A set of open questions originates in the conclusions of Cremades, Ibanez and Marchesano (JHEP 07 (2002) 009, arXiv:hep-th/0201205), which introduced quasi-supersymmetric (Q-SUSY) intersecting D-brane models as a stringy realization of a two-loop protection of scalar masses for a modest hierarchy between a low string scale and the electroweak scale. The acknowledged open issues are: whether the uncancelled NS-NS tadpoles of these non-supersymmetric toroidal orientifold compactifications can be consistently treated; whether a parametrically large Planck-to-string-scale hierarchy can be achieved in toroidal constructions where no transverse volume is available to all branes at once; what the phenomenological consequences are of the approximately extended N=4 supergravity living in the bulk; up to what cutoff the two-loop scalar-mass protection of the explicit three-generation supersymmetric Standard Model construction holds; and what the alternative of cancelling tadpoles with explicit RR fluxes instead of a non-factorizable brane implies for the massless spectrum. The authors' companion papers completed the general phenomenology of these models, and adjacent work classified the induced moduli dynamics of these compactifications, but the background redefinition for realistic Q-SUSY models, the toroidal hierarchy problem, and the two-loop cutoff of the explicit model remain open.

Background

In Type II string theory, Dp-branes are extended objects on which open strings end, carrying gauge theories on their worldvolumes. In intersecting D-brane models one compactifies on a six-torus and wraps, for example, D6-branes (branes with six-dimensional worldvolumes) around three-cycles of the torus; chiral fermions then appear at the intersections of two branes, with multiplicities given by intersection numbers, which makes Standard-Model-like spectra possible. To obtain a consistent closed-string theory the torus must additionally be quotiented by an orientifold action (an identification of the geometry combined with a reversal of the string orientation), which introduces orientifold planes (negatively charged mirror images of the branes under this quotient) and Ramond-Ramond (RR) tadpole cancellation conditions; these conditions are the string-theoretic analog of anomaly cancellation. A quiver diagram encodes such constructions as a graph: each brane stack (gauge group factor) is a node, and each intersection (bifundamental chiral matter) is a link.

A separate thread is the idea of a low string scale of roughly 1-100 TeV with large extra dimensions, in which the observed hierarchy between the Planck scale and the weak scale is replaced by a smaller gap. Cremades, Ibanez and Marchesano studied what they called the modest hierarchy problem: the residual, empirically suggested gap of a few orders of magnitude between a fundamental scale of order 10-100 TeV and the electroweak scale. They observed that certain intersecting-brane configurations, which they called quasisupersymmetric (Q-SUSY), are globally non-supersymmetric while every brane-pair sector preserves its own local N=1 supersymmetry (N=1 supersymmetry being the minimal four-dimensional supersymmetry, with one gravitino); quiver nodes then carry different N=1's and the links are chiral multiplets. In such theories the scalar masses are protected against quadratic divergences up to two loops, so a three-generation supersymmetric-Standard-Model-like model with a string scale of order 30 TeV could keep the Higgs light. The paper also computed the Fayet-Iliopoulos terms (D-term supersymmetry-breaking masses for scalars generated by displaced complex-structure moduli) and the gauge couplings at the intersections.

The price of this construction is that the compactification is not supersymmetric as a whole. Non-supersymmetric brane configurations leave uncanceled Neveu-Schwarz-Neveu-Schwarz (NS-NS) tadpoles: one-point functions for the massless closed-string dilaton and geometric moduli. Equivalently, the tree-level (disk-level) scalar potential for the real parts of the NS moduli SS and UIU^I (the dilaton-axion and the complex-structure moduli of the tori) does not vanish; in the Q-SUSY models this potential, equation (5.43) of the source paper, is remarkably simple, linear in those fields, and some NS tadpoles cancel when the corresponding RR tadpoles do. The Fischler-Susskind mechanism is the standard response to tadpoles: the background fields are redefined order by order so that the tadpoles are absorbed, in simple cases producing warped (non-trivially scaled) metrics, as shown by Dudas and Mourad and by Blumenhagen and Font for toy examples.

Two structural obstacles specific to these models were left open. First, in a toroidal compactification whose branes wrap cycles in all tori, there is no cycle transverse to all branes simultaneously, so the usual trick of making the Planck mass large by taking a large transverse volume is unavailable; the authors speculated that a Randall-Sundrum-type gravity localization (gravity trapped near a brane by warp factors of the metric, as in warped compactifications) induced by the non-vanishing NS potential might be a way out. Second, the bulk of such toroidal orientifolds preserves N=4 supergravity (the maximal supersymmetry of four-dimensional supergravity that can arise on tori, with four gravitini, N=8 in the purely toroidal case), so the graviton's superpartners can only acquire masses from supersymmetry breaking on the branes, and the phenomenological consequences of this approximately extended bulk supergravity (for the cosmological constant and for soft supersymmetry-breaking terms, meaning mass and coupling parameters added by hand that break supersymmetry without reintroducing quadratic divergences) were unexplored. The field's later standard solution to hierarchies, flux compactification of Giddings, Kachru and Polchinski, uses Calabi-Yau manifolds (compact Ricci-flat internal spaces) rather than toroidal intersecting-brane geometry and does not address these questions.

Problem Statement

Can quasi-supersymmetric (Q-SUSY) intersecting D6-brane models on toroidal Type II orientifolds — in particular the explicit three-generation supersymmetric-Standard-Model-like construction of the source paper — be completed into consistent string compactifications that realize a controlled modest hierarchy between the string scale and the electroweak scale? Concretely, the source leaves open, and later literature has not closed: (1) the uncancelled NS-NS tadpoles, equivalently the non-vanishing tree-level scalar potential linear in the real parts of the NS moduli SS and UIU^I (equation (5.43) of the source), signal an instability of the flat background — can the closed-string background be consistently redefined in a way that is feasible for realistic multi-brane models, rather than only for the simple known examples?; (2) since in these toroidal constructions no volume is transverse to all branes simultaneously, a large Planck-to-string-scale hierarchy cannot come from a large transverse volume — can a parametrically large Planck-to-string-scale hierarchy nevertheless be generated?; (3) the bulk preserves N=4 supergravity (N=8 in the purely toroidal case) and the graviton's superpartners acquire masses only from brane supersymmetry breaking — what are the phenomenological consequences of this approximately extended bulk supergravity, for example for the cosmological constant and soft supersymmetry-breaking terms?; (4) the scalar (Higgs) masses are protected only up to two loops — what is the quantitative stability of the explicit three-generation model against loop corrections, and up to what cutoff (string scale) is the hierarchy stable?; and (5) as an alternative to adding a non-factorizable brane to cancel tadpoles, what are the consequences of cancelling them with explicit RR fluxes, both for the massless spectrum and for the protection of the Higgs masses? These five questions are linked components of one viability assessment of the Q-SUSY construction. A decisive negative result on any of these components — for instance, a proof that the background redefinition or the hierarchy mechanism is impossible in these toroidal constructions — resolves the viability question negatively. Item (5), the RR-flux alternative, is partially discussed in the source itself and in Uranga's work on D-branes, fluxes and chirality, and item (4) has toy-model and one-loop analogs in the literature.

References

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