)B symmetry will also break QCD SU(3). The model has only one chiral family, plus a vector (i.e., a chiral and an anti-chiral) family. One may identify SU(4)9 SU(2)5 SU(2) 5 (the subscripts indicate which sectors each comes from) as the Pati-Salam gauge group. The QCD SU(3) comes from the spontaneous symmetry breaking of the SU(4)9 SU(2) 5 by a bi-fundamental matter eld, while the remaining SU(2)5 is identi ed with the weak SU(2)L . The SU(4)5 may get strong and induce both dynamical supersymmetry breaking and electroweak symmetry breaking. The gauge group SU(2)9 SU(2) 9 is also broken. At the string scale, g3 = g9 while g2 = g5 . Now there are two families of quarks and leptons, coming from the 95 sector, while the Higgs elds come from the 55 sector. Again, the perturbative couplings obey the baryon quantum number conservation because of U(1)B . The rst scenario has one chiral family in the 99 sector while the second scenario has two chiral families in the 59 sector. It turns out that there is another scenario where there are three chiral families. Under diagonal spontaneous symmetry breaking, SU(2)9 SU(2)5 becomes SU(2). This Higgs mechanism is permitted by the presence of the appropriate bi-fundamental elds, resulting in a model which contains SU(4) SU(2)L SU(2)R with 3 chiral families: one from the 99 sector and two from the 59 sector. Weak interaction universality automatically follows, independent of the relative values of g9 and g5 . The standard model couplings are all di erent. In the case g = g9 = g5 , we have at the string scale, 82 2 2 g 2 = g3 = 2g2 = gY . 3 (2) Hence the electroweak Weinberg mixing angle satis es sin2 W = 3/7 at the string scale. This scenario has one practical advantage. Since g3 > g2 > gY at the string scale, the running couplings need power-like behavior for only a relatively short range of energies; that is, they do not have to grow much. As a consequence, the string coupling will stay weak, and perturbative Type I string theory should be valid for analysis. It is clear that, among other properties, the presence of U(1) s (associated with the centers of mass of the branes), the presence of bi-fundamental matter elds, and the identical nature of 9-brane gauge group and 5-brane gauge groups (if present) are generic features of many D = 4, N = 1 supersymmetric, chiral Type I string models. These properties are quite compatible with experiments and the extra large dimension scenario. Further investigation along this direction will certainly be worthwhile. This paper is organized as follows. In Section 2, the basic idea of the TeV scale string scenario is reviewed. The Type I string picture with extra large dimensions is discussed. As 4 an illustration, the Pati-Salam like Type I string model is presented in Section 3. Among other issues, we see how the proton decay bound can be resolved. Section 4 contains some discussions on the various issues in the scenario. Section 5 contains the comments. The details of the construction of the Type I string model is contained in Appendix A. It is relegated to an appendix because of its technical nature. In Appendix B, we review how to calculate amplitudes to determine the terms in the superpotential of the model. II. BRANE PICTURE The idea of extra large dimensions is most conveniently realized in terms of Type I string theory and D-branes [10]. The graviton (coming from the closed string sector) lives in the bulk, while the gauge and charged matter elds (coming from the open string sector) live on the branes (which is p + 1-dimensional for a Dp-brane). Since gravity and gauge elds see di erent numbers of dimensions, it is possible to have extra large dimensions without making the gauge couplings extremely small at low energies. In the worldsheet construction of heterotic string model, both gravity and gauge elds live in the same space, and so the idea of extra dimensions is di cult to implement. It is possible to realize the extra dimensions with the solitonic 5-branes in heterotic string theory. However, the techniques in constructing heterotic string model with these solitonic 5-branes are not very well developed. As a result, Type I string theory and D-branes provide the most natural setting to understand the generic features of extra dimensions. Here, we review and expand on the earlier discussions [8,7]. A. Supersymmetric Type I String Various N = 1, D = 4 Type I string models have been studied in the past two years [11 18]. They are especially suitable for realizing the idea of extra large dimensions. Gravity lives in the bulk while the gauge elds live on the branes. There are 9-branes, which overlap completely with the bulk. If a model has both p-branes and q-branes, then supersymmetry imposes the restriction p q = 0 (mod 4). To keep the Lorentz property of 4-dimensional spacetime, only 5-branes and 9-branes are permissible. So, for some models, there can be 5-branes as well. The 4-dimensional string has the usual 4 spacetime dimensions (x0 , . . . , x3 ), and 6 compacti ed dimensions. We shall treat this 6 dimensions T 6 as composed of 3 two-tori: T1 (with coordinates x8 , x9 ), T2 (with coordinates x6 , x7 ) and T3 (with coordinates x4 , x5 ), the volumes of which are v1 , v2 and v3 respectively. So the volume of the 6 compacti ed dimensions is v1 v2 v3 . Crudely speaking, the volume vi can be expressed in terms of the compacti ed radius ri , vi = (2 ri)2 . 1 So the low energy e ective action is given by radius ri here does not necessarily have to be the radius Ri of the torus. It is simply a characteristic length scale of the compacti ed dimension. In the case of a ZN orbifold, the volume 1 is given by i vi = N i (2 Ri )2 i (2 ri )2 . 1 The 5 S= m8 v1 v2 v3 1 m6 v1 v2 v3 2 1 3 m2 vi 2 s s s d4 x g F + ... R+ F+ 7 2 7 (2 ) 4 (2 ) 4 i=1 (2 )3 i (3) where ms is the string scale. F is the eld strength of the gauge elds in the 9-branes while the Fi is the gauge eld strength on di erent types of 5-branes (the worldvolumes of which are M4 T1 , M4 T2 and M4 T3 respectively; M4 being the 4 dimensional Minkowski space-time). Here is the string coupling, i.e., e , where is the dilaton eld. The relative normalization of the Newton s constant and the gauge coupling (which is related to the D-brane tension) is obtained by factorizing scattering amplitudes into open and closed string channels [10,19]. (In Type I string, the N-point open-string one-loop amplitude is equivalent to the closed string scattering to N open strings at the tree level. This relation follows from unitarity). This should be compared to heterotic string theory where all states are closed string states. The precise numerical factors are determined once we de ne the string coupling to be the ratio of the fundamental string and D-string tensions in Type IIB string theory [10]. For simplicity, we will consider only one type of 5-branes in what follows. Let G9 (G5 ) be the gauge group of the 9-brane (5-brane). The 5-branes are compacti ed on T3 , while the 9-branes are compacti ed on T 6 . The branes and the bulk have a common 4-dimensional uncompacti ed spacetime. The 4-dimensional Planck mass MP and the Newton s constant GN are given by 2 G 1 = MP = N 8m8 v1 v2 v3 s (2 )6 2 m2 v3 s (2 )3 (4) and the gauge couplings of G9 and G5 are 2 g9 = m6 v1 v2 v3 s , (2 )7 2 g5 = (5) These relations are subject to quantum corrections, which we shall ignore for the moment. Recall that the gauge couplings of the standard model are of order 1. In string theory, there is a T-duality symmetry, i.e., physics is invariant under a T-duality transformation. If any of the volume vi is much smaller than the string scale, i.e., vi less than m 2 , the T-dual s description is more convenient: vi m2 s 1 vi (2 )4 vi m4 s (2 )2 (6) In this dual picture, the new volume (2 )4 /(vi m4 ) of the dual Ti torus is large. Under s this duality transformation, the Dirichlet and Neumann boundary conditions of the open strings are interchanged, and so the branes are also mapped to other types of branes. For example, for i = 1, i.e., we T-dual the T1 torus; the 9-branes become 7-branes (x0 , . . . , x7 ), while the 5-branes become 7-branes (x0 , . . . , x5 , x8 , x9 ). Therefore, they are orthogonal in the compacti ed space. The e ective action becomes S= 1 m4 v2 v3 2 1 m4 v1 v3 2 m8 v1 v2 v3 s s s F + ... R+ F+ d4 x g (2 )7 2 4 (2 )5 4 (2 )5 6 (7) If the standard model gauge group is in G9 , then, in this 7-brane picture, 2 g9 = m4 v2 v3 s 1 (2 )5 (8) Now, suppose ms is 1 TeV. To satisfy Eq.(4), i.e., to obtain the MP = 1019 GeV, at least one of the 2-volumes must be large. Since the G9 gauge coupling must be of order 1, the only choice is to take v1 large. This means the G5 gauge coupling becomes extremely small, i.e., the gauge elds decouple. The conserved currents that couple to G5 will appear like those of global symmetries. We can also keep both G9 and G5 gauge couplings of order 1. This can be achieved if we T-dual both the T1 and T3 tori to end up with orthogonal (in the compacti ed space) 5-branes with the following e ective action: S= 1 m2 v2 2 1 m2 v1 2 m8 v1 v2 v3 s s d4 x g F + ... R+ F+ 7 2 3 (2 ) 4 (2 ) 4 (2 )3 (9) In this case, we can take v3 large to satisfy Eq.(4). We shall use this 55 -brane picture later, but we shall still refer to the 5-branes coming from T-dualizing the 9-branes as the 9-branes. To summarize, there are 8 inequivalent scenarios one can entertain: 95-, 77-, 55and 73-brane con guration, and their T 6 -duals (i.e., T-dual along all 6 dimensions): 59-, 77-, 55- and 37-brane con gurations. The 95-, 77- and 55-brane e ective actions are given above. We have kept the two radii in each torus to be the same. To build an N = 1 supersymmetric model, we need to orbifold the compacti ed dimensions in the complexi ed basis. Equal radii in each torus yield discrete symmetries that can be gauged in orbifolds. If G9 is identical to G5 , and the matter elds and couplings are symmetric under the interchange of 9- and 5- sectors, then the above 8 cases reduce to 4 inequivalent cases. This seems to be the generic situation in simple Type I model-building. To keep at least one sector of gauge elds visible (i.e., gauge coupling of order 1), we can take at most two T 2 s, say T1 and T3 , with large radii. Eq.(9) implies that the product of the two radii is 2 MP = 8m8 v1 v2 v3 s 22 32 2 m6 r2 r3 s (2 )6 2 (10) If they are equal, then the radius is around 10 12 m for ms = 1 TeV. If we want both G9 and G5 to be observable, we can take only one T 2 to have large radius. In the e ective action (9), we can take v3 large. This scenario is necessary in any one of the following situations: (i) the standard model is contained in one sector, say G9 , while a large gauge coupling from G5 may be needed for a strong interaction to generate dynamical supersymmetry breaking. (ii) the standard model is contained in both G9 and G5 , (for example QCD SU(3) in G9 while weak SU(2)L in G5 ). In this case, the G9 and G5 gauge couplings are in general di erent even at the string scale: 7 2 2 g3 = g9 = m2 v2 s , (2 )3 2 2 g2 = g5 = m2 v1 s (2 )3 (11) (iii) QCD SU(3) is inside the 9-branes while the weak SU(2)L comes from the diagonal spontaneous symmetry breaking of a SU(2) inside the 9-branes and a SU(2) inside the 5branes. In this case, the standard model gauge couplings g3 , g2 and g1 are in general di erent at the string scale, even if g9 = g5 . We shall illustrate each of these possibilities in the next section. From equation (4), r3 10 4 g9 g5 ms TeV 2 meter (12) If both g5 and g9 are of order unity, and ms is 1 TeV, then r is 10 4 meter. If g5 becomes small (equivalent to large radius r1 ), r3 just becomes even smaller. Let us go back to the general case with three types of 5-branes (as in Eq.(3)). Similar analysis is easy to carry out, so we shall simply restrict ourselves to a few comments. It is easy to see that under T-duality, the rule p p = 0 (mod 4) is preserved. If v3 gets large, we see that the gauge couplings of both the 9-brane gauge sector and the third type of 5-brane gauge sector become vanishingly small. As a consequence, the matter elds in this particular 59 sector will essentially decouple from all gauge interactions. They will still couple to other elds via other interactions, including gravity. So they are suitable candidates for dark matter. B. Non-supersymmetric String and the Cosmological Constant Supersymmetry was introduced originally to solve the hierarchy problem. Since this hierarchy problem disappears when the string scale is close to the weak scale, we should also consider non-supersymmetric Type I models. Generically, besides 9-branes, 7-, 5- and/or 3-branes may be present in a speci c model, depending on the details. We also expect a cosmological constant 4 to be present. Again, we can consider the inequivalent scenarios when the various tori become large. Besides the 9753-brane con guration, duality can bring us to the 7975-, 7575- and 7535-brane con gurations. Depending on the choice, taking one torus volume large will decouple gauge elds from one or more sectors (if they are present). The analysis is similar to that given for the supersymmetric case and will not be repeated here. There is one important di erence between the supersymmetric case and the nonsupersymmetric case. For supersymmetry to be unbroken, the 6 dimensional manifold must be a complex manifold. This means that T 6 can always be written as T 6 = T 2 T 2 T 2 , where the two radii in each T 2 are the same, as required by orbifold symmetry. (The exception is the case in which some of the T 2 s are twisted only by Z2 and not by other ZN twists. The two radii are unconstrained and one can choose freely only one of them to be large. However, orbifold compacti cations of Type I string theory involving this type of twists always give rise to non-chiral models [18]). In the non-supersymmetric case, it is possible that only one dimension has large radius (this breaks the complex structure). Let us comment on the cosmological constant. We have seen how a large Planck mass MP can be generated from a much smaller string scale ms . Naively, the same e ect happens to the cosmological constant. If there is a 10-dimensional cosmological constant 10 = m10 , then s 8 4 = 10 v1 v2 v3 , which is obviously unacceptable. Fortunately, this argument is incorrect. Recall the construction of the string model. We start from a 4-dimensional supersymmetric model toroidally compacti ed from 10 dimensions; it has no cosmological constant. We reduce the number of supersymmetries by orbifolding/orientifolding. The orbifolding of each of the three tori is needed to break the spacetime supersymmetry and generate chiral fermions, so the mechanism is intimately tied to D = 4 spacetime. This suggests that 4 = m4 . This is substantially smaller than the previous naive estimate. Unfortunately, s this is still unacceptably large, so we need to nd some mechanism to suppress it further. Now that we have seen how extra large dimensions can blow up the Planck mass, we are naturally led to ask if the reverse can suppress the cosmological constant. Suppose we construct a non-supersymmetric string model in 3 spacetime dimensions. (In the construction of non-supersymmetric models, we do not need to complexify the compacti ed dimensions.) So generically 3 = m3 . Now, let us take the radius r of one of s the compacti ed direction to be large, i.e., decompactify that direction. So the theory essentially describes a 4-dimensional spacetime. The 4-dimensional cosmological constant is given by 4 3 m3 s. r r (13) For ms = 1 TeV and r the size of the universe, 4 is small enough to be acceptable. This means the supersymmetry breaking mechanism within the string model-building must be intrinsically 3-dimensional. This imposes a strong constraint in non-supersymmetric string model-building. Generically, the theory can decompactify in other directions in the eld space, so that 4 ends up of the order of m4 . However, 4 measures the vacuum energy s density, so it is natural for it to choose the minimum energy path of decompacti cation. This imposes a strong constraint in model-building. Notice that this mechanism will not work if the string scale is around the GUT scale, as is the case in the old scenario. The above scenario is di erent from Witten s suggestion [20], which also utilizes the 3 spacetime dimensional picture. In 3 dimensional globally supersymmetric theories, the fermion-boson mass splitting m is zero, as naively expected, but becomes non-zero in supergravity models. This implies that m m2 /M, where m is the typical mass and M is the 3-dimensional Planck mass [21]. So the fermion-boson mass splittings are nonzero while 3 is zero. As we decompactify a direction with radius r, 4 clearly remains 2 zero. However, the 4-dimensional MP = M/r, so, for nite MP , M goes to in nity as r goes to in nity, and m goes to zero. This seems to imply that the decompacti cation of the 3-dimensional supergravity model yields 4-dimensional supergravity. So we believe that non-supersymmetric 4-dimensional models can come from the decompacti cation of 3-dimensional non-supersymmetric models, but not supersymmetric models. III. AN EXPLICIT STRING MODEL In this section, we use an explicit 4-dimensional chiral N = 1 supersymmetric Type I string model as an illustration of some of the ideas discussed above. Toroidal compacti cation of Type I string theory on a six dimensional torus T 6 gives rise to a four dimensional model with N = 4 supersymmetry. One can reduce the number of supersymmetries to 9 N = 1 by orbifolding. For example, take T 6 = T 2 T 2 T 2 , where each of the T 2 has a Z3 and a Z2 rotational symmetry. The Z3 generator g and the Z2 generator R acts on the complex coordinates z1 , z2 , z3 of the compacti ed dimensions as follows: gz1 = z1 , Rz1 = z1 , gz2 = z2 , Rz2 = z2 , gz3 = z3 Rz3 = z3 (14) (15) where = exp(2 i/3). The elements g and R generates the group Z6 . If we identify points in T 6 under this discrete rotational symmetry, the resulting orbifold M = T 6 /Z6 has SU(3) holonomy; only 1 of the 4 gravitinos are kept under the orbifold action. As a result, Type I string theory compacti ed on M has N = 1 supersymmetry in 4 dimensions. To compute the spectrum, it is convenient to view Type I string theory as Type IIB orientifold. Type IIB string theory has a worldsheet reversal symmetry. The orientifold projection reverses the parity of the closed string worldsheet (and hence interchanges the role of left- and right-movers in Type IIB theory). Gauging this worldsheet parity symmetry results in a theory of unoriented closed strings. Open strings and D-branes are introduced to cancel the divergences (tadpoles) from the Klein bottle amplitude (a one-loop amplitude for unoriented closed strings). The orientifold group O (the discrete symmetries of Type IIB theory that we are gauging) contains the elements and R. Tadpole cancelation requires introducing both D9- and D5-branes. Global Chan-Paton charges associated with the D-branes manifest themselves as gauge symmetry in space-time. As a result, there are gauge elds from both D9- and D5-branes. The details of the tadpole cancelation conditions and the construction of Z6 orientifolds can be found in appendix A. First, consider the case where the untwisted NS NS sector B eld background is zero; tadpole cancelation implies that n9 = n5 = 32, where n9 (n5 ) is the number of 9-branes (5-branes). This means that the total rank of the gauge group (which comes from both 9-branes and 5-branes) is 32. This model has gauge group [SU(6) SU(6) SU(4) U(1)3 ]2 and was rst constructed in Ref [15]. Although the gauge group contains the standard model gauge group SU(3) SU(2) U(1), the residual gauge symmetry is too large for the model to be phenomenologically interesting. In the presence of the untwisted NS NS sector B- eld background, it was shown [22,23] that the rank of the gauge group is reduced to 32/2b/2 . Here, b is the rank of the matrix Bij (i, j labels the complex coordinates of T 6 ). Since we are compactifying Type I string theory on a 6 dimensional manifold, b = 0, 2, 4, 6. The details of the construction of these models can be found in Appendix A. For b = 2, the model has [SU(2) SU(2) SU(4) U(1)3 ]2 gauge symmetry, which can be considered as a Pati-Salam like model with some extra global/gauge symmetry depending on the gauge coupling of the 9-brane and 5-brane gauge group. This is the model we are going to study in more details in this paper. For b = 4, the gauge group is [SU(2) SU(2) U(1)2 ]2 which is too small to contain the standard model. For b = 6, the gauge group is [SU(2) U(1)]2 , again does not contain the standard model. Let us discuss in more details the spectrum of the model with [SU(2) SU(2) SU(4) U(1)3 ]2 gauge symmetry. Open strings start and end on D-branes. Since there are two kinds of D-branes (9-branes and 5-branes), there are three types of open strings that we need to consider: 99, 55 and 59 open strings. The open string spectrum of this model is given in Table I. Here, we consider all D5-branes sitting at the same orbifold xed point. The fact that the 99 and the 55 sector have the same spectrum follows from T-duality. Since 10 open string has only two end-points, the charged matter elds are either bi-fundamentals or symmetric (or anti-symmetric) representations of the gauge groups. Notice that the rst and the second U(1) of both the 99 and the 55 gauge groups are anomalous, with U(1) anomaly equals 16 and +16 respectively. We can form a linear combination of these U(1) s such that only one of them is anomalous (this combination is given by Q1 Q2 where Q1,2 are the rst and the second U(1) charge respectively). By the generalized Green-Schwarz mechanism [24], some of the elds charged under the anomalous U(1) will acquire vevs to cancel the Fayet-Illiopoulos D-term. In addition to the open string spectrum, there are also closed string states. Since they do not carry Chan-Paton factors, they are singlets under the gauge group. We see that the model has enough realistic features so that we can use it to study various scenarios discussed earlier. Here we shall consider three di erent possible ways that the model may be interpreted as an approximate way to describe nature. There is one chiral family in the rst scenario, two chiral families in the second scenario, and three chiral families in the third scenario. Our description is sketchy and we shall simply assume the dynamics needed to behave in the way we like. Our purpose is to illustrate some of the features of brane-physics, and draw attention to the model s features that are generic to other Type I string models. We shall not worry about which (if any) of the three scenarios is actually realized by the string dynamics. A. Scenario 1 To describe this scenario, let us go to the T-dual picture where there are two di erent types of 5-branes (as in Eq.(9)). For convenience, we will still refer to the 5-branes coming from T-dualizing the 9-branes as the 9-branes. Suppose the standard model SU(3) SU(2) U(1) gauge group comes from the 9-brane sector only. In this model, the gauge group is SU(4) SU(2)L SU(2)R , and the 99 sector matter elds are singlet under the 5-brane gauge group. We can make the 5-brane gauge coupling relatively strong, so that SU(4)5 gets strong and may trigger dynamical supersymmetry breaking. It may also cause spontaneous symmetry breaking of SU(2)5 SU(2)5 so that the 55 and the 59 sector matter elds become heavy. In any case, let us focus our attention on the 99 sector. Here some of the low dimension terms in the superpotential is given by (see Table I for notations) W = (U1 Q2 + U2 Q1 ) H + (U1 S2 + U2 S1 ) U3 + (Q1 S4 + Q2 S3 ) Q3 + . . . (16) where we have suppressed the dependence and the exact coe cients of the couplings. (The dependence of N-point couplings is g N 2 (N 2)/2 ). To break the gauge group down to SU(3) SU(2) U(1), we can move some of the 9-branes away from each other. This mechanism is equivalent to the spontaneous symmetry breaking (SSB) action of the Higgs eld in the e ective eld theory; that is, we can give vacuum expectation value to the Higgs superpartner of one of the U elds. Since the U elds are charged under U(4) SU(4) U(1), it is more appropriate to consider SU(4) SU(2)L SU(2)R U(1) SU(3) SU(2)L U(1) U(1) U(1), 1 1 (4, 1, 2)( 1) = (3, 1)( 3 , 1 , 1) (3, 1)( 3 , 1 , 1) (1, 1)(1, 1 , 1) (1, 1)(1, 1 , 1) 2 2 2 2 (17) 11 Here, the rst U(1) charge is the B L number, the second U(1) charge is IR = SU(2)R isospin, and the third U(1) charge is 3B + L which comes from the decomposition U(4) SU(4) U(1). Notice that the U(1) hypercharge Y = B L + 2IR and the baryon number B = (B L + 3B + L)/4. Therefore, under SU(4) SU(2) SU(2) U(1) SU(3) SU(2)L U(1)Y U(1)B U(1)3B+L 4 , (4, 1, 2)( 1) = (3, 1)( 2 , 1 , 1) (3, 1)( 3 , 1 , 1) (1, 1)(2, 0, 1) (1, 1)(0, 0, 1) 3 3 3 (18) Here the U(1)s are independent but not orthogonal. If the scalar (1, 1)(0, 0, 1) acquires a vev, U(1)3B+L is broken, and the elds Qi and Ui become 11 (4, 2, 1)(+1) = (3, 2)( 3 , 3 ) (1, 2)( 1, 0) 1 (4, 1, 2)( 1) = (3, 1)( 2 , 1 ) (3, 1)( 4 , 3 ) (1, 1)(2, 0) (1, 1)(0, 0) 3 3 3 (19) We see that the Qi and Ui yield precisely one chiral and one vector (i.e., one chiral plus one anti-chiral) family of the standard model SU(3) SU(2)L U(1)Y U(1)B . This also splits the SU(2)R doublet H into two standard models doublets H1 and H2 : (1, 2, 2)(0) = (1, 2)(1, 0) (1, 2)( 1, 0) (20) The term H1 H2 does not appear as lower order terms in the superpotential. In this scenario where there are only 9-branes, g3 = g2 = g9 , and gY = 3 g9 at the string scale. 5 Consider the chiral fermions in the 99 sector before the electroweak symmetry breaking. There is 1 chiral family and 1 vector (chiral plus anti-chiral) family. Generically, a linear combination of U1 and U2 will pair up with U3 to become heavy, while the other linear combination will remain massless. After the SSB to SU(3) SU(2) U(1), this U1 + U2 combination gives the right-handed quarks and leptons. Similarly, a linear combination of Q1 and Q2 may pair up with Q3 to become heavy, while the other linear combination will remain massless. They yield the weak isodoublets of quarks and leptons. So we see that the model has only one chiral family of quarks and leptons. Now, notice that there are no baryon number violating terms in the superpotential. This is due the third U(1) symmetry. The quarks have U(1)3 charge +1, while the antiquarks have charge 1. The presence of such a U(1) associated with the SU(4) is a generic feature of brane physics (the U(1) factor is the center of mass of the D-branes). So we should expect the conservation of the baryon number as a generic feature. Suppose, in Eq.(9), it is v1 , not v3 , that is becoming very large. In this case, the 5-brane sector gauge coupling becomes vanishingly small. So the 5-brane matter elds essentially decouple and can be candidates for dark matter. Notice that these 5-brane gauge elds do not couple to the visible matter elds (except by very weak gravitational interaction), so they are essentially invisible. B. Scenario 2 Suppose the QCD SU(3) comes from the 9-brane sector while the weak SU(2) comes from the 5-brane sector. To be speci c, the gauge group is SU(4)9 SU(2)5 SU(2)5 . The 12 quarks and leptons come from the 59 sector while the Higgs eld comes from the 55 sector. There is a Z2 symmetry under which all matter elds are odd while the Higgs eld is even. The superpotential is given by W= (U1 Q2 + U2 Q1 ) H + (U1 S2 + U2 S1 ) U3 + (Q1 S4 + Q2 S3 ) Q3 + (u1 q2 + u2 q1 ) h + (u1 s2 + u2 s1 ) u3 + (q1 s4 + q2 s3 ) q3 2 2 2 4 2 2 2 2 + i=1 j=1 2 4 Ui Qj h + i j u3 + i j H + i=1 j=3 2 2 i=1 j=1 i Uj U3 + i=1 j=1 i Qj Q3 (21) + i=1 j=3 i j q3 + . . . i=1 j=1 Again, we have suppressed the dependence and the exact coe cients of the couplings. As before, vev for one of Ui elds induces SSB: SU(4) SU(2)L SU(2)R SU(3) SU(2) U(1)Y . There are two families of quarks and leptons. As in the previous scenario, conservation of the third U(1) charge prevents any perturbative baryon number violating term. The analysis is quite similar to the above scenario, so we shall not repeat. A crucial di erence is that, even at the string scale, the QCD coupling g3 = g9 and the weak coupling g2 = g5 ; they need not be the same. From Eq.(11), we see that their relative values depend on the compacti cation volumes. The hypercharge U(1) coupling is a function of g3 and g2 : 3g9 g5 gY = (22) 2 2 3g9 + 2g5 If g9 = g5 = g, then gY = 3 g 5 at the string scale. C. Scenario 3 We see that the model has 1 chiral family in the 99 sector and 2 chiral families in the 59 sector. Furthermore, there is a Z2 symmetry between the 9-brane and the 5-brane. We can construct a new model by gauging this Z2 symmetry, or part of it, i.e., a Z2 orbifold of the original model. The Z2 symmetry we want to orbifold is an outer-automorphism. In terms of current algebra in conformal eld theory, such an orbifold converts level-1 current algebra to level-2 current algebra. Similar procedures can be carried out in the e ective eld theory without having to impose the condition that g9 = g5 [25]. The basic idea is as follows. We start from a product gauge group SU(N) SU(N), with gauge couplings g and g respectively. By giving vev to the bi-fundamental eld = (N, N) along the at direction = vIN (where IN is an N N identity matrix), the gauge group is broken to SU(N). In the speci c model that we consider in this paper, the elds 1 , 2 are bi-fundamentals under the U(2)9 U(2)5 gauge group. Similarly, 1 , 2 are bi-fundamentals under U(2) 9 U(2) 5 . By giving vevs to i s and i s of the above form (with N = 2): U(2)9 U(2)5 SU(2)L U(1) U(2) 9 U(2) 5 SU(2)R U(1) 13 (23) The gauge couplings of SU(2)L , SU(2)R and the accompanying U(1) s are given by g = 2 2 g9 g5 / g9 + g5 . The U(1)s are broken by the Green-Schwarz mechanism, so the resulting model has Pati-Salam gauge group SU(4)9 SU(2)L SU(2)R with additional custodial SU(4)5 U(1)2 symmetry. There are three families of chiral fermions under the PatiSalam gauge group. Two of them come from the 59 sector: Ui give rise to two families of right-handed quarks and leptons, while Qi give rise to two families of left-handed quarks and leptons. The remaining family comes from the 99 sector: the right-handed quarks and leptons come from a linear combination of U1 and U2 , and the left-handed quarks and leptons come from a linear combination of Q1 and Q2 . It is interesting to note that one of the three families has a di erent origin. Whether this will o er an explanation to the fact that there is one heavy family deserves further investigation. Note that weak interaction universality is automatic. The SSB of SU(4) SU(2)L SU(2)R SU(3) SU(2)L U(1)Y is essentially the same as in the rst scenario, that is, giving a vev to the one of the U elds. The gauge couplings of the standard model gauge groups do not need to meet at the string scale and are given by g3 = g9 g2 = gY = so that sin2 W is given by 2 3g3 sin W = 2 (25) 2 6g3 + 2g2 If g9 = g5 = g, we see that g3 = g, g2 = g/ 2, gY = 3 g and sin2 W = 3/7 at the string 8 scale. What about the U(1)B gauge boson associated with the baryon number conservation? Even if its coupling is very weak, it certainly must pick up a mass for the model to be phenomenologically viable. Suppose spontaneous symmetry breaking takes place when one of the scalar eld charged under this U(1)B develops a vev. This vev v must be small so that the baryon number violating terms are suppressed by powers of v/ms . The exact amount of suppression depends on the details. Consider the scenario in which this spontaneous symmetry breaking happens simultaneously when QCD SU(3) is broken as well, which implies that free quarks and gluons can exist. Suppose the U(1)B boson picks up a mass , then, following Ref [26], there are free quarks and gluons with mass about (1GeV)2 / . For = 10 keV, we see that a free quark or a free gluon will have a mass around 100 TeV. 2 g9 g5 2 2 g9 + g5 3g3 g2 2 2 3g3 + 2g2 (24) D. Another String Model Let us consider another N = 1, D = 4 chiral Type I model, namely the Z3 Z2 Z2 model recently constructed by Kakushadze [27]. This model has 9-branes and three types of 14 5-branes as given in Eq.(3), all of them have identical gauge groups, so the resulting gauge group is [U(6) SO(5)]4. Let us assume that QCD SU(3) comes from one of the U(6), while the weak SU(2) comes from one of the SO(5). It seems there are enough Higgs elds to break one of the SU(6) down to SU(4) and then to SU(3), and one of the SO(5) to SU(2). Again, we see that the U(1) carrying baryon numbers is present. However, this U(1) is anomalous, so it will pick up a mass via the Green-Schwarz mechanism automatically. Consider the situation where the torus T3 is very large. Following Eq.(3), we see that both the gauge couplings of the 9-brane and the third 5-brane sectors become vanishingly small. In particular, the 9-brane matter elds essentially decouple and can be candidates for dark matter. IV. DISCUSSION It is clear that, among other properties, perturbative D = 4, N = 1 supersymmetric, chiral Type I string models have some very attractive features for the study of the TeV scale string scenario : (i) Gravitons live in the bulk while gauge and charged matter elds live on the branes. (ii) The presence of U(1)s (associated with the centers of mass of the branes) which help to stabilize the proton. (iii) The identical nature of 9-brane gauge group and 5-brane gauge groups (if present) allows di erent standard model gauge couplings at the string scale. (iv) the presence of bi-fundamental matter elds allows diagonal spontaneous symmetry breaking; again this mechanism allows di erent standard model gauge couplings at the string scale. This feature may validate the weak string coupling description of Type I string. These properties are quite compatible with present experiments and allow the future tests of the extra large dimension scenario. There are a number of reasons why this TeV scale superstring scenario was not seriously considered earlier. In the old string phenomenology framework, (i.e., pre-string-duality days), gravity and gauge interactions live in the same space. Since gauge interactions clearly live in an e ective 4 spacetime dimensions, at least up to the electroweak scale, the largest 1 the extra dimensions can be is MEW , as considered in [2]. However, generically, the string scale is above MGU T to satisfy the proton decay bound. The reason is following. Before our understanding of string duality, all phenomenologically interesting string models are within the heterotic string theory in the conformal eld theory framework, where the original rank of the gauge group is 22. Although the rank of the massless gauge symmetry can be substantially reduced, the massive sector retains (at least some of) the original large group feature. A typical heterotic string model that contains the standard model of strong and electroweak interactions in its low energy sector will contain massive bosons that can mediate proton decay. Since these massive bosons have masses of the string scale, we must keep the string scale high enough, say around MGU T , to satisfy the proton decay bound. Generically, the proton decay bound requires the absence of dimension-4 and -5 baryon-number violating operators. If the string scale is around 1 TeV, the higher-dimensional (up to dimension-18) baryon-number violating operator terms can be dangerous. To prevent their appearance, some discrete symmetry or custodial gauge symmetry is necessary. However, the presence of such symmetry is not generic in the old heterotic string theory. In comparison, the U(1) s 15 in Type I strings are very generic; they correspond to the center of mass of the D-branes. As we have seen in some cases, the di culty is how to make them massive. Suppose we consider the heterotic string beyond the world-sheet construction. For example, solitonic 5-branes can contribute to the massless spectrum in non-perturbative heterotic string, which may have properties that are suitable for phenomenology. However, the analysis of non-perturbative heterotic string is di cult. Hopefully, duality between the Type I and the heterotic string [28] allows us to treat more fully the non-perturbative e ects. The string model that we have presented here is constructed from perturbative Type I string theory. If the gauge coupling is of order 1, and we expect ms R > 1, Eq.(9) implies that the string coupling is small. One would still like to know the energy regime where the perturbative Type I picture may become invalid [29]. Naively, one may expect the 4dimensional low energy e ective eld theory to be valid at momentum scales below r 1 . This is because the low energy e ective couplings are small (except for the strong QCD coupling). Quantum corrections coming from the massive string modes are negligible at low energies. Above this scale, one expects the (4 + n)-dimensional e ective eld theory to be valid. At scales above R 1 but below ms , we should move from e ective eld theory to string theory, where perturbative Type I string theory is likely to be valid. When the energy-momentum scale is around the string scale ms , the Type I string perturbative description may or may not remain valid. This may depend on the particular scenario and the particular process one is interested in. In view of Type I heterotic duality, one would ask if the weakly-coupled heterotic string description should take over in this regime. However, the techniques in constructing heterotic string vacua with NS 5-branes (the NS 5-branes are dual to the D5branes in the Type I theory) are not well developed. Since Type I string theory provides a natural setting to realize the idea of extra large dimensions, it is likely that the scenarios that we presented here capture the important features which persist in the large regime. V. COMMENTS It is interesting to compare the merits of the two scenarios of string phenomenology: the old scenario with string scale around the GUT scale, and the new scenario [8,5,6] with the string scale around the electroweak scale. Experimentally, the new scenario is clearly superior. High energy scatterings can probe the extra small dimensions while gravity can probe the extra large dimensions. These experiments are coming in the near future. If this scenario is correct, we can expect a lot of experimental information on the detailed structure, which can provide valuable guidance on the precise way nature is realized within string theory. At this moment, before the availability of the experimental data, we can still ask which scenario is more appealing from the theoretical perspective. Without detailed realistic models, any comparison is quite subjective. Nevertheless, we believe the exercise can be illuminating. A scenario may be deemed more natural than another if it has fewer number of disparate scales. Let us give a naive counting of the number of scales in each scenario. In the old scenario where the string scale is around the Planck scale MP , we also have the electroweak scale. The Planck scale MP is about three orders of magnitude above the GUT scale; this discrepancy is di erent enough to require some new physics ingredients to explain. Let us count this situation as 3 scales. The quark and lepton masses are very 16 di erent. For example, the mass of the top quark is more than 105 that of the electron. Let us assume that the fermion mass splitting introduces another scale that needs understanding. Including the cosmological constant, we have 5 di erent scales. Let us take one of them, say the electroweak scale, to set the overall normalization. Uni cation of the gauge couplings provides a nice explanation of the GUT scale, so there remains 3 scales that remain to be understood. If one wants to treat the GUT scale and the Planck scale as close enough to be considered as one, we still have two scales that beg for an explanation. In the new scenario, we have the string scale around 1 TeV, which is close enough to the electroweak scale to be considered as a single scale. Similarly, the small compacti cation radii between the electroweak and the string scale should not be treated as new scales. Suppose the standard model gauge couplings are uni ed at the string scale. Since the gauge and matter elds are living in extra dimensions, say 8 total spacetime dimensions, the gauge couplings are irrelevant operators. So these couplings run as powers and diverge rapidly as we move to lower energies. Once the energies involved go below the scale of the small radii, they become marginal operators and vary only logarithmically. Suppose the Yukawa couplings at the string scale are di erent but comparable. Again, as irrelevant operators, they diverge rapidly, so they can easily di er by orders of magnitude at scales below the electroweak scale [6]. This provides a qualitative explanation for the fermion mass hierarchy. So we shall not count the fermion mass splittings as an extra scale. Now we can count the number of scales in this scenario : using the string/electroweak scale to set the overall normalization, we have only two scales that beg for an explanation: the cosmological constant and the large radius of r = 1 mm= 1016 /TeV. (In fact, a cosmological constant of the order r 4 is quite compatible with observations.) Theoretically, it seems that the new scenario looks slightly better than, or at least comparable to, the old scenario. Experimentally, the new scenario is much more testable/reachable and hence superior. So overall, the new scenario certainly deserves further investigation. ACKNOWLEDGMENTS We would like to thank Philip Argyres, Chong-Sun Chu, Keith Dienes, Alon Faraggi, Piljin Yi, and especially Zurab Kakushadze for valuable discussions. The research of G.S. and S.-H.H.T. was partially supported by the National Science Foundation. G.S. would like to thank the kind hospitality of the Institute for Theoretical Physics at Stony Brook during his stay. G.S. would also like to thank Joyce M. Kuok Foundation for nancial support. APPENDIX A: CONSTRUCTION OF THE MODEL In this appendix, we give the details of how to construct from D-branes and orientifolds the N = 1, D = 4 chiral string model with [SU(4) SU(2) SU(2) U(1)3 ]2 gauge symmetry presented in Section III. The model also exhibits some novel features [22] of the untwisted NS NS sector B- eld background recently discussed in [23]. We start from Type IIB string theory compacti ed on T 6 = T 2 T 2 T 2 , where each of the two-tori has a Z3 and a Z2 rotational symmetry. The Z3 generator g and the Z2 generator R acts on the complex coordinates z1 , z2 , z3 of T 6 as follows: 17 gz1 = z1 , Rz1 = z1 , gz2 = z2 , Rz2 = z2 , gz3 = z3 Rz3 = z3 (A1) (A2) where = exp(2 i/3). The elements g and R together generate the Abelian group Z6 . Let us consider Type I string theory compacti ed on the toroidal orbifold M = T 6 /Z6 . It is convenient to view Type I compacti cation as Type IIB orientifold. The orientifold projection reverses the parity of the closed string worldsheet. This results in a theory of unoriented closed strings. One-loop niteness generically requires introducing open strings starting and ending on D-branes, so that the divergences (tadpoles) coming from the cylinder, Mobius strip and Klein bottle amplitudes cancel. The orientifold group O = { a Rb g c | a = 0, 1 ; b = 0, 1 ; c = 0, 1, 2} contains both the elements and R. Therefore, one has to introduce both D9- and D5-branes to cancel the tadpoles. The global Chan-Paton charges associated with the D-branes manifest themselves as gauge symmetries in space-time. Hence, there are gauge bosons from both 99 and 55 open strings. The orbifold action on the Chan-Paton factors is described by unitary matrices k,p that acts on the string end-points (k labels the orbifold group element, p labels the type of branes). Let | , ij be an open string state, where is the state of the worldsheet elds and i, j are the Chan-Paton factors of the string end-points (the open string starts on a p-brane and ends on a q-brane). The action of the orbifold element k is given by k: 1 | , ij ( k,p)ii |k , i j ( k,q )j j (A3) Tadpole cancelation determines the form of the k,p matrices. There are two types of constraints that we need to consider. The rst one comes from the cancelation of the untwisted tadpoles for the D9-branes and the D5-branes respectively. This type of constraint determines the number of D9- and D5-branes. In the general case where the untwisted NS NS sector B- eld can be non-vanishing (with b equals the rank of the matrix Bij , which is always even), tadpole cancelation for the untwisted R-R 10-form potential gives [22,23] 2b (Tr( 1,9 ))2 2b/2 64Tr( 1,9 ) + (32)2 = 0 . (A4) Therefore, the number of D9-branes is given by n9 = 32/2b/2 . Similarly, tadpole cancelation condition for the untwisted R-R 6-form potential gives [23] (Tr( 1,5 ))2 64 1 Tr( 1,5) + b (32)2 = 0 . b/2 2 2 (A5) Therefore, the number of D5-branes is given by n5 = 32/2b/2 . This was also expected from T-duality between D9- and D5-branes. The other constraint comes from tadpole cancelation of the twisted R-R 6-form potential. Since the twisted closed string states propagating in the tree-channel do not have momentum or winding, the twisted tadpoles remain the same in the presence of the untwisted NS NS sector B- eld background (the e ect of which is to shift the left- plus right-moving momentum lattice). The twisted tadpoles for ZN orientifolds in 6 dimensions have been computed in [30 32] and generalized to 4 dimensions in Ref [12,14,15]. Here, we state the results for the Z6 case: 18 Tr ( g,p ) = ( 1)b/2 32 [cos( /3)]3 = ( 1)b/2 4 Tr ( R,p ) = Tr ( Rg,p ) = 0 (A6) (A7) Let us consider the solutions to the above tadpole cancelation conditions for all possible values of b: For b = 0, n9 = n5 = 32 R,p = diag (iI16 , iI16 ) , 2 (A8) 2 g,p = diag I6 , I6 , I4 , I6, I6 , I4 . (A9) where IM is an M M identity matrix. The gauge group from the 99 open strings is SU(6) SU(6) SU(4) U(1)3 . The 55 open strings also give rise to the gauge group SU(6) SU(6) SU(4) U(1)3 if the D5-branes are located at the same xed point. The total rank of the gauge group is 32. This model was rst constructed in Ref [15]. For b = 2, n9 = n5 = 16. R,p = diag (iI8 , iI8 ) , (A10) . (A11) g,p = diag I2 , 2I2 , I4 , I2, 2 I2 , I4 The gauge group (from both 99 and 55 open strings) is [SU(2) SU(2) SU(4) U(1)3 ]2 . The total rank of the gauge group is 16. This is the model that we study in this paper. For b = 4, n9 = n5 = 8. R,p = diag (iI4 , iI4 ) , 2 (A12) 2 g,p = diag I2 , I2 , I2, I2 . (A13) The gauge group (from both 99 and 55 open strings) is [SU(2) SU(2) U(1)2 ]2 . The total rank of the gauge group is 8. For b = 6, n9 = n5 = 4. R,p = diag (iI2 , iI2 ) , g,p = I4 . (A14) (A15) The gauge group (from both 99 and 55 open strings) is [SU(2) U(1)]2 . The total rank of the gauge group is 4. The gauge groups of the models for b = 4, 6 are too small to accommodate the standard model, which make them phenomenologically uninteresting. We will focus on the b = 2 model with [SU(2) SU(2) SU(4) U(1)3 ]2 gauge symmetry. To construct the open string spectrum, we keep all physical states that are invariant under the orbifold action. There are contributions from 99, 55 and 59 open strings. As pointed out in Ref [23], the 59 open string sector states come with a multiplicity = 2b/2 . (Recall that without B- eld, the multiplicity of states in the 59 sector was one per con guration of ChanPaton charges [31,32]). The open string spectrum of the [SU(2) SU(2) SU(4) U(1)3 ]2 model is given in Table I. 19 APPENDIX B: H-CHARGES, SCATTERINGS AND COUPLINGS In this appendix, we review the conformal eld theory techniques in calculating scattering amplitudes (and hence couplings) in orbifold models. In Type I string theory, closed string sector only gives rise to gauge singlets. We will therefore focus on the couplings between open string states. In the standard orbifold formalism, the internal part of the worldsheet supercurrent can be written as i3 a a i 3 i a TF = X + H.c. = e X a + H.c. , 2 a=1 2 a=1 where a are complex world-sheet fermions, which can be bosonized: a = exp(i a ) = exp(iH ) , a = exp( i a ) = exp( iH ) . (B1) (B2) Here, H (known as the H-charge) equals (1, 0, 0), (0, 1, 0) or (0, 0, 1) for a = 1, 2, 3. The supercurrent is therefore a linear combination of terms with well de ned H-charges. In the covariant gauge, we have the reparametrization ghosts b and c, and superconformal ghosts and [33]. It is most convenient to bosonize the , ghosts: = e , = e , where and are auxiliary fermions and is a bosonic ghost eld obeying the OPE 1 (z) (w) log(z w). The conformal dimension of eq is 2 q(q + 2). In covariant gauge, vertex operators are of the form V (z) ij , where V (z) is a dimension 1 operator constructed from the conformal elds (which include the longitudinal components as well as the ghosts), and ij is the Chan-Paton wavefunction. The vertex operators for space-time bosons carry integral ghost charges (q Z) whereas for space-time fermions the ghost charges are halfintegral (q Z + 1 ). Here, q speci es the picture. The canonical choice is q = 1 for 2 1 space-time bosons and q = 2 for space-time fermions. We will denote the corresponding vertex operators by V 1 (z) and V 1 (z), respectively. Vertex operators in the q = 0 picture 2 (with zero ghost charge) is given by picture changing : V0 (z) = lim e TF (z)V 1 (w) . w z (B3) One can see that besides the supercurrent, open string states also carry H-charges. The vertex operator for gauge bosons in the 1 picture is given by ij where is the spacetime index. Therefore, they do not carry H-charges. On the other hand, the vertex operator for matter elds in 99 and 55 sector is given by a ij . Hence, in the 1 picture, H = (1, 0, 0), (0, 1, 0) or (0, 0, 1) depending on which worldsheet fermion is excited. The moding of the worldsheet fermions in the 59 sector is di erent from that in the 99 sector. Therefore, in the 1 picture, matter elds in the 59 sector carry half-integral H-charges instead of integral H-charges. The H-charges of the massless elds of the [SU(4) SU(2) SU(2) U(1)3 ]2 model is given in Table I. Having constructed the vertex operators for the massless states, one can in principle compute the scattering amplitudes, or the corresponding couplings in the superpotential. The coupling of M chiral super elds in the superpotential is given by the scattering amplitude of 20 the component elds in the limit when all the external momenta are zero. Due to holomorphicity, one needs to consider only the scatterings of left-handed space-time fermions, with vertices V 1/2 (z), and their space-time superpartners. Since the total ghost charge in any tree-level correlation function is 2, it is convenient to choose two of the vertex operators in the 1/2-picture, one in the 1-picture, and the rest in the 0-picture. Using the SL(2, C) invariance, the scattering amplitude is therefore M AM = gst 2 Tr 1 2 M dz4 dzM V 1 (0)V 1 (1)V 1 ( )V0 (z4 ) V0 (zM ) , 2 2 (B4) where we have normalized the c ghost part of the correlation function c(0)c(1)c( ) to 1. To obtain the open string scattering amplitudes, we have to take the integration variables zi to the real axis, with zi > zi+1 . Now the terms in the superpotential can be read o directly from the resulting scattering amplitudes. For a non-zero coupling, the sum of the H-charges must be zero in the corresponding scattering amplitude. Note that the supercurrent carries terms with di erent H-charges. Because of picture changing, H-charges are not global charges even though they must be conserved exactly. In additional to the H-charge conservation, there is also a discrete Z2 symmetry coming from the orbifold twist. For the couplings to be non-zero, the total twist in the scattering amplitude (B4) must be an integer. 21 TABLES Sector Field S1 S2 S3 S4 U1 U2 U3 Q1 Q2 Q3 H s1 s2 s3 s4 u1 u2 u3 q1 q2 q3 h , 1 2 1 , 2 Q1 , Q2 U1 , U2 1 , 2 3 , 4 [SU (2) SU (2) SU (4) U (1)3 ]2 (3, 1, 1; 1, 1, 1)(+2, 0, 0; 0, 0, 0)L (3, 1, 1; 1, 1, 1)(+2, 0, 0; 0, 0, 0)L (1, 3, 1; 1, 1, 1)(0, 2, 0; 0, 0, 0)L (1, 3, 1; 1, 1, 1)(0, 2, 0; 0, 0, 0)L (2, 1, 4; 1, 1, 1)( 1, 0, 1; 0, 0, 0)L (2, 1, 4; 1, 1, 1)( 1, 0, 1; 0, 0, 0)L (2, 1, 4; 1, 1, 1)( 1, 0, +1; 0, 0, 0)L (1, 2, 4; 1, 1, 1)(0, +1, +1; 0, 0, 0)L (1, 2, 4; 1, 1, 1)(0, +1, +1; 0, 0, 0)L (1, 2, 4; 1, 1, 1)(0, +1, 1; 0, 0, 0)L (2, 2, 1; 1, 1, 1)(+1, 1, 0; 0, 0, 0)L (1, 1, 1; 3, 1, 1)(0, 0, 0; +2, 0, 0)L (1, 1, 1; 3, 1, 1)(0, 0, 0; +2, 0, 0)L (1, 1, 1; 1, 3, 1)(0, 0, 0; 0, 2, 0)L (1, 1, 1; 1, 3, 1)(0, 0, 0; 0, 2, 0)L (1, 1, 1; 2, 1, 4)(0, 0, 0; 1, 0, 1)L (1, 1, 1; 2, 1, 4)(0, 0, 0; 1, 0, 1)L (1, 1, 1; 2, 1, 4)(0, 0, 0; 1, 0, +1)L (1, 1, 1; 1, 2, 4)(0, 0, 0; 0, +1, +1)L (1, 1, 1; 1, 2, 4)(0, 0, 0; 0, +1, +1)L (1, 1, 1; 1, 2, 4)(0, 0, 0; 0, +1, 1)L (1, 1, 1; 2, 2, 1)(0, 0, 0; +1, 1, 0)L 2(2, 1, 1; 2, 1, 1)(+1, 0, 0; +1, 0, 0)L 2(1, 2, 1; 1, 2, 1)(0, 1, 0; 0, 1, 0)L 2(1, 1, 4; 1, 2, 1)(0, 0, +1; 0, +1, 0)L 2(1, 1, 4; 2, 1, 1)(0, 0, 1; 1, 0, 0)L 2(1, 2, 1; 1, 1, 4)(0, +1, 0; 0, 0, +1)L 2(2, 1, 1; 1, 1, 4)( 1, 0, 0; 0, 0, 1)L (H1 , H2 , H3 ) 1 (+1, 0, 0) (0, +1, 0) (+1, 0, 0) (0, +1, 0) (+1, 0, 0) (0, +1, 0) (0, 0, +1) (+1, 0, 0) (0, +1, 0) (0, 0, +1) (0, 0, +1) (+1, 0, 0) (+1, 0, 0) (+1, 0, 0) (0, +1, 0) (+1, 0, 0) (0, +1, 0) (0, 0, +1) (+1, 0, 0) (0, +1, 0) (0, 0, +1) (0, 0, +1) (+ 1 , + 1 , 0) 2 2 (+ 1 , + 1 , 0) 2 2 (+ 1 , + 1 , 0) 2 2 (+ 1 , + 1 , 0) 2 2 (+ 1 , + 1 , 0) 2 2 (+ 1 , + 1 , 0) 2 2 (H1 , H2 , H3 ) 1/2 1 (+ 1 , 1 , 2 ) 2 2 1 ( 1 , + 1 , 2 ) 2 2 1 (+ 1 , 1 , 2 ) 2 2 1 1 1 ( 2 , + 2 , 2 ) 1 (+ 1 , 1 , 2 ) 2 2 1 1 1 ( 2 , + 2 , 2 ) 1 ( 1 , 1 , + 2 ) 2 2 1 (+ 1 , 1 , 2 ) 2 2 1 1 1 ( 2 , + 2 , 2 ) 1 ( 1 , 1 , + 2 ) 2 2 1 1 1 ( 2 , 2 , + 2 ) 1 (+ 1 , 1 , 2 ) 2 2 1 (+ 1 , 1 , 2 ) 2 2 1 1 1 (+ 2 , 2 , 2 ) 1 ( 1 , + 1 , 2 ) 2 2 1 (+ 1 , 1 , 2 ) 2 2 1 1 1 ( 2 , + 2 , 2 ) 1 ( 1 , 1 , + 2 ) 2 2 1 1 1 (+ 2 , 2 , 2 ) 1 ( 1 , + 1 , 2 ) 2 2 1 ( 1 , 1 , + 2 ) 2 2 1 1 1 ( 2 , 2 , + 2 ) 1 (0, 0, 2 ) 1 (0, 0, 2 ) 1 (0, 0, 2 ) 1 (0, 0, 2 ) 1 (0, 0, 2 ) 1 (0, 0, 2 ) Open 99 Open 55 Open 59 TABLE I. The massless open string spectrum of the 4-dimensional Type I Z6 orbifold model with N = 1 space-time supersymmetry and gauge group [SU (2) SU (2) SU (4) U (1)3 ]2 . The U (1) s come from the traces of U (2) , U (2) and U (4) respectively. The H-charges in both the 1 picture and the 1/2 picture for states in the open string sector are also given. The vector multiplets are not shown. The closed string sectors give rise to the gauge singlets and the gravity supermultiplet. The H-charges are explained in Appendix B. 22 REFERENCES [1] U. Amaldi, W. de Boer and H. Furstenau, Phys. Lett. B260 (1991) 447; C. Giunti, C.W. Kim and U.W. Lee, Mod. Phys. Lett. A6 (1991) 1745; P. Langacker and M. Luo, Phys. Rev. D44 (1991) 817; J. Ellis, S. Kelley and D.V. Nanopoulos, Phys Lett. B249 (1990) 441. For a review of the string perspective, see e.g., K.R. Dienes, Phys. Rept. 287 (1997) 447, hep-th/9602045. [2] I. Antoniadis, Phys. Lett. 246B (1990) 377; I. Antoniadis, C. Munoz and M. Quiros, Nucl. Phys. 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