Uplift-mechanism dependence of the joint statistics of SUSY breaking and the cosmological constant in the string flux landscape

In type IIB flux compactifications, the joint distribution of the gravitino mass $m_{3/2}$ (the supersymmetry-breaking scale) and the cosmological constant $\Lambda$ has been derived only for vacua whose de Sitter uplift comes from an anti-D3 brane at the tip of a warped throat, in both KKLT and Large Volume Scenario moduli stabilization, where the distribution is tilted toward low supersymmetry-breaking scales as $\Lambda\to 0$. The open problem, posed explicitly by the authors of that computation, is to extend the joint-statistics analysis to the other known uplift mechanisms — D-term uplifting, Kähler uplifting, dilaton-dependent non-perturbative effects, T-branes, and related de Sitter constructions — and to determine the relative abundance of vacua with different uplift sectors, so that a landscape-level joint distribution of $m_{3/2}$ and $\Lambda$ can be obtained. The source additionally gives a parametric argument that the $\Lambda\to 0$ distribution forms are stable against low-energy quantum corrections, and notes as a caveat that translating vacuum statistics into low-energy predictions depends on unknown selection principles from early-universe cosmology.

Background

String theory admits an enormous number of metastable four-dimensional vacua (the 'landscape') because background three-form fluxes (quantized field strengths of the ten-dimensional form fields, threading the compact space) in type IIB string compactifications on Calabi-Yau orientifolds (Calabi-Yau manifolds quotiented by an orientation-reversing symmetry, as required for consistent type IIB compactifications with D-branes) generate a superpotential for the moduli — the massless scalar fields parameterizing the compactification geometry — while admitting a combinatorially enormous number of discrete flux choices. Denef and Douglas initiated the statistical study of this ensemble, computing how flux vacua distribute over moduli space and over the values of the flux superpotential. A central question is the statistics of supersymmetry (SUSY) breaking and of the cosmological constant $\Lambda$ across vacua. The SUSY-breaking scale is measured by the gravitino mass $m_{3/2}$ (the mass of the gravitino, the superpartner of the graviton, which sets the size of the auxiliary F- and D-term vacuum expectation values that break SUSY). Early analyses based on generic expectations for F- and D-term sizes found power-law or logarithmic distributions favoring high SUSY-breaking scales, and a later study showed that incorporating Kähler-moduli stabilization changes these conclusions because the structure of the underlying anti-de Sitter (AdS) vacua enters the distribution.

In controlled constructions, fluxes stabilize the complex-structure moduli and the axio-dilaton (the field whose real part sets the string coupling and whose imaginary part is a universal axion), while the Kähler moduli are fixed either by non-perturbative effects such as gaugino condensation, the formation of a condensate of the gauge fields' superpartners in a hidden sector that generates an exponentially small superpotential term (the KKLT scenario of Kachru, Kallosh, Linde and Trivedi) or by the interplay of non-perturbative effects with perturbative $\alpha'$ corrections to the Kähler potential (the Large Volume Scenario, LVS). Both yield supersymmetric AdS vacua, which must be raised by an 'uplift' sector to obtain de Sitter (dS) vacua with small positive $\Lambda$. The uplift analyzed in the source paper is an anti-D3 brane (an anti-brane, whose presence breaks SUSY spontaneously) placed at the tip of a strongly warped throat — a cone-like, strongly redshifted region of the compactification. At the effective-field-theory level this sector is described by the nilpotent goldstino formalism: the anti-brane's Goldstone fermion is encoded in a Volkov-Akulov superfield whose auxiliary component squares to zero (nilpotency), contributing a positive potential energy without introducing new propagating fields. Cicoli, Licheri, Maharana, Singh and Sinha derived the joint distribution of $m_{3/2}$ and $\Lambda$ (in Planck units) over ensembles in which every sampled point corresponds to an actual vacuum with all moduli stabilized, taking as inputs the distributions of the flux superpotential, the string coupling, and the warp-factor hierarchies of the throats. They found that as $\Lambda\to 0$ the $m_{3/2}$ distribution is tilted toward lower scales in both KKLT and LVS — opposite to earlier generic expectations — that adding a small number of further throat sectors preserves the tilt, and that the $\Lambda\to 0$ form should be stable against low-energy quantum corrections because it relies only on the parametric conditions $\Lambda\ll m_{3/2}^{2}$ (KKLT) or $\Lambda\ll m_{3/2}^{3}$ (LVS).

The result is, however, specific to the anti-D3 uplift, and the authors themselves flag this as the main limitation. Other controlled dS uplift mechanisms exist: D-term uplifting, in which field-dependent Fayet-Iliopoulos terms (D-term contributions associated with gauged symmetries, generated by gauge fluxes on D7-branes, higher-dimensional extended membranes in string theory on which open strings end) raise the AdS minimum; Kähler uplifting, in which the leading perturbative $\alpha'$ correction to the Kähler potential competes with non-perturbative effects to produce dS vacua at parametrically large volume; de Sitter vacua from dilaton-dependent non-perturbative effects; T-brane uplifting, where non-diagonal internal gauge-flux backgrounds on hidden-sector D7-branes generate a positive contribution to the moduli potential; and further LVS-type dS constructions. Whether the joint statistics of $m_{3/2}$ and $\Lambda$ in vacua built on these mechanisms resembles the anti-D3 case, and how numerous such vacua are relative to anti-D3-uplifted ones, is unknown: the field currently lacks a landscape-level, uplift-sector-resolved joint distribution of the SUSY-breaking scale and the cosmological constant.

Problem Statement

For type IIB flux compactifications with full moduli stabilization, derive the joint distribution of the supersymmetry-breaking scale (gravitino mass $m_{3/2}$) and the cosmological constant $\Lambda$ for de Sitter vacua obtained by uplift mechanisms other than the anti-D3-brane-in-warped-throat construction — namely D-term uplifting, Kähler uplifting, dilaton-dependent non-perturbative effects, and T-brane uplifting — in the same sense as the existing anti-D3 analysis of KKLT and LVS: sampling only over points corresponding to vacua, with the distributional inputs (flux superpotential, string coupling, and uplift-sector variables) stated together with their provenance, and including the behavior of the distribution as $\Lambda\to 0$, in particular whether the tilt toward lower $m_{3/2}$ persists. These four mechanisms constitute the required set: a joint distribution derived for any one of them is a resolving sub-result for that sector, and the objective is resolved by the four sector-specific distributions together with the uplift-sector abundance. In combination with these sector-specific distributions, determine the relative abundance of vacua with different uplift sectors, with the counting measure and its assumptions stated, so that the sector-specific results can be combined into a joint distribution of $m_{3/2}$ and $\Lambda$ over the flux landscape as a whole, establishing whether the low-scale tilt is an artifact of the anti-D3 sector or a property of the landscape. A related refinement with the same target is to determine whether low-energy quantum corrections modify the derived $\Lambda\to 0$ distribution forms beyond the parametric stability of the $\Lambda\ll m_{3/2}^{2}$ and $\Lambda\ll m_{3/2}^{3}$ regimes.

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