Proposes that the weak/Planck hierarchy problem is solved if there are n≥2 large (sub-mm to mm scale) flat extra dimensions into which gravity, but not the Standard Model, propagates, diluting gravity’s apparent strength in 3+1D. This lowers the true (higher-dimensional) Planck scale to as low as ~1 TeV. It is the model (arXiv:hep-ph/9803315) that makes TeV-scale quantum gravity — and hence microscopic black-hole production at colliders — a live theoretical possibility at all; without it there is no mechanism-level route to LHC black holes. relevance_note: root paper of the ONLY class of models (TeV-scale gravity) under which the LHC could conceivably form black holes.
§1 — The proposal
O-43 - As of 1998 gravity had been accurately measured only down to about a centimeter - 33 untested orders of magnitude above the Planck length
The uncontested empirical opening that makes TeV-scale gravity a live possibility at all: nothing measured excluded large gravity-only extra dimensions below ~1 cm (by 1998 sub-millimeter torsion experiments were only planned). This is the gate on the whole is-LHC-black-hole-production-possible branch of the analysis.
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H-30 - The hierarchy problem is solved by n of 2 or more large gravity-only extra dimensions with the true Planck scale at about a TeV
The root model (ADD, arXiv:hep-ph/9803315) of the only mechanism class under which LHC collisions could form black holes: with the true Planck scale at ~TeV, trans-Planckian parton collisions become possible at colliders. The framework abandons supersymmetric unification, trades the m_EW/M_Pl hierarchy for a mm-vs-weak-scale hierarchy (argued stable, a property of a solution rather than a fine-tuned Lagrangian parameter), and requires unknown TeV-scale quantum gravity to suppress dangerous proton-decay and flavor operators by assumption. Primordial nucleosynthesis survives (MeV energies excite no gravitons); an explicit 6D vortex construction with Pati-Salam bulk group is given as existence proof.
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A-61 - Gauss law in 4+n dimensions ties a TeV fundamental scale to R of 1e30overn minus 17 cm - excluding n=1 and hiding n of 2 or more from all existing tests
Reasoning (the calculation):
- For two masses within r << R, gravitational flux spreads through all 4+n dimensions: V(r) ~ m1 m2 / (M_(4+n)^(n+2) r^(n+1)). For r >> R flux lines cannot penetrate the compact dimensions further, giving V(r) ~ m1 m2/(M_(4+n)^(n+2) R^n r): the effective 4D coupling is M_Pl^2 ~ M_(4+n)^(2+n) R^n (eq. 3).
- Setting M_(4+n) = m_EW ~ 1 TeV and solving for R: R ~ 1e(30/n) x 1e-17 cm x (1 TeV/m_EW)^(1+2/n) (eq. 4). n=1 → 1e13 cm: gravity would deviate from Newton at solar-system distances — excluded. n=2 → 0.1-1 mm; n=6 → ~1e-12 cm. Since gravity was untested below ~1 cm, every n ≥ 2 was consistent with all data.
- Because the SM is measured to weak-scale distances, its fields cannot propagate in the extra dimensions and must be localized to a 4D wall of thickness ~1/m_EW; only the graviton lives in the bulk. Charge conservation still holds on the wall (massless gauge fields are localized; an escaping charge stretches a flux tube of tension ~m_EW^2 that pulls it back or breaks into charge pairs), while energy can be carried into the bulk above threshold — the sharp p_T cutoff and fireworks signatures.
Step 6 verdict
Verdict: approved (checked). Traced: flux spreading in 4+n dimensions gives V ~ 1/(M^(n+2) r^(n+1)) for r << R; matching at r ~ R yields M_Pl^2 ~ M^(2+n) R^n; solving with M = 1 TeV reproduces R ~ 1e(30/n - 17) cm. n=1 → 1e13 cm (solar-system-scale deviation, excluded); n=2 → ~0.1 mm, below the ~1 cm 1998 gravity frontier (O-43), so all n >= 2 were untested. O(2pi)^n compactification-convention factors shift R by less than an order and do not touch the conclusion. The wall-localization corollary (SM fields measured to weak-scale distances must be brane-bound; charge conservation via the flux-tube argument) is coherent. Valid.
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§2 — Phenomenological and astrophysical constraints
A-62 - Graviton emission scales as ΔE to the n+2 over m_EW - so 1998 lab and astrophysical data fail to exclude TeV-scale gravity
Reasoning:
- Emission rate: amplitude ∝ 1/M_Pl per KK mode, rate ∝ 1/M_Pl^2 x (ΔE R)^n modes below ΔE; using M_Pl^2 = m_EW^(2+n) R^n this collapses to ΔE^n/m_EW^(2+n) (eq. 6) — equivalently the 4+n-dimensional graviton couples as 1/m_EW^((2+n)/2). Branching ratio ~(ΔE/m_EW)^(2+n) (eq. 7): steeply suppressed for ΔE << 1 TeV, which is why a TeV cutoff for gravity survives keV-MeV astrophysics.
- Lab: largest ΔE processes dominate — Upsilon → X + graviton ~1e-8 (unobservable), Z → X + graviton ~1e-5, the strongest laboratory constraint, easily satisfied given the steep m_EW sensitivity.
- Astrophysics via the goldstone dictionary 1/F^2 ↔ ΔE^n/m_EW^(2+n) (eq. 8), worst case n=2, m_EW=1 TeV: Sun (ΔE ~ keV) F_eff ~ 1e12 GeV, totally safe; red giants (~100 keV) F_eff ~ 1e10 GeV, an order of magnitude above the limit; SN 1987A (20-70 MeV) F_eff ~ 1e8 GeV vs claimed 1e10 GeV lower limit — the one real constraint, met by n > 2 or m_EW ≳ 10 TeV; for n ≳ 7, 1/R ≳ 100 MeV exceeds all relevant temperatures and no astrophysical bound applies.
- Cosmic rays at 1e15-1e19 eV (CM 1-100 TeV on protons) do not already probe the exotic regime: they are accelerated smoothly, without hard interactions, and interact dominantly by soft QCD — so no significant constraint arises from them. (Nucleosynthesis is likewise untouched: MeV per particle cannot excite gravitons.)
Step 6 verdict
Verdict: approved (checked). Traced the collapse: per-mode amplitude 1/M_Pl, multiplicity (Delta-E R)^n, so rate ~ (1/M_Pl^2)(Delta-E R)^n = Delta-E^n / m_EW^(2+n) using M_Pl^2 = m_EW^(2+n) R^n - checks; branching ratio (Delta-E/m_EW)^(n+2) follows. The dataset comparisons then follow from the goldstone-dictionary premise (eq. 8) and the quoted 1998 limits; the SN 1987A tension is honestly retained inside the statement with its two escape valves (m_EW >~ 10 TeV or n > 2). The cosmic-ray point - acceleration is smooth and interactions are soft QCD, so no hard TeV-regime probe exists - is a valid non-exclusion argument. Conclusion “unexcluded as of 1998” follows conditional on the quoted limits. Valid.
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