Dark dimension and the grand unification of forces

The dark dimension scenario, predicting one extra mesoscopic dimension in the micron range, has emerged by applying various swampland principles to the dark energy. In this note we find that realizing the grand unification of gauge forces is highly constraining in this context. Without actually constructing any grand unified theory (GUT) models, we argue that the mere assumption of grand unification of forces in this scenario, together with the experimental bounds on massive replicas of the Standard Model gauge bosons, predicts an upper bound for the GUT scale, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>M</a:mi><a:mi>GUT</a:mi></a:msub><a:mo>≲</a:mo><a:msup><a:mn>10</a:mn><a:mn>16</a:mn></a:msup><a:mtext> </a:mtext><a:mtext> </a:mtext><a:mi>GeV</a:mi></a:math>. Combined with the experimental bound on the proton lifetime, this predicts that the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>X</c:mi></c:math> gauge boson mediating proton decay is a 5D solitonic string of Planckian tension stretched across a length scale <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>L</e:mi><e:mo>∼</e:mo><e:mo stretchy="false">(</e:mo><e:mn>1</e:mn><e:mi>–</e:mi><e:mn>10</e:mn><e:mtext> </e:mtext><e:mtext> </e:mtext><e:mi>TeV</e:mi><e:msup><e:mo stretchy="false">)</e:mo><e:mrow><e:mo>−</e:mo><e:mn>1</e:mn></e:mrow></e:msup></e:math> ending on gauge branes of the same diameter <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mo>∼</i:mo><i:mi>L</i:mi></i:math>. This leads to a mass of <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msub><k:mi>M</k:mi><k:mi>X</k:mi></k:msub><k:mo>∼</k:mo><k:msup><k:mn>10</k:mn><k:mn>15</k:mn></k:msup><k:mi>–</k:mi><k:msup><k:mn>10</k:mn><k:mn>16</k:mn></k:msup><k:mtext> </k:mtext><k:mtext> </k:mtext><k:mi>GeV</k:mi></k:math>. In particular assuming grand unification in the dark dimension scenario results in a tower of Kaluza-Klein excitations of Standard Model gauge bosons on the gauge branes in the 1–10 TeV range. This suggests that the diameter/separation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>L</m:mi></m:math> of the gauge branes correlates with both the weak scale <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mo>∼</o:mo><o:mn>1</o:mn><o:mo>/</o:mo><o:mi>L</o:mi></o:math> near a TeV the GUT scale <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mo>∼</q:mo><q:msubsup><q:mi>M</q:mi><q:mn>5</q:mn><q:mn>2</q:mn></q:msubsup><q:mi>L</q:mi></q:math> at <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msup><s:mn>10</s:mn><s:mn>16</s:mn></s:msup><s:mtext> </s:mtext><s:mtext> </s:mtext><s:mi>GeV</s:mi></s:math>.