H-2 - TeV-scale type-I string theory with large extra dimensions solves the hierarchy problem
String embedding of the ADD large-extra-dimensions proposal (companion to hep-ph/9803315). The observed weakness of gravity at long distances is attributed to the large volume of the extra dimensions in which only closed-string gravity propagates, while SM matter is automatically localised on a D3-brane. The scenario requires an O(1) string coupling and new dimensions much larger than the weak scale.
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H-4 - A single warped extra dimension (RS1) generates the weak-Planck hierarchy exponentially
The second principal TeV-gravity model family (alongside ADD). Unlike ADD it needs only one extra dimension, whose size is only ~50 fundamental lengths; equivalently, from the visible-brane viewpoint the effective strong-gravity scale is ~TeV because of the small overlap of the graviton wave function with our brane, while the fundamental 5D scales all sit near M_Pl.
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H-27 - Under standard gravity, collider collisions are at least 22 orders of magnitude too weak to form a black hole
Candidate answer to the black-hole-production sub-question under the assumption that gravity is standard; the rival branch (TeV-scale gravity with large extra dimensions, under which production becomes possible) postdates this analysis’s frame — the paper’s estimate quantifies exactly how far standard gravity is from criticality, which is why the entire later black-hole debate runs through extra-dimension models. The paper adds that RHIC is actually less effective at raising nuclear density than lower-energy machines (less stopping power), and that no phenomenon suggestive of gravitational clumping has ever been observed in any collision.
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H-33 - The short-distance structure of gravity is something not listed here
Residual catch-all minted at step 4b so the cluster’s members cover the answer space; not one more named scenario, but the escape hatch “some completion of short-distance gravity not proposed in this literature”.
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The members carve the space of short-distance completions of gravity: the ordinary Planckian-scale answer (on which collider collisions are hopelessly far from gravitational collapse), the two low-scale scenarios the corpus proposes (large flat extra dimensions with a TeV string scale; one warped dimension), and a residual for any completion not stated in this literature. Exclusivity holds because these are incompatible claims about the same underlying geometry; the cluster is not exhaustive by construction (other low-scale mechanisms exist in principle), so the residual closes it. Torsion-balance limits, the absence-of-constraints situation of the late nineties, and collider null results are the discriminating evidence.
Prior
# HC-1 prior — short-distance structure / fundamental scale of gravity: is the QG scale within LHC reach?
# Members in HC-1.hypotheses order:
# [H-2 ADD / TeV-string large flat extra dimensions, H-4 RS1 single warped dimension,
# H-27 ordinary Einstein gravity with a Planckian fundamental scale, H-33 residual].
# Inputs: the outside view on weak-scale naturalness solutions; one base-rate observation pulled into the
# prior (O-43 via E-25, now used_for_prior: true — gravity untested below ~1 cm in 1998, 33 orders above the
# Planck length — so step 8 skips it); and the cluster's no-observation arguments from
# `no_observation_arguments.py <dir> --cluster HC-1` (A-3, A-4, A-5, A-54, A-62; all approved/corrected).
# The case-specific discriminators — the torsion-balance null (E-28) and the 1998 survival-of-constraints
# datum (E-29) — are left UNMARKED for step 8.
# H-27 ordinary Einstein gravity, Planckian fundamental scale — the anchor (unit weight). Outside view: the
# Planck scale (~1e19 GeV) is the default fundamental scale, and O-43 (E-25) records that no gravitational
# measurement reached below ~1 cm in 1998 — so nothing in the data distinguished a Planckian scale from a
# low one, and the split below rests on theory/naturalness priors, not evidence. A-54 (corrected, -> H-27)
# confirms the internal physics: even on wildly conservative overestimates the collapse parameter is ~1e-33,
# so this is the well-understood, physically default member. Anchored at 1.
w_standard = 1.0
# Combined low-scale (H-2 + H-4) mass as odds vs standard. Reference class: bold weak-scale naturalness
# solutions to a hierarchy problem (low-scale SUSY, technicolor, extra dimensions) proposed 1970s-2000s —
# a minority ever realized, none confirmed as of this corpus. These two are serious, calculable proposals
# by leading theorists (A-3 gives H-2 a concrete type-I string embedding; A-4 shows RS1 needs only an O(10)
# input), and A-62 (approved) shows the ADD premise was not excluded by any 1998 dataset — so the low-scale
# odds are small but not negligible. ~0.12 : 1 puts combined low-scale near 10%.
p_lowscale = 0.12 # odds that the fundamental gravity scale is ~TeV rather than Planckian
# Split of the low-scale mass between H-2 (ADD/TeV-string) and H-4 (RS1). Roughly even; a modest edge to ADD
# because within HC-1 it is the broader target — a family spanning field-theoretic ADD, type-I string, and
# n = 2-6 (A-3). RS1 is the more economical single-dimension mechanism and A-5 (approved) notes it is exempt
# from the ADD-type astrophysical/cosmological bounds, so it is not a priori disfavored; hence only a slight
# tilt. The discriminating torsion-balance truncation (E-28) is NOT baked in here — it is step-8's job.
share_add = 0.55
w_add = p_lowscale * share_add # H-2
w_rs1 = p_lowscale * (1 - share_add) # H-4
# H-33 residual — a short-distance completion that is neither Planckian-standard, nor ADD, nor RS1: e.g. some
# other low-scale mechanism, a different brane/warping setup, or a genuinely different QG scale nobody in this
# literature named. Its own argued weight (rule 4), not a leftover. The theory space of QG completions is not
# exhausted and O-43 leaves 33 untested orders of magnitude of physical room; but O-43 also excludes any
# completion that would have modified gravity above ~1 cm, and by ~2005 the proposed-completion space was
# well canvassed by a large community — so modest. ~0.14 : 1 vs standard, ~11% once normalised.
w_residual = 0.14
prior("HC-1", [w_add, w_rs1, w_standard, w_residual])