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Brane Supersymmetry Breaking

We show how to construct chiral tachyon-free perturbative orientifold models, where supersymmetry is broken at the string scale on a collection of branes while, to lowest order, the bulk and the other branes are supersymmetric. In higher orders, supersymmetry breaking is mediated to the remaining sectors, but is suppressed by the size of the transverse space or by the distance from the brane where supersymmetry breaking primarily occurred. This setting is of interest for orbifold models with discrete torsion, and is of direct relevance for low-scale string models. It can guarantee the stability of the gauge hierarchy against gravitational radiative corrections, allowing an almost exact supergravity a millimeter away from a non-supersymmetric world.

The breaking of supersymmetry in String Theory is a long-standing fundamental problem with many ramifications. It is related to the selection of the correct vacuum state, to the cosmological constant problem, to the lifting of flat directions for string moduli, and is a necessary ingredient of a realistic string phenomenology. Unfortunately, despite the recent progress in the understanding of non-perturbative phenomena based on string dualities, little was done on the problem of supersymmetry breaking.
In perturbation theory, closed string vacua with spontaneously broken 1 supersymmetry can be constructed generalizing the Scherk-Schwarz mechanism, and in particular resorting to freely acting orbifolds [1,2]. The breaking scale is then fixed by a compactification radius, and realistic scenarios ask for radii of the order of a few TeV [2]. This approach is therefore likely to be relevant if the string scale is far below the Planck mass [3], and possibly close to electro-weak energies [4,5,6]. A natural framework for such models is the type-I string theory, where gauge interactions are localized on D-branes while gravity propagates in the bulk [5].
Scherk-Schwarz compactifications were recently extended to type I string models in [7], where a new feature was pointed out: the massless spectra of D-branes orthogonal to the coordinate used for the breaking remain supersymmetric at the tree level. As a result, in this case the scale of supersymmetry breaking in the observable world is not directly proportional to the compactification scale, and can have lower values, making this class of constructions more flexible and potentially relevant even if the string scale is moved to intermediate values [8,9].
The main problem with this mechanism is the cosmological constant. The reason is that the bulk energy density behaves generically as ρ bulk ∼ 1/R 4+d ⊥ , where R is the radius of the coordinate used to break supersymmetry and d ⊥ is the dimensionality of the space transverse to our brane world, assumed large with respect to the string scale. The projection 1 In the sense that it can be restored by tuning a continuous parameter.
-3of this cosmological constant on the brane is enhanced by the volume of the transverse space r d ⊥ , and is far above O(TeV 4 ). In fact, the radius R of a longitudinal direction can not be far from the (TeV) type I string scale M I in a perturbative setting, and as a result the brane energy density acquires a quadratic sensitivity to the four-dimensional On the other hand, if R is transverse (R ∼ r << O(TeV −1 )), one obtains ρ brane ∼ 1/r 4 . In both cases, the energy density on the brane is far above the TeV scale, and this destabilizes the hierarchy that one tries to enforce. One way out is to resort to special models with broken supersymmetry but with a vanishing or exponentially small cosmological constant [11].
Alternatively, one could conceive a different scenario, with supersymmetry broken primordially on our brane world and a string scale of a few TeV. In this case the brane cosmological constant would be, by construction, O(M 4 I ), while the bulk, only affected by gravitationally suppressed radiative corrections, would be almost supersymmetric [5].
In particular, one would expect that the gravitino mass and the other soft masses in the bulk be extremely small O(M 2 I /M P l ) ∼ 10 −4 eV for M I ∼ 1 TeV. Such small masses for scalar moduli and gauge fields might also induce deviations from Newtonian gravity in the (sub)millimeter region that can be experimentally tested [12,5]. Moreover, the cosmological constant induced in the bulk would be ρ bulk ∼ M 4 i.e. of order (10 MeV) 6 for d ⊥ = 2. Alternatively, brane supersymmetry breaking could also be of interest in models with an intermediate string scale ∼ 10 11 GeV [9], if it occurs on a brane distant from our world and is therefore mediated to us by gravitational interactions.
The purpose of this letter is to show that it is possible to construct perturbative orientifold models [13] where supersymmetry breaking originates from a collection of branes, while both the bulk spectrum and the spectrum of other branes are supersymmetric at tree-level. Whereas models with bulk supersymmetry can naturally be constructed in the effective field theory, for instance appealing to non-perturbative super-Yang-Mills dynam--4ics, we believe that a direct string construction is of some interest, in particular to attain a better comparison with field theory supersymmetry breaking mechanisms.
Brane supersymmetry breaking can also be induced turning on internal magnetic fields [14]. The mechanism we are proposing shares some properties with this setting, although supersymmetry is broken at the string scale, but appears to avoid some of its problems, namely the presence of tachyons and the generic lack of gaugino masses for the unbroken gauge group.
The rules for constructing perturbative type-I orientifolds [15,16] rest on the modular invariance of the closed string spectrum and on some conditions linking its Klein-bottle projection to the open and unoriented sector. These are to be supplemented by Ramond-Ramond (RR) tadpole conditions, that are directly related to anomaly cancellations and may be regarded as global neutrality conditions for RR charges in a compact internal space [17]. Tadpole cancellations result from opposite contributions of boundaries and crosscaps or, equivalently, of branes and orientifolds.
It is possible to construct models where some of the orientifold contributions are inverted, so that the necessary RR cancellations require some care. A simple example of this phenomenon is afforded by the T 4 /Z 2 orientifold where, compatibly with the "crosscap constraint" [18], all twisted Klein-bottle contributions are reversed. These exotic Klein-bottle projections are quite interesting, and have already led to tachyon-free non-supersymmetric open-string vacua [19]. In our case, the resulting unoriented closed spectrum is supersymmetric and contains, aside from the (1, 0) gravitational multiplet, 17 tensor multiplets and 4 hypermultiplets, but its RR tadpoles can not be canceled in the usual way. A more sophisticated, but physically very interesting, set of examples, is provided by the open descendants of Z 2 × Z 2 models, and in particular of those with discrete torsion, where some of the orientifold charges are necessarily reversed. In all these cases, the tadpole conditions may be solved unpairing NS and R contributions, and thus inducing brane supersymmetry -5breaking. In this letter, for the sake of brevity, we confine our attention to the T 4 /Z 2 case, where one obtains a chiral 6D spectrum that is free of tachyons and satisfies all usual anomaly cancellation constraints. In this model, that contains 32 D9 and 32 anti-D5 branes, the absence of tachyons can also be understood as in recent studies [20] of stable non-BPS states of the type IIB string.
The same mechanism can be applied to the Z 2 × Z 2 models, that will be discussed elsewhere, and in principle should offer the possibility to deal with other orientifold models where tadpole conditions do not admit naive supersymmetric solutions.

A six-dimensional example
We now present an explicit 6D model containing D9 and D5 (anti-D5) branes, where at tree level supersymmetry breaking is induced on the55 and59 states, while the 99 states and the bulk (closed) spectrum are supersymmetric. The starting point in this construction is a modification of the Ω projection in the twisted sector of the T 4 /Z 2 model. This inverts the charge of the O5 planes, and is actually compatible with the perturbative rules of orientifold models, but the cancellation of RR tadpoles requires 32 D5 branes. Omitting for brevity the contributions of the transverse bosons, the torus partition function is where Λ denotes the compactification lattice. In T , we have introduced the convenient combinations of SO(4) characters defined as -6-Turning to the Klein bottle, let us consider the two inequivalent choices 2 where P (W ) denotes the momentum (winding) lattice sum and ǫ = ±1. For both choices of ǫ, the closed string spectrum has (1, 0) supersymmetry, but the two resulting projections are quite different. The usual choice (ǫ = 1) leaves 1 gravitational multiplet, 1 tensor multiplet and 20 hypermultiplets, while ǫ = −1 leaves 1 gravitational multiplet, 17 tensor multiplets and 4 hypermultiplets 3 . The projected closed spectrum coincides with the one considered in ref. [21], but our open spectrum is markedly different. The transverse-channel Klein bottle amplitude reads where P e (W e ) denotes the lattice of even momenta (windings) and v is the volume of the compact space. The reversal of the O5 charge respects the positivity structure at the origin of the T 4 lattice, and indeed the coefficients combine into perfect squares: The case ǫ = 1 leads to the familiar U(16) × U(16) model [15,16], and therefore from now on we restrict our attention to the other choice.
2 The world-sheet moduli for the various amplitudes, implicit in the following, are defined as in [7]. 3 A similar projection with a non-zero B ab [15] would result in 13 or 11 tensor multiplets [22]. -7- where we have also introduced primed characters, related by a chirality change S 4 ↔ C 4 to the unprimed ones in (1): As usual, at the origin of the lattice the different terms organize into perfect squares: Here, for simplicity, all D5 branes, whose geometry is neatly displayed by the breaking terms, have been placed at the origin of the compact space. The direct-channel annulus is obtained by an S-transformation, and reads Finally, the Möbius amplitude at the origin of the lattices (for the definition of the hatted characters, see [7]), is easily obtained combiningK 0 andÃ 0 , and allows one to reconstruct the full transverse Möbius amplitudẽ -8- Eq. (11) shows some marked differences with respect to the more familiar model with ǫ = 1. These may be given a neat physical interpretation, sinceM describes the propagation between holes and crosscaps, or equivalently between branes and orientifold planes.
Therefore, one can see that all D9-O9 terms, the D5-O5 terms in the R-R sector and the Parametrizing the Chan-Paton charges as The massless matter representations may be read from: The 99 spectrum is supersymmetric, and comprises the (1,0) vector multiplets for the  16)] of the D9 gauge group. As a result, the non-superymmetric ND spectrum looks effectively supersymmetric, and indeed all 95 terms do not contribute to the vacuum energy. However, as in the 6D temperature breaking discussed in [7], the chirality of the fermions in Q ′ s is not the one required by 6D supersymmetry. This chirality flip is a peculiar feature of six-dimensional models, and does not persist in the reduction to four dimensions 4 . Thus, in four dimensions supersymmetry would be strictly unbroken on the D9 branes before turning on D5 gauge interactions. This setting shows some marked 4 In view of the present results, the 6D model presented in the first reference in [7] actually contains, in addition to the D9 branes, one set of 16 D5 branes and one set of 16 D5 branes. This justifies both the chirality changes in the twisted spectrum of the D5 branes and the presence of the tachyonic mode in the 55 annulus amplitude.
-10differences with respect to the mechanism of bulk supersymmetry breaking (or brane supersymmetry), where the massless ND sector was supersymmetric from both the N and D viewpoints [7].
It is easy to check that all the irreducible gauge and gravitational anomalies cancel in this model as a result of the tadpole conditions for the RR fields. The residual anomaly polynomial does not factorize, and reveals the need for a generalized Green-Schwarz mechanism [24], with couplings of a more general type than those found in supersymmetric models: Thus, the spectrum is tachyon free, all RR tadpoles are canceled and there are no massless gauginos.
The breaking of supersymmetry gives rise to a vacuum energy localized on the D5 branes, and thus to a tree-level potential for the N S moduli, that can be extracted from the corresponding uncancelled NS tadpoles in eqs. (5), (8)  associated to the 6D dilaton φ 6 and to the internal volume v: Using factorization and the values N = D = 32 needed to cancel the RR tadpoles, the potential (in the string frame) is: where φ 10 is the 10D dilaton, that determines the Yang-Mills coupling g YM on the D5 branes, and c is some positive numerical constant. The potential (18) is clearly localized -12on the D5 branes, and is positive. This can be understood noticing that the O9 plane contribution to vacuum energy, identified from (17), is negative and exactly cancels for N = 32. This fixes the D5 brane contribution to the vacuum energy, that is thus positive, consistently with the interpretation of this mechanism as global supersymmetry breaking.
The potential (18) has the usual runaway behavior, as expected by general arguments.
An interesting application of the above mechanism is to the Z 2 × Z 2 orbifold model with discrete torsion, where nontrivial two-form fluxes at the orbifold fixed points [26] invert the sign of the lattice independent terms in the torus amplitude. It is interesting to notice that the corresponding supersymmetric spectra, which contain chiral fermions, are inconsistent because of uncancelled RR tadpoles. The solution to the RR tadpoles can again be obtained introducing anti-D5 branes, along the lines described previously. We supersymmetry is also broken in the sector 5 i5j , where a tachyon appears, in agreement with the analysis of non-BPS states of [20]. Finally, the remaining NS-NS tadpoles give rise to a scalar potential localized on the anti-D5 branes, as in eq. (18).
It would be interesting to investigate along the same lines other models, as the 4d Z 4 orientifold, where the tadpole conditions have no naive solution.