Moduli Stabilization for Intersecting Brane Worlds in Type 0 String Theory

Physics Letters B 532 (2002) 141–151 www.elsevier.com/locate/npe

Moduli stabilization for intersecting brane worlds in type 0 string theory

Ralph Blumenhagen a,BorisKörsb, Dieter Lüst a

a Humboldt Universität zu Berlin, Institut für Physik, Invalidenstr. 110, 10115 Berlin, Germany b Spinoza Institute, Utrecht University, Utrecht, The Netherlands Received 20 February 2002; accepted 27 February 2002 Editor: M. Cveticˇ

Abstract Starting from the non-supersymmetric, tachyon-free orientifold of type 0 string theory, we construct four-dimensional brane world models with D6-branes intersecting at angles on internal tori. They support phenomenologically interesting gauge theories with chiral fermions. Despite the theory being non-supersymmetric the perturbative scalar potential induced at leading order is shown to stabilize geometric moduli, leaving only the dilaton tadpole uncanceled. As an example we present a three generation model with gauge group and fermion spectrum close to a left–right symmetric extension of the Standard Model.  2002 Published by Elsevier Science B.V.

1. Introduction no-force law. The most radical signal of an inherent instability is the presence of tachyons in the spectrum, scalars with a negative tree level mass squared. Since it was discovered that one can have string ex- By now, the open string tachyons localized on some citations as light as a few TeV by allowing for large lower-dimensional defect, a D-brane, in the entire internal compactification spaces [1,2], string models ten-dimensional spacetime are rather well understood without supersymmetry received new attention as phe- [3–5]. They are unstable modes of the gauge theory nomenologically plausible scenarios. This may either sector and their condensation translates into a decay involve string theories without any supersymmetry al- of these D-branes into stable configurations but does ready in ten dimensions or compactifications of orig- not affect equally drastically the background space inally supersymmetric theories with supersymmetry which the branes are wrapped on. The latter is in- breaking at the string scale. A crucial problem in these deed the case for closed string tachyons, unstable grav- constructions is the dynamical stability of the back- itational modes, which directly relate to a decay of ground which is no longer protected by any kind of the spacetime. In field theoretic language, open string tachyon condensation and the induced D-brane recom- E-mail addresses: [email protected] bination has its natural interpretation as the Higgs ef- (R. Blumenhagen), [email protected] (B. Körs), fect, namely the spontaneous breaking of the corre- [email protected] (D. Lüst). sponding gauge group to that subgroup, which is asso-

0370-2693/02/$ – see front matter  2002 Published by Elsevier Science B.V. PII:S0370-2693(02)01504-6 142 R. Blumenhagen et al. / Physics Letters B 532 (2002) 141–151 ciated to the stable D-brane configuration at the end- world models. To be concrete we will construct a brane point of the decay.1 Hence open string tachyons may world scenario of type 0 string theory [24–30] with indeed occur but are not harmful as they can play the intersecting D6-branes, where we will closely follow role of the Higgs bosons of the effective gauge theory. the type I and type II D-brane models described be- While one can often read the statement that a non- fore. The low energy description of our type 0 brane supersymmetric configuration is already considered worlds has many appealing features close to Standard stable if tachyons are absent, this is of course still a Model or GUT physics and at the same time stabilizes very simplistic view. In the absence of supersymmetry, at least some of the closed string geometric moduli. quantum corrections will usually tend to generate po- In fact, while one may expect that these type 0 com- tentials for the other massless scalar fields that very of- pactifications could lead to models with worse stabil- ten drive the background either to some extreme limit ity properties compared to the type II and type I D- or to a point where tachyons reappear. In particular the brane models, the perturbative analysis appears to sug- dilaton can be pushed to zero or infinite coupling. It gest the opposite: the formerly constructed toroidal will be one of the main objectives of this Letter to con- intersecting brane worlds of type I string theory suf- struct non-supersymmetric D-brane models in which fer from a perturbative instability that drives the com- I the geometrical moduli fields are stabilized. plex structure moduli U2 of the torus to a degener- There exist two possibilities to construct non- I →∞ ate limit, U 2 , which implies an unacceptable supersymmetric brane world models in the first place. squashing of the internal space. This instability could (i) One starts from a supersymmetric theory in ten di- only be avoided by freezing these fields to particular mensions and only breaks supersymmetry in the open values by imposing orbifold symmetries [15,19]. We string, gauge theory sector on the branes. (ii) Super- shall find that the type 0 brane worlds do not display symmetry is already broken right from the very begin- this problem, but, on the contrary, their scalar poten- ning in the closed string sector. Most of the previous I tial stabilizes the U 2 at finite specific values, leaving work follows the first pattern, namely one is consid- only the dilaton tadpole uncanceled at leading order. ering supersymmetric type I and type II string theory At the same time they share all the generic proper- where D6-branes intersect at relative angles on an in- ties of the former intersecting brane worlds of type ternal torus or orbifold, which has been first proposed I strings and only introduce slight modifications in in [8,9] and developed further in [10–21]. These mod- the explicit construction. Of course, all of these state- els display many attractive, generic features such as ments can be subject to higher order perturbative or chiral fermion spectra in the four-dimensional effec- even non-perturbative corrections, which are hoped to tive theory [22,23], which are localized at intersec- stabilize the dilaton somehow, but may also affect the tions of two D6-branes and therefore mostly trans- other moduli. Finally, open string tachyon condensa- form in the bifundamental representation of the uni- tion may again occur and play the role of the Higgs tary gauge groups living on the respective D6-branes. effect in the spontaneously broken gauge sector. How- Scalar fields either decouple or become tachyonic ever now the D-brane configuration after tachyon con- and may serve as Higgs bosons, as their condensa- densation will still be non-supersymmetric. tion closely resembles standard spontaneous symme- The Letter is organized as follows: in Section 2 we try breaking patterns. In this way intersecting D-brane briefly review the construction of the ten-dimensional scenarios can be constructed which come very close tachyon-free type 0 string theory and its basic prop- to the non-supersymmetric Standard Model. However erties. We then discuss the modifications that come in the simplest case of D6-branes wrapped around into play by employing the intersecting brane world homology 3-cycles of the six-dimensional torus, the concept within this theory in Section 3. Next we background geometry is generically unstable. present the spectra of massless fermions, study the re- In this Letter we will discuss the second option (ii) quirement of anomaly cancellation as a consistency as an alternative, novel approach for intersecting brane check and discuss the scalar potential of the complex structure moduli in Section 4. Finally, in Sec- tion 5, we present an example with the gauge group 1 See also [6,7] in this context. R. Blumenhagen et al. / Physics Letters B 532 (2002) 141–151 143

of a left–right symmetric extension of the Standard +|D(p − 1), η RR (4) Model and corresponding spectrum of chiral fermions, with η = η = 1foraD(p − 1)-brane and η = η =−1 whose complex structure moduli are being frozen at for a D(p − 1)-brane. The freedom of choice of the speciﬁc values. sign η is due to the boundary condition of a worldsheet fermion at the position of a brane 2. Type 0 string theory i − ˜ i = µ + ˜ µ = ψr iηψr 0,ψr iηψr 0, (5) The starting point is the non-supersymmetric ten- with indices i referring to directions transverse to the dimensional type 0B string theory [31,32], which is brane and indices µ along the brane. Note, that in type known to be perturbatively unstable due its tachyonic II string theory only the superposition D(p − 1) + ground state. D(p − 1) is invariant under the GSO projection, so that only one D-brane of each even dimensionality 2.1. Type 0B string theory survives.

Type 0 string theory can either be defined as a 2.2. Type 0 string theory quotient of type II string theory by the spacetime fermion number (−1)Fs . Alternatively, one can start In [24,26] it was noted for the first time that by with the ten-dimensional superstring theory equipped taking a particular orientifold of type 0 string theory with the modified GSO projection one can get rid of the closed string tachyon. Dividing + by the worldsheet parity Ω alone does not remove + − FL FR 1 ( 1) = − FR PGSO = (1) the tachyon but the combination Ω Ω( 1) [28] 2 does. Let us review what the effect of this projection leading to the type 0B torus amplitude ± on the RR-forms and D-branes is. The RR-forms Cp transform as T ∼ 1 | |2 +| |2 +| |2 +| |2 O8 V8 S8 C2 − |η|16 : ± → − (p 2) ± Ω Cp ( 1) 2 Cp , ± ± 1 0 8 0 8 1/2 8 − FR : →± ∼ |ϑ | +|ϑ | +|ϑ | . (2) ( 1) Cp Cp , (6) 2|η|24 0 1/2 0 so that the combined action gives We have denoted by O8 etc. the characters of the SO(8) affine Lie-algebra at level k = 1. One pecu- ± (p−2) ± Ω : C →±(−1) 2 C . (7) liarity of this model is that compared to type II su- p p perstring theory all Ramond–Ramond (RR) forms are Thus, the following forms survive the orientifold doubled. Type 0B string theory therefore contains even projection ± RR-forms C originating from the (R+,R+), respec- + − + − + − C ,C ,C ,C ,C ,C . (8) tively, (R−,R−) sector, where the sign indicates the 10 8 6 4 2 0 left-, respectively, right-moving worldsheet fermion Summarizing, in type 0 string theory there exist only number. Since each RR-form appears twice, there ex- one RR-form of each even degree. The action of Ω on ist also two kinds of D(p − 1)-branes, which couple to the D-branes is C± the fields p , respectively. These D-branes are most conveniently be described in another basis, namely Ω |Dp,η,η =|Dp,−η,−η (9) 1 + − 1 + − implying that the symmetry of the brane spectrum C = √ C + C ,C= √ C − C , p p p p p p under Ω requires any respective Dp-brane to be 2 2 (3) accompanied by a Dp -brane. where the two D(p − 1)-branes are given by the Using these inputs, in [24,28] the complete string boundary states [33–35] amplitudes at one loop order have been evaluated and the tadpole cancellation conditions deduced. They |D(p − 1), η, η =|D(p − 1), η NSNS demand the introduction of N + N = 64 branes, i.e., 144 R. Blumenhagen et al. / Physics Letters B 532 (2002) 141–151

32 of each type, and imply a maximal gauge symmetry leading to spacetime fermions. Note, that even though U(32). In the type I superstring theory Ω leaves the the closed string sector is purely bosonic the open branes invariant and thus imposes a projection upon string sector contains spacetime fermions. Moreover, the open string excitations that leads to a gauge group the force between two D-branes of opposite type is SO(32). Here, Ω permutes the two kinds of branes attractive, as they only interact via exchange of closed and identifies their degrees of freedom, so that one is string modes from the NSNS sector. This is clear, since left with a unitary gauge group U(32). the two D-brane are not charged under a common RR- A property of this theory which will become form. However, adding the two annulus amplitudes important later on is that the orientifold planes do only yields a vanishing force similar to the no-force BPS couple to closed string modes from the RR-sector, situation, from which we can deduce that the tension implying that they have vanishing tension. This can be of the type 0 branes is related to the tension of the seen from the tree-channel Klein-bottle and Möbius type II branes via strip amplitudes = √TII T0 . (14) ∞ 4 2 ϑ 1/2 K =−210c dl 0 , (10) 12 The RR-tadpole cancellation condition is satisfied η 0 for N = N = 32 leading to a gauge group U(32) ∞ 1/2 4 with additional massless Majorana–Weyl fermions ϑ ⊕ M= 25(N + N )c dl 0 , (11) in the 496 496 representation of U(32).Note, η12 that the annulus is also the only amplitude which 0 receives contributions from the NSNS sector in the with arguments q˜ = exp(−4πl) and q˜ = tree channel. In ten dimensions this is attributed − exp(−4πl), respectively, and the normalization c = to the dilaton tadpole exclusively, which leads to 2 5 Vo l 10/(8π α ) . This leads to a modification of the a run-away behavior for the string coupling. An scalar potential for the moduli fields in the way that effective way to deal with it may consist in a suitable the orientifold tension of type I string theory does not adaption of the Fischler–Susskind mechanism [36– appear there. 39]. In lower dimensions the scalar potential will also The interaction between the D-branes at leading involve geometric moduli fields. order are given by the annulus amplitudes. In the loop channel the amplitude between two D-branes of the same kind looks like 3. Type 0 intersecting brane worlds

∞ 4 4 2 2 0 − 0 N + N dt ϑ ϑ / In [8] magnetic background fields on internal tori A = c 0 1 2 (12) (Dp,Dp) 8 t6 η12 were shown to provide effective means to construct 0 interesting type I string compactifications with chiral fermions in unitary gauge groups. By an exact with argument q = exp(−2πt). This leads to a repul- perturbative duality transformation, a T-duality along sive force between the two D-branes, which means three of the internal circles, this setting of (non- that the tension of the type 0 branes is not any longer commutative) D9-branes with magnetic background balanced against their RR-charge. Moreover, it is evi- fluxes is transformed into a completely equivalent pic- dent from (12) that between two D-branes of the same ture with (commutative) D6-branes intersecting at rel- type there are only open string excitations which are ative angles on the dual tori [40,41]. The later in- bosonic in spacetime. Contrary, for the loop channel vestigations of the prospects and properties of these annulus amplitude for two D-branes of opposite type models revealed them to be among the most promis- one obtains ing candidates to achieve a bottom-up construction ∞ 4 of a Standard Model or GUT field theory out of NN dt ϑ 1/2 A =− 0 string theory. While most effort was spent on study- (Dp,Dp ) c 6 12 (13) 4 t η ing non-supersymmetric models, some attention was 0 R. Blumenhagen et al. / Physics Letters B 532 (2002) 141–151 145 also devoted to supersymmetric constructions [16,42, 43] which have close relations to M-theory vacua with a background space of G2 holonomy. In this fashion we now consider a compactification on an internal six-dimensional complex torus

T6 = T2 × T2 × T2 1 2 3 (15) I I T2 with coordinates (X ,Y ) on each I . Each two- dimensional torus is defined by its complex structure I = I + I I = I + Fig. 2. D6-brane configurations. U U 1 iU2 and its Kähler structure T T1 I iT2 . In order to apply the methods of intersecting brane worlds to the type 0 theory, one also needs Thus the symmetry of the brane spectrum requires to switch to a T-dual theory. The world sheet parity to combine any D6-brane at some angle projection gets mapped to ΩR where R : Y I →−Y I m¯ I is a reflection along the three circles that have been ϕI = arctan a (18) a I I dualized. The theory with maximal gauge group now naU2 simply contains 32 D6- and 32 D6 -branes located relative to the XI axis with another D6-brane at along the fixed circles of R, i.e., along XI ,to − I an opposite angle ϕa . We henceforth employ the cancel the tadpoles. Note, that the discrete parameters convention to use letters a = 1,...,K for the branes I = I I ∈{ } U1 b can take two values b 0, 1/2 in order of type η = η = 1anda = 1,...,K for those of type R to maintain the symmetry of the background, η = η =−1. Since the two sets of stacks are identified corresponding to the two tori shown in Fig. 1. under Ω R, there will be a gauge group U(Na) on any The most general D6-brane of type a wrapped on individual stack with N = N being the number of T6 T2 a a the and of codimension one on any single I is branes. Similar to the type I models we expect chiral I I characterized by its winding numbers (na,ma) on the fermions at any intersection point of such branes. six elementary circles. It is appropriate to use a basis In order to get a precise quantitative understanding we need to embark on the computation of the contri- nI , m¯ I = nI ,mI + bI nI a a a a a (16) butions to the tadpole divergencies at one loop order denoting the wrapping around the cycles along XI and in string perturbation theory. The three relevant dia- Y I , respectively. The parity operation ΩR acts also grams can most easily be found by combining the re- on the winding numbers sults of [28] and [8] and have been collected in the appendix. The upshot of this computation is summarized R → in the following: the Klein bottle does not involve any Ω :D6(nI ,m¯ I ) D6(nI ,−¯mI ). (17) a a a a open strings. The oscillator part is thus given by (10) This is illustrated in Fig. 2. but momentum integrations need to be replaced by Kaluza–Klein (KK) and winding sums. One then gets ∞ 3 K = 11 I + 2 c4 U2 dl finite (19) = I 1 0 for the tree channel contribution to the RR tadpole. 2 2 We have defined c4 = Vo l 4/(8π α ) . Note again that there is no NSNS contribution whatsoever. The open string amplitudes involve (D6, D6) and (D6, D6) open strings separately. The Möbius strip receives only contributions from open strings invariant under ΩR, Fig. 1. Torus with bI = 0, 1/2. i.e., those stretching between branes of opposite type. 146 R. Blumenhagen et al. / Physics Letters B 532 (2002) 141–151

Its RR tadpole reads 4. Chiral fermion spectra and scalar potential ∞ 3 6 I I The most important feature for the phenomenolog- M =−2 c N n U dl + finite. (20) aa 4 a a 2 ical relevance of intersecting branes is the fact that I=1 0 the open string quantization of strings between two Again, there is no NSNS tadpole. Finally, the annulus D-branes that intersect in a point on the internal space diagram involves all kinds of open strings, where the leads to a single chiral fermion in the effective lower- contribution to the RR tadpole arises merely from the dimensional theory [23]. (D6, D6) and (D6, D6) open strings ∞ 3 4.1. Chiral fermions (RR) I I I I I 1 A = c4NaNb n n U +¯m m¯ dl ab a b 2 a b U I This feature gets slightly modified in type 0 theory. I=1 2 0 As was pointed out above, only open strings stretched + finite. (21) between D6-branes of opposite type give rise to space- The NSNS tadpole arises from all open string sectors time fermions, whereas open string stretched between in the annulus and will be dealt with in Section 4, when two brane of the same sort give rise to spacetime the scalar potential is discussed. bosons. If the two intersecting branes are not identified Since in type 0 string theory there exists one RR- by R, i.e., at an (ab) or (ab) intersection, the fermion form of each even degree, in the original T-dual picture will simply transform in the bifundamental representa- with magnetic fluxes on the D9- and D9 -branes we tion of the respective gauge groups, e.g., the (Na, Nb ) have to cancel the D9- as well as the effective D7-, of U(Na)×U(Nb ). If we are facing an (aa ) intersec- D5- and D3-brane charges, leading to eight separate tion instead, a further distinction has to be made. For tadpole cancellation conditions. Note, that in type I intersection points that are fixed under R, one needs to string theory the D7- and D3-brane charges were regard the projection by ΩR. The Möbius strip am- automatically canceled due to the symmetry under Ω. plitude (11) implies the solution Analogously, here the charges of the second set of RR- K forms in type 0B are projected out by Ω . Thus, the 0 1Na γΩR = (24) cancellation conditions which derive from the above 1N 0 a=1 a RR tadpole computation are very similar to those of R type II string theory, except for a doubled background for the action of Ω on the Chan–Paton labels. This D9-brane charge: leads to a single fermion in the antisymmetric representation of U(Na) at such an invariant intersection of K 3 K 3 type (aa). On the contrary, for intersections not in- N nI = 32, N m¯ I = 0, a a a a variant under R no projection applies and they pro- = = = = a 1 I 1 a 1 I 1 vide a symmetric and an antisymmetric representation K K for any doublet of intersection points. After defining I J ¯ K = I ¯ J ¯ K = Nanana ma 0, Nanama ma 0. (22) a=1 a=1 3 3 I I I I (R) I I = n m¯ − n m¯ ,I = 2m¯ It is understood that the conditions are to be satisfied ab b a a b aa a = = = I=1 I=1 for any combination of indices I J K I.The (25) gauge group for the total and R-invariant intersection numbers of K branes a and b or a, the spectrum of chiral fermions G = U(Na) (23) is summarized in Table 1. a=1 In contrast to the intersecting brane world models may have been reduced in its rank, where rk(G) < of type I strings, where it was necessary to have bI > 0 I 32 is possible for na > 1. Note, that in contrast to for at least one torus in order to achieve odd numbers intersecting brane world models of type I string theory of generations, one can now also get three generation one cannot obtain SO(2Na) or Sp(Na) gauge groups. models with purely imaginary complex structures. It R. Blumenhagen et al. / Physics Letters B 532 (2002) 141–151 147

Table 1 mechanism invoking the coupling of four-dimensional Chiral massless fermions bulk RR-fields. This issue is more conveniently ad- Representation Multiplicity dressed in the original type 0 theory, before any R (A ) I ( ) T-duality. The spectrum of RR-forms has been given a L aa 1 (R) in (8). In detail, in type 0 string theory one has the fol- (Aa ⊕ Sa)L I − I 2 aa aa lowing Wess–Zumino terms in the Born–Infeld action (Na, Nb)L Iab − 5 + 4 − 3 C0 Fa , C2 Fa , C4 Fa , is interesting to note that the charged scalar fields D9a D9a D9a can transform in different representations than the fermions. They arise in the sector of open strings + 2 − + C6 Fa , C8 Fa, C10. (29) with both ends on the same kind of brane and never D9 D9 D9 experience the projection by ΩR. Thus, spacetime a a a bosons and in particular the Higgs fields can never These forms give rise via dimensional reduction to appear in antisymmetric or symmetric representations the following two-forms and axionic scalars in four dimensions of the gauge group. − + CI = C ,CI = C , 4.2. Anomaly cancellation 2 4 0 2 T2 T2 I I It is not difficult to check that the spectrum of Ta- I = + I = − ble 1 satisfies the cancellation of non-abelian anom- B2 C6 ,B0 C4 , alies. By using (22) the sum of all contributions to the T2 ×T2 T2 ×T2 J K J K triangle anomaly of the factor U(Na) vanishes: − + B = C ,B= C (30) (R) 1 (R) 2 8 0 6 Nb Iab + (Na − 4)I + 2Na Iaa − I aa 2 aa T6 T6 b=a = + = − = 0. (26) and C2 C2 ,C0 C0 with the following Hodge- duality relations in four dimensions This may serve as a consistency check, but is, of = I = I course, guaranteed by the tadpole cancellation any- dC0 2dB2,dB0 2dC2 , way. I = I = dC0 2dB2 ,dB0 2dC2. (31) As usual, the mixed U(1) − SU(N)2 and U(1)3 Thus, by integrating (29) over the internal six- anomalies do not cancel right away but require a dimensional torus, one obtains the following couplings suitable Green–Schwarz mechanism. For the U(1)a − 2 SU(Nb) anomaly one obtains 1 2 3 1 2 3 2 Nam¯ m¯ m¯ C2Fa,nn n B0F , a a a b b b b 1 (Na − 2) (R) = + M4 M4 Aaa Nb Iab 2Iaa 2 2 b =a I ¯ J ¯ K I J K ¯ I I 2 Nanama ma C2 Fa,nb nb mb B0 Fb , 1 (R) + 2Na I − I , M M 2 aa aa 4 4 1 J K ¯ I I I ¯ J ¯ K I 2 Aab = NaI for b = a. (27) Nana na ma B2 Fa ,nbmb mb C0 Fb , 2 ab M4 M4 Using the relation (27) this can be written as 1 2 3 ¯ 1 ¯ 2 ¯ 3 2 1 Nananana B2Fa, mbmbmb C0Fb , (32) Aab = NaIab for all a,b. (28) 2 M4 M4 Analogous to the type I string, we expect that this which combine into the usual tree diagrams to con- anomaly is canceled by a generalized Green–Schwarz tribute to the respective anomaly. Adding up all terms 148 R. Blumenhagen et al. / Physics Letters B 532 (2002) 141–151

2 FaFb we ﬁnds that the resulting GS amplitude is pro- that at least to the next to leading order, in the annulus portional to diagram, the Kähler moduli may stabilize as well.

AGS = NaIab (33) which has the correct form to cancel the field theory 5. A left–right symmetrically unified model anomaly (28). In this section we present a concrete model which 4.3. The disc level scalar potential contains the gauge group of a left–right symmetric extension of the Standard Model with some additional The scalar potential can be read off from the dilaton charged chiral matter. Since, in contrast to the type II tadpole divergence of the closed string tree channel orientifold models, the type 0 orientifold models only amplitude which is essentially the square of the disc have chiral fermions for ab intersections, the anomaly expectation value cancellation conditions for the U(Na) gauge factors ∂V(φ,UI ) (including Na = 1, 2) prevent the realization of a pure φ ∼ 2 . (34) disc ∂φ three generation Standard Model. Recall that for the type II orientifold models two of the left-handed quark Due to the absence of the NSNS contributions in the doublets transformed in the (3, 2) representation and Klein bottle and Möbius strip diagrams, the resulting one in the (3, 2) representation. In this way, the (ab) potential is identical to the one obtained in type I string and (ab) intersections could combine to provide three theory [15] except for the absence of the orientifold generations, which is now impossible, as only (ab) tension, will provide fermions at all. K We now choose five stacks of D6–D6 -branes, bI = − I = φ4 T2 V φ,U2 T6e (2Na) 1/2 on all three two-dimensional tori I and the = a 1 wrapping numbers shown in Table 2. 3 It can easily be checked that these wrapping num- (m¯ I )2 × nI 2U I + a , (35) bers satisfy the RR-tadpole cancellation conditions. a 2 U I I=1 2 The resulting massless chiral spectrum is presented in Table 3. where φ4 is the four-dimensional dilaton φ4 = φ − This is the matter content of the minimal left–right ln(Vo l 6)/2andT6 the D6-brane tension. The normalization was fixed by comparing to the DBI effective symmetric extension of the Standard Model plus three ⊕ ¯ × action. Already from the scaling behavior of the po- generations of a charged (A, 1) (1, A) of SU(2)L tential SU(2)R. Note, that the fourth U(1) factor decouples completely from the chiral spectrum and therefore I →∞ I → ∞ V φ,U2 for U2 0, (36) can be considered as resulting from a spectator brane which is only there to satisfy the RR cancellation it is clear that there must exist a global minimum I conditions. Two of the five U(1) factors are anomalous at which the U2 are stabilized at tree level. We will demonstrate this explicitly by presenting an example with stabilized complex structure in the following Table 2 section. Wrapping numbers The potential (35) does not depend on the Kähler 1 ¯ 1 2 ¯ 2 3 ¯ 3 moduli T I . It was explained in [15] that it is to be (na, ma)(na, ma)(na, ma)Na − 1 expected that this feature remains true perturbatively (2, 0)(1, 2 )(4, 1) 3 1 3 1 if the D6-branes intersect in points. Only if there are (1, 2 )(1, 2 )(1, 2 ) 2 − 1 3 1 also branes that are parallel on circles of the internal (1, 2 )(1, 2 )(1, 2 ) 2 space the propagation of KK and winding modes will 7 − 1 (2, 0)(1, 2 )(1, 2 ) 1 introduce a dependence on the T I . It was estimated (2, 0)(1, − 1 )(1, − 7 ) 1 from their respective proportionality to 1/T I and T I 2 2 R. Blumenhagen et al. / Physics Letters B 532 (2002) 141–151 149

Table 3 Table 4 Chiral massless spectrum Ground state energies 5 SU(3) × SU(2)L × SU(2)R × U(1) Multiplicity Sector E0

(3, 2, 1)(1,1,0,0,0) 3 (12) 0.05 (13) 0.05 (3, 1, 2)(−1,0,−1,0,0) 3 (24) 0.02 (1, 2, 1)(0,−1,0,0,−1) 3 (34) 0.02 (1, 1, 2)(0,0,1,0,1) 3 (25) 0.09 (35) 0.09 (1, A, 1)(0,2,0,0,0) 3 ¯ (1, 1, A)(0,0,−2,0,0) 3 6. Conclusions and besides U(1)4 the anomaly free ones are In this Letter we have constructed a class of non- 1 U(1)B−L = U(1)1 + U(1)5, supersymmetric string models whose geometric com- 3 plex structure moduli are stabilized at finite values by = + + U(1)K U(1)2 U(1)3 2U(1)5. (37) the leading order perturbative potential. One may fur- It is beyond the scope of this Letter to follow the ther argue, that the stabilization can be extended to the phenomenological properties of this model further. Kähler moduli as well, hence leaving only the dila- Instead, we investigate the disc level scalar potential ton unstable perturbatively. Our novel construction is for this concrete model to provide an explicit example intrinsically non-supersymmetric, as it starts from the for the stabilization of the complex structure moduli. non-supersymmetric but tachyon-free ten-dimensional By performing a numerical analysis of the resulting type 0 string theory compactified on a torus. To this potential (35) we find that there exist a unique global theory we have applied the techniques of intersecting minimum for the values of the complex structures brane worlds finding a set of solutions for the effective four-dimensional field theory which displays per- U 1 = 0.086,U2 = 1.045,U3 = 0.486. (38) 2 2 2 spectives for obtaining semi-realistic models that are We have also determined the ground state energies comparative to the earlier studied type I and type II E0 in the various bosonic open string sectors at compactifications. One may consider various exten- this minimum where depending on the values of the sions and generalizations of the present program. It complex structures tachyons can potentially arise. It is may for instance be interesting to construct orbifolds another nice feature of these type 0 models that only of the purely toroidal models along the lines of [44, (ab) type intersections need to be considered, thus the 45]. number of dangerous sectors is essentially halved. If two D6-branes are parallel on at least one T2,then we can move the two D-branes apart on this torus Acknowledgements and avoid the tachyonic instability classically. For the remaining open string sectors we find the following We are supported in part by the EEC RTN pro- ground state energies: gramme HPRN-CT-2000-00131. We would like to All ground state energies are positive implying that thank A. Uranga for discussion. this model, in the approximation we used, is perturbatively stable (see Table 4). Note, that the possible tachyon in the (23) open string sector transforms in Appendix A. Amplitudes of type 0 with branes at the (2, 2) representation and is needed for breaking angles the left–right symmetric down to the Standard Model. However, as we mentioned already there can be no In this appendix we summarize the more technical tachyons and therefore no Higgs particles in sym- results for the three amplitudes that contribute to the metric or anti-symmetric representations of the gauge tadpole divergence at the χ = 0 level of perturbation group. theory. They are obtained most directly by combin- 150 R. Blumenhagen et al. / Physics Letters B 532 (2002) 141–151 ing the results of [28] and [8]. The Klein bottle am- Together this combines into the annulus diagram of plitude of the original ten-dimensional type 0 theory type II strings, which in particular guarantees the was given in (10). The modification due to the toroidal absence of any tachyon contribution to the tadpole compactification is standard and does not take refer- divergence. Finally, the Möbius strip diagram gives the ence to the novel issue of intersecting branes, contribution ∞ ∞ / 1/2 3 1/2 4 ϑ 1 2 ϑ ϑ (R) 0 I ϕI /π 7 I 0 M = 2 a K =−2 c4 U dl aa 2 c4NaI dl . (A.5) 2 12 aa 3 1/2 = η η I ϑ I I 1 0 1/2ϕa /π 0 3 2 2 We have used the standard definitions −4πl(r2RI +s2/RI ) × e 1 2 . (A.1) α ϑ[ ](q) 2 β 2πiαβ α −1/24 I=1 r,s∈Z = e q 2 η(q) For the open string amplitude the peculiarities of type ∞ − + 0 as well as the modifications due to the relative × 1 + qn 1/2 αe2πiβ angles of the D6-branes need to be taken into account. n=1 The open string states involve only the NS sector but n−1/2−α −2πiβ one needs to include the D6- and D6-branes now. The × 1 + q e , relative angles lead to shifts in the oscillator spectrum ∞ of string excitations by η(q) = q1/24 1 − qn . 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