String theory and grand unification suggest a submicroelectronvolt QCD axion

Axions, grand unification, and string theory are each compelling extensions of the Standard Model. We show that combining these frameworks imposes strong constraints on the QCD axion mass. Using perturbative unitarity arguments and explicit string compactifications—such as those from the Kreuzer-Skarke (KS) type IIB ensemble—we find that the axion mass is favored to lie within the range <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msup><a:mn>10</a:mn><a:mrow><a:mo>−</a:mo><a:mn>11</a:mn></a:mrow></a:msup><a:mtext> </a:mtext><a:mtext> </a:mtext><a:mi>eV</a:mi><a:mo>≲</a:mo><a:msub><a:mi>m</a:mi><a:mi>a</a:mi></a:msub><a:mo>≲</a:mo><a:msup><a:mn>10</a:mn><a:mrow><a:mo>−</a:mo><a:mn>8</a:mn></a:mrow></a:msup><a:mtext> </a:mtext><a:mtext> </a:mtext><a:mi>eV</a:mi></a:math>. This range is directly relevant for near-future axion dark matter searches, including ABRACADABRA/DMRadio and CASPEr. We argue that grand unification and the absence of proton decay suggest a compactification volume that keeps the string scale above the unification scale (<c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mo>∼</c:mo><c:msup><c:mn>10</c:mn><c:mn>16</c:mn></c:msup><c:mtext> </c:mtext><c:mtext> </c:mtext><c:mi>GeV</c:mi></c:math>), which in turn limits how heavy the axion can be. The same requirements limit the KS axiverse to have at most <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mo>∼</e:mo><e:mn>47</e:mn></e:math> axions. As an additional application of our methodology, we search for axions in the KS axiverse that could explain the recent Dark Energy Spectroscopic Instrument hints of evolving dark energy but find none with high enough decay constant (<g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi>f</g:mi><g:mi>a</g:mi></g:msub><g:mo>≳</g:mo><g:mn>2.5</g:mn><g:mo>×</g:mo><g:msup><g:mn>10</g:mn><g:mn>17</g:mn></g:msup><g:mtext> </g:mtext><g:mtext> </g:mtext><g:mi>GeV</g:mi></g:math>); we comment on why such high decay constants and low axion masses are difficult to obtain in string compactifications more broadly.