Saltatory Relaxation of the Cosmological Constant

Frank Wilczek; John March-Russell; Jonathan L. Feng; Savdeep Sethi

REVIEW

Saltatory Relaxation of the Cosmological Constant

Jonathan L. Feng, John March-Russell, Savdeep Sethi, Frank Wilczek

High Energy Physics - Theory · Submitted 30 May 2000 · 41 references

Pith's verdict: Two genuinely new string-theoretic ingredients for Brown-Teitelboim relaxation, wrapped around a headline relation that is conditional on an assumption the authors themselves flag.

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arXiv IDhep-th/0005276 (version 2)

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CitationFrank Wilczek, John March-Russell, Jonathan L. Feng, Savdeep Sethi. "Saltatory Relaxation of the Cosmological Constant." arXiv:hep-th/0005276 (2000). https://pith.science/paper/MPDZZ6AV

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The thing to know up front: the two new ingredients are real, and the paper is candid that its headline relation is conditional rather than derived. Judge it on that basis and it holds up reasonably well; judge it as a solution to the cosmological constant problem and it does not, which the authors also say (abstract:s5, and Sec. 6). What is new is worth the read. Sec. 3 makes the jump size in the effective cosmological term depend on compactification, so the quantized unit is not stuck at the Planck scale. Sec. 4 gives coincident branes an entropy, so nucleation rates pick up an exp(k^beta A T^2) enhancement. Neither is in Brown-Teitelboim, and both are the sort of thing only string theory supplies. The BT review in Sec. 2 is clean and the bounce formulas are transcribed correctly. Sec. 3.3 is honest in a way I appreciate: they try the conifold, find the charge density is only log-suppressed, conclude that the degenerating-cycle route does not actually deliver rho_2 ~ 10^-90, and then hope for a worse singularity in an N=1 setting. Footnote 2 records a correction from an earlier version. Sec. 6 lists the gaps without prompting: no FRW treatment, ambiguous temperature, uncertain beta, unreliable bounce action at high curvature. The soft spots, in order of weight. Eq. 41, lambda M^2 ~ M_SUSY^4, is asserted, not derived, and Eq. 47 inherits that. Eq. 47 also sits very close to the observed coincidence Lambda_obs^{1/4} ~ 10^-3 eV restated, so it reads as a consistency check on the scenario rather than a prediction that could have come out otherwise. The stability bound on lambda itself, Eq. 39, is the real dynamical result and survives without Eq. 41. Second, the naturalness condition rho_2 <~ Lambda_obs |lambda|^{1/2} is the hardest number in the paper, and Sec. 3 reaches a hope rather than a construction. Third, the density-of-states estimates are order-of-magnitude; the k-scaling of the bubble area against the exponent deserves more care than it gets. This is for people working on the cosmological term in string theory and on vacuum selection. It deserves referees.

Referee report

Significance

If the chain holds, the paper connects three otherwise unrelated scales — the observed cosmological term, the Planck scale, and the supersymmetry breaking scale — in a single inequality that is in principle testable at colliders, and it does so without invoking a new symmetry or a continuous relaxation field. The reframing of the stability requirement from absolute stability to stability over the age of the universe is a genuine conceptual move, and the observation that near-flat vacua inherit the stability of the adjacent anti-de Sitter vacua through the Coleman–de Luccia barrier is a useful one. The paper is also candid about its own weak points: it states the hidden-sector caveat in Sec. 6, discloses the concurrent Bousso–Polchinski work in the Note, and labels the temperature choice as an ambiguity rather than a result. The string-theory content, however, is at present a program rather than a derivation: the two computable routes to a small charge density are shown by the authors' own estimates to fall short by many orders of magnitude, so the claim that string theory 'provides' the required scale should be read as a conjecture about unexplored N=1 compactifications.

Major objections (5)

[Abstract; Sec. 5.1, Eq. (41); Sec. 6, Eq. (47)] The abstract states without qualification that the authors 'derive a non-trivial relation between the supersymmetry breaking scale and the value of the cosmological term', and Eq. (47) is presented as that relation. The derivation has four steps, and only the first three are dynamical: naturalness (Eq. 13) fixes ρ2 ∼ Λobs/√|λ|, the leading bounce action (Eq. 38) gives B ∼ x⁴Λobs/λ², and B ≳ 1 gives the bound |λ| ≲ x²√Λobs (Eq. 39). The fourth step, |λ|M² ∼ M_SUSY^4 (Eq. 41), is an assumption justified only by the remark that 'it is reasonable to expect' the bare term to be set by the supersymmetry breaking scale. The body is honest about this — Sec. 6 concedes that a hidden sector at a higher scale makes the simplest scenario's vacuum unstable — but the abstract is not. Since Eq. (47) is the paper's headline claim, the abstract should state the conditionality explicitly, and Sec. 5.1 should show how the bound on M_SUSY moves if λ is set at an intermediate scale, so the reader can see how much of the TeV conclusion is dynamics and how much is the identification in Eq. (41).
[Sec. 3.2–3.3, Eqs. (25)–(26)] The paper's stated purpose is to show that string/M theory can supply the small jump size that BT had to postulate, but the two cases the authors can actually compute both fail by enormous margins, and the failure is not displayed as a single quantitative statement. Direct descent requires gs ≲ 10^-44 for canonical ℓs, or gs ≲ 10^-23 in the best large-extra-dimension case, which the authors themselves note is hard to reconcile with Standard Model gauge couplings. Wrapped branes on conifold-like degenerating cycles give ρ2 ∼ 1/√|ln Z|, i.e. only logarithmic suppression, so satisfying Eq. (13) requires Z exponentially small. The mechanism therefore rests entirely on hypothesized N=1 singularities whose moduli-space metric is 'worse than logarithmic', and no example is given. This is load-bearing for the claim that the relaxation scale is a string-theory output rather than a re-parametrization of BT. I would ask for a short table or explicit display comparing the required ρ2 from Eq. (13) against what each of the three routes (direct descent, conifold wrapping, hypothesized N=1 singularity) can deliver, so the size of the gap is visible at a glance, and for the abstract and introduction to describe the small unit as a conjecture rather than a consequence of string theory.
[Sec. 2.3 and Sec. 5] Section 2.3 identifies the excessive relaxation time as one of the two principal obstacles to BT — the time to descend the staircase is so long that even the slow inflation of the penultimate vacuum would leave the universe devoid of matter — and the density-of-states factor of Sec. 4 is introduced precisely to cure it. But Sec. 5 never computes a relaxation time. It compares the degeneracy factor D to the bounce action B at a single step and concludes that 'tunneling proceeds rapidly', without ever forming a rate per unit four-volume, integrating it over the staircase from Λ_O ∼ |λ| down to Λobs, and comparing the result to the Hubble rate. Given that the number of steps required is of order √|λ|/ρ2 ∼ |λ|/Λobs, which is astronomically large for λ ∼ 10^-60, a per-step comparison is not by itself sufficient to establish that relaxation completes. This is the one place where the new ingredient is supposed to do real work, and the quantitative check is missing. A short subsection estimating Γ/H⁴ per step and the total time, even at the level of power counting, would substantially strengthen the paper.
[Sec. 4.1, Eq. (30); Sec. 5.2, Eqs. (44)–(46)] The exponent β in N_k ∼ exp(k^β A T²) is not computed; the authors argue it 'likely lies between β = 2 and β = 3/2' and adopt 3/2 as conservative. The qualitative conclusions are sensitive to this choice. In Sec. 5.2 the single-membrane endpoint scales as Λ_O ≳ |λ|^{2-1/β}, so β = 3/2 gives |λ|^{4/3} (Eq. 46) and requires three brane species, while β = 2 gives |λ|^{3/2} and requires only two. The multi-brane scenario further requires several distinct 2-branes whose charge densities are hierarchically separated by many orders of magnitude, which is asserted but not shown to arise from any compactification. Since the claim that the anthropic principle 'may be avoided' rests on this multi-stage picture, the paper should state explicitly which conclusions survive for β = 2 and for β = 3, and should say what in the compactification would produce the required spread of charge densities.
[Sec. 5.1, Eqs. (35)–(38)] The statement that 'D is maximized for kρ2 ∼ c_O' is the pivot of the one-step scenario, but it does not follow from the formulas as written. The degeneracy factor D = k^β A T² grows monotonically with k, while the bounce action saturates at B = 24π²M²/Λ_O as k → ∞ (Eq. 35), so the unconstrained tunneling probability P ∼ N_k e^{-B} increases without bound with k. The maximum at kρ2 ∼ c_O is obtained only after imposing by hand a restriction to interiors that 'live a long time and even remotely resemble ours'. That restriction is a selection postulate, not a consequence of the rate, and it is the same issue that Sec. 4.3 flags as a 'catastrophe' and resolves by asserting that the thin-wall action breaks down as b ∼ 1/k → 0. The paper should either state the long-lived-interior filter explicitly as a postulate at the point where it is used, or show that the short-distance cutoff of Sec. 4.3 actually suppresses large k before D overtakes B. As written, the 'exponentially most probable' language in Sec. 5.1 overstates what the rate calculation delivers.

Minor objections (8)

[Abstract and throughout] The abstract misspells 'Teitelboim' as 'Teitelb oim', and the sentence 'It is plausibly, though not certainly, satisfied in Nature' should read 'It is plausible, though not certain, that it is satisfied in Nature'. Similar OCR-level breaks ('quantifie d', 'tha t', 'l eng on the elliptic fiber') appear in the body and should be cleaned up in the final version.
[Sec. 5.1, Eqs. (37)–(38)] Equation (37) is the basis of the headline bound, but the approximations leading from it to Eq. (38) are not shown. Please state explicitly that c_O ∼ √|λ| and ρ2 ∼ Λobs/√|λ| are being substituted, and display the intermediate expression, so that a reader can verify the scaling B ∼ x⁴Λobs/λ² without reconstructing it.
[Sec. 5.2, Eq. (46)] The displayed exponent in the sentence preceding Eq. (46) is garbled in the manuscript ('ΛO >∼ |λ|2−β−1'); it should presumably read Λ_O ≳ |λ|^{2-1/β}. Please correct, and check that the β = 2 and β = 3/2 endpoints quoted in the surrounding text follow from the corrected formula.
[Sec. 4.2, Eq. (34)] The black-hole analogy (de Sitter ↔ ordinary hole, flat ↔ extremal, anti-de Sitter ↔ naked singularity) is suggestive but heuristic, and the geometric mean T ∼ (T_O T_I)^{1/2} is a postulate rather than a derived result. A sentence saying so explicitly, at the point where Eq. (34) is introduced, would prevent the reader from mistaking it for a consequence of the near-extremal-hole analysis of Refs. [39, 40].
[Sec. 3.1, Eq. (18)] The assertion that the four-form contribution to the effective cosmological term is positive for both M2-brane and wrapped-M5-brane effective 2-branes is important, since it fixes the sign of λ in Eq. (6). Please add a line showing how the sign follows from the sign convention in Eq. (18), rather than leaving it as an assertion.
[Sec. 2.1, Euclideanization discussion] The paragraph explaining that the sign of the F² term in Eq. (3) depends on the Euclideanization prescription, and that taking ⟨F4⟩ pure imaginary leaves the analysis unchanged, is careful but compressed. A footnote with the explicit substitution would make the invariance checkable.
[References] Reference [24] gives 'Phys. Lett. B367 97 (103)'; the year should presumably be 1996. Please also check that all hep-th numbers in the bibliography are current.
[Sec. 6] The acknowledgment that the mechanism has not been embedded in a realistic Friedmann–Robertson–Walker model, and that a later matter phase transition could drive the effective term negative with no obvious recovery path, is well placed. It would help to say in one sentence whether the authors regard this as a technical gap or as a potential obstruction to the scenario.

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The paper

The paper claims

String/M-theory brane properties and coincident-brane density of states can make saltatory relaxation of the cosmological constant fast enough, and requiring stability on cosmological timescales then implies a supersymmetry breaking scale below a few TeV.

The reading

The cosmological constant problem asks why the vacuum energy of the universe is some 120 orders of magnitude smaller than naive theoretical estimates. Brown and Teitelboim proposed that the vacuum energy relaxes in discrete jumps as membranes nucleate, each jump reducing the effective cosmological term by a quantized amount. The problem was that the required jump size is absurdly small, and the tunneling rate is exponentially suppressed. This paper argues that string/M-theory naturally supplies both fixes. First, the charge and tension of branes depend on the size and shape of extra dimensions, so the effective four-dimensional jump size can be exponentially small without fine-tuning fundamental parameters. Second, when many branes nucleate together, their internal degrees of freedom contribute a large density-of-states factor that can overcome the tunneling suppression. With these ingredients, the authors study whether the vacuum can relax to the observed value and then stay there. They find that requiring stability on cosmological timescales, rather than absolute stability, yields a relation: the supersymmetry breaking scale must be below about a few TeV. This is a conditional statement, resting on the assumption that the bare cosmological constant is set by the supersymmetry breaking scale. The authors are explicit that the work is incomplete: no realistic cosmological model with matter is constructed, and the temperature controlling the density-of-states factor is not under theoretical control.

Extended reading notes

Core claim

The central load-bearing assertion is Eq. (47): requiring that the effective cosmological term be stable on cosmological timescales, rather than absolutely stable, yields the relation M_SUSY^2 less than about (10^-3 eV)(M_Planck), equivalently a supersymmetry breaking scale below a few TeV. If correct, this connects the observed cosmological constant, the Planck scale, and the scale of supersymmetry breaking in a single inequality, and it makes the saltatory relaxation mechanism testable at colliders.

Load-bearing premise

The load-bearing premise is the identification of the bare cosmological constant with the supersymmetry breaking scale, Eq. (41): lambda M^2 ~ M_SUSY^4. The authors justify it only by saying it is 'reasonable to expect' (Sec. 5.1). If the bare cosmological term is instead set by a hidden sector at a much higher scale, the stability bound B > 1 fails and the vacuum is unstable in the simplest scenario. The authors acknowledge this explicitly: 'Unless the hidden sector contribution to the bare cosmological term is somehow suppressed to this TeV scale, in our simplest scenario the vacuum will be unstable.'

Original abstract

We modify and extend an earlier proposal by Brown and Teitelboim to relax the effective cosmological term by nucleation of branes coupled to a three-index gauge potential. Microscopic considerations from string/M theory suggest two major innovations in the framework. First, the dependence of brane properties on the compactification of extra dimensions may generate a very small quantized unit for jumps in the effective cosmological term. Second, internal degrees of freedom for multiply coincident branes may enhance tunneling rates by exponentially large density of states factors. These new features essentially alter the relaxation dynamics. By requiring stability on the scale of the lifetime of the universe, rather than absolute stability, we derive a non-trivial relation between the supersymmetry breaking scale and the value of the cosmological term. It is plausibly, though not certainly, satisfied in Nature.

The compiled record

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,492 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.87 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.11 · compares with prior results 0.67 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We modify and extend an earlier proposal by Brown and Teitelboim to relax the effective cosmological term by nucleation of branes coupled to a three-index gauge potential. Microscopic considerations from string/M theory suggest two major innovations in the framework. First, the dependence of brane properties on the compactification of extra dimensions may generate a very small quantized unit for jumps in the effective cosmological term. Second, internal degrees of freedom for multiply coincident branes may enhance tunneling rates by exponentially large density of states factors. These new features essentially alter the relaxation dynamics. By requiring stability on the scale of the lifetime of the universe, rather than absolute stability, we derive a non-trivial relation between the supersymmetry breaking scale and the value of the cosmological term. It is plausibly, though not certainly, satisfied in Nature.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We modify and extend an earlier proposal by Brown and Teitelboim to relax the effective cosmological term by nucleation of branes coupled to a three-index gauge potential.	contribution 0.98	0.88	0.08	asserted 0.99
s1	Microscopic considerations from string/M theory suggest two major innovations in the framework.	contribution 0.86	0.18	0.03	hedged possibility 0.86
s2	First, the dependence of brane properties on the compactification of extra dimensions may generate a very small quantized unit for jumps in the effective cosmological term.	contribution 0.92	0.21	0.07	hedged possibility 1.00
s3	Second, internal degrees of freedom for multiply coincident branes may enhance tunneling rates by exponentially large density of states factors.	contribution 0.96	0.21	0.06	hedged possibility 0.99
s4	These new features essentially alter the relaxation dynamics.	interpretation 0.88	0.53	0.09	asserted 0.98
s5	By requiring stability on the scale of the lifetime of the universe, rather than absolute stability, we derive a non-trivial relation between the supersymmetry breaking scale and the value of the cosmological term.	result 0.92	0.93	0.62	asserted 1.00
s6	It is plausibly, though not certainly, satisfied in Nature.	interpretation 0.84	0.39	0.11	hedged possibility 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from argument 0.59.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
C citations
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

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By dimensionally reducing Type II supergravity with fluxes, the authors show that all axion dependence of the flux potential flows through Minkowski 4-form auxiliary fields, which they argue forces corrections to appe...
Building an explicit de Sitter
hep-th 2012-08 conditional novelty 6.0 of 10

An explicit type IIB compactification on CP^4_{11169}[18] is constructed in which all geometric moduli are stabilized and the Kähler sector lands in a metastable de Sitter vacuum at volume about 52.
Minimal Simple de Sitter Solutions
hep-th 2008-10 conditional novelty 6.0 of 10

A four-dimensional de Sitter vacuum in type IIA string theory requires only O6-planes, a nonzero Romans mass, form fluxes, and negative internal curvature, and an explicit two-modulus example exists on a compact hyper...
Simple de Sitter Solutions
hep-th 2007-12 conditional novelty 6.0 of 10

A type IIA compactification on two nilmanifolds with orientifolds, KK5-branes, and fluxes yields metastable de Sitter minima whose curvature, coupling, and Hubble scale can be tuned small.
Retrofitting O'Raifeartaigh Models with Dynamical Scales
hep-th 2006-08 accept novelty 6.0 of 10

Small mass parameters in O'Raifeartaigh and Fayet supersymmetry-breaking models can be generated dynamically by coupling them to a hidden Yang-Mills sector, with discrete R symmetries enforcing the structure.
Signals of Inflation in a Friendly String Landscape
astro-ph 2006-04 conditional novelty 6.0 of 10

In a friendly string landscape with freely scanned mass parameters, inverted hybrid inflation predicts ns≈0.97, a <3% chance of just-so curvature, and an intermediate inflation scale near 10^12 GeV.
Spontaneous Inflation and the Origin of the Arrow of Time
hep-th 2004-10 conditional novelty 6.0 of 10

Empty de Sitter space is unstable to spontaneous eternal inflation, so total entropy can grow without bound in both time directions and the arrow of time arises dynamically rather than from a fine-tuned initial condition.
The Type IIA Flux Potential, 4-forms and Freed-Witten anomalies
hep-th 2018-02 conditional novelty 5.0 of 10

The full classical flux potential of type IIA Calabi-Yau orientifolds factorizes into a bilinear form whose axion dependence is fixed by Freed-Witten anomalies of 4d string defects.
Backreaction Issues in Axion Monodromy and Minkowski 4-forms
hep-th 2016-11 conditional novelty 5.0 of 10

Reformulating axion-monodromy backreaction in terms of Minkowski 4-forms shows the closed-string field range is capped near the Planck mass, while open-string moduli may allow a flux-tunable escape.
String Cosmology: A Review
hep-th 2007-10 accept novelty 3.0 of 10

String cosmology's central obstacle is the eta problem: moduli stabilization generically generates inflaton mass terms of order the Hubble scale, and no fully explicit, correction-controlled string inflation model yet exists.
Eternal inflation and its implications
hep-th 2007-02 accept novelty 3.0 of 10

Inflation is generically eternal forward in time, producing infinitely many pocket universes, but the inflating region cannot be eternal into the past, so some non-inflationary physics must set its past boundary.
String Cosmology: from the Early Universe to Today
hep-th 2023-03 conditional novelty 2.0 of 10

A comprehensive survey that organizes moduli stabilisation, string inflation, post-inflationary moduli domination, dark energy, and alternative early-universe scenarios into a single reference with observational targets.
Les Houches Lectures on Constructing String Vacua
hep-th 2008-03 unverdicted novelty 2.0 of 10

The paper is a pedagogical review of how F-theory and type IIB flux compactifications are used to construct string vacua with stabilized moduli.
The Cosmological Constant and the String Landscape
hep-th 2006-03 accept novelty 2.0 of 10

This paper argues that the cosmological constant is environmental and selected anthropically within the string landscape, a position that remains influential but unproven.
Lectures on Naturalness, String Landscape and Multiverse
hep-th 2020-08 unverdicted novelty 1.0 of 10

Lecture notes providing a technical introduction to naturalness problems and the string theory landscape for graduate students.

How the literature uses it

45 arXiv papers cite this in 56 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 32 sentences without a confident label

Contests it 1 paper

However the analysis in [ref] does not take moduli stabilization into account and the resultant distribution of quantized fluxes does not imply the preference of a small cosmological constant.

Probability of Vacuum Stability in Type IIB Multi-K\"ahler Moduli Models · Introduction · contest 0.84

Builds on it 1 paper

In its simplest incarnation, our approach is a generalization of the idea pioneered in [ref] about discharging cosmological constant by nonperturbative membrane nucleation processes, and also benefits from results of [ref] .

Implications of Weak Gravity Conjecture for de Sitter Decay by Flux Discharge · Introduction · 0.96

Uses its method or tool 1 paper

This rate may be derived from a worldline formalism that has the advantage of being easily generalized to higher-dimensional objects [ref] .

De Sitter space constraints on brane tensions and couplings · Nucleation of 2-branes · 0.88

Compares with it 4 papers

However there are important technical differences when we compare to those models since in our theory the bulk cosmological constant depends on the $4$-form dual magnetic fluxes (bi)linearly, as opposed to quadratically [ref] , as is clear from Eq. ([ref]).

Pancosmic Relativity and Nature's Hierarchies · Lg · 0.99
This is in sharp contrast to previous works [ref] where the flux sector does not include linear $4$-form terms but starts with $F^2$.

Implications of Weak Gravity Conjecture for de Sitter Decay by Flux Discharge · Introduction · 0.98
This picture is similar in looks to that proposed in Ref. [ref] , where a saltatory relaxation (i.e., relaxation by jumps) of the cosmological constant takes place as the universe tunnels repeatedly down the potential.

The Cosmological Constant Problem and Inflation in the String Landscape · Inflation in the landscape · 0.99

Cites it as background 17 papers

The instanton taxonomy was proffered in [ref] for the theories with $4$-form fluxes screening the cosmological constant [ref] .

A Quantum-Mechanical Mechanism for Reducing the Cosmological Constant · Introduction · 0.97
Their classification was initiated in [ref] for the theories with $4$-form fluxes screening the cosmological constant [ref] .

Hidden Variables of Gravity and Geometry and the Cosmological Constant Problem · Introduction · 0.89
A complementary line of inquiry, starting with the seminal work [ref] , has been to frame the question in terms of statistical distributions in the landscape of flux vacua [ref] .

Moduli Stabilisation and the Statistics of SUSY Breaking in the Landscape · Introduction · 0.95

Discussion

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