The dark dimension and the Swampland

Abstract

A bstract Motivated by principles from the Swampland program, which characterize requirements for a consistent UV completion of quantum gravity, combined with observational data, we are led to a unique corner of the quantum gravity landscape. In particular, using the Distance/Duality conjecture and the smallness of dark energy, we predict the existence of a light tower of states and a unique extra mesoscopic dimension of length $$ l\sim {\Lambda}^{-\frac{1}{4}}\sim {10}^{-6}m $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>l</mml:mi> <mml:mo>∼</mml:mo> <mml:msup> <mml:mi>Λ</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>4</mml:mn> </mml:mfrac> </mml:mrow> </mml:msup> <mml:mo>∼</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> </mml:msup> <mml:mi>m</mml:mi> </mml:math> , with extra massless fermions propagating on it. This automatically leads to a candidate for a tower of sterile neutrinos, and an associated active neutrino mass scale $$ {m}_{\nu}\sim {\left\langle H\right\rangle}^2{\Lambda}^{-\frac{1}{12}}{M}_{\textrm{pl}}^{-\frac{2}{3}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>m</mml:mi> <mml:mi>ν</mml:mi> </mml:msub> <mml:mo>∼</mml:mo> <mml:msup> <mml:mfenced> <mml:mi>H</mml:mi> </mml:mfenced> <mml:mn>2</mml:mn> </mml:msup> <mml:msup> <mml:mi>Λ</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>12</mml:mn> </mml:mfrac> </mml:mrow> </mml:msup> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mi>pl</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mfrac> <mml:mn>2</mml:mn> <mml:mn>3</mml:mn> </mml:mfrac> </mml:mrow> </mml:msubsup> </mml:math> . Moreover, assuming the mechanism for stabilization of this dark dimension leads to similar masses for active and sterile neutrinos we are led to the prediction of a Higgs vev $$ \left\langle H\right\rangle \sim {\Lambda}^{\frac{1}{6}}{M}_{\textrm{pl}}^{\frac{1}{3}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mi>H</mml:mi> </mml:mfenced> <mml:mo>∼</mml:mo> <mml:msup> <mml:mi>Λ</mml:mi> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>6</mml:mn> </mml:mfrac> </mml:msup> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mi>pl</mml:mi> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>3</mml:mn> </mml:mfrac> </mml:msubsup> </mml:math> . Another prediction of the scenario is a species scale $$ \hat{M}\sim {\Lambda}^{\frac{1}{12}}{M}_{\textrm{pl}}^{\frac{2}{3}}\sim {10}^9\hbox{--} {10}^{10} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>M</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> <mml:mo>∼</mml:mo> <mml:msup> <mml:mi>Λ</mml:mi> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>12</mml:mn> </mml:mfrac> </mml:msup> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mi>pl</mml:mi> <mml:mfrac> <mml:mn>2</mml:mn> <mml:mn>3</mml:mn> </mml:mfrac> </mml:msubsup> <mml:mo>∼</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>9</mml:mn> </mml:msup> <mml:mo>–</mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>10</mml:mn> </mml:msup> </mml:math> GeV, corresponding to the higher-dimensional Planck scale. This energy scale may be related to the resolution of the instability of the Higgs effective potential present at a scale of ~10 11 GeV. We also speculate about the interplay between this energy scale and the GZK limit on ultra-high energy cosmic rays.

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