Tunneling away the relic neutrino asymmetry

The Earth acts as a matter potential for relic neutrinos which modifies their index of refraction from vacuum by <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>δ</a:mi><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:math>. It has been argued that the refractive effects from this potential should lead to a large <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mrow><c:mi mathvariant="script">O</c:mi><c:mo stretchy="false">(</c:mo><c:msqrt><c:mi>δ</c:mi></c:msqrt><c:mo stretchy="false">)</c:mo></c:mrow></c:math> neutrino-antineutrino asymmetry at the surface of the Earth. This result was computed by treating the Earth as flat. In this work, we revisit this calculation in the context of a perfectly spherical Earth. We demonstrate, both numerically and through analytic arguments, that the flat-Earth result is only recovered under the condition <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:msup><h:mi>δ</h:mi><h:mrow><h:mn>3</h:mn><h:mo>/</h:mo><h:mn>2</h:mn></h:mrow></h:msup><h:mi>k</h:mi><h:mi>R</h:mi><h:mo>≫</h:mo><h:mn>1</h:mn></h:math>, where <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" display="inline"><j:mi>k</j:mi></j:math> is the typical momentum of the relic neutrinos and <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mi>R</l:mi></l:math> is the radius of the Earth. This condition is required to prevent antineutrinos from tunneling into classically inaccessible trajectories below the Earth’s surface and washing away the large asymmetry. As the physical parameters of the Earth do not satisfy this condition, we find that the asymmetry at the surface should only be <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:mrow><n:mi mathvariant="script">O</n:mi><n:mo stretchy="false">(</n:mo><n:mi>δ</n:mi><n:mo stretchy="false">)</n:mo></n:mrow></n:math>. While the asphericity of the Earth may serve as a loophole to our conclusions, we argue that it is still difficult to generate a large asymmetry even in the presence of local terrain. Published by the American Physical Society 2024