Irreducible Bhabha background in the detection of muonium-antimuonium conversion

Experiments such as MACS and the proposed MACE study muonium-antimuonium conversion by the energies of the final state <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msup> <a:mi>e</a:mi> <a:mo>±</a:mo> </a:msup> </a:math> . The <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msup> <c:mi>e</c:mi> <c:mo>+</c:mo> </c:msup> </c:math> and <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mrow> <e:msup> <e:mrow> <e:mi>e</e:mi> </e:mrow> <e:mrow> <e:mo>−</e:mo> </e:mrow> </e:msup> </e:mrow> </e:math> from an antimuonium decay tend to be nonrelativistic and relativistic, respectively, and vice versa for muonium. However, these <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:msup> <g:mi>e</g:mi> <g:mo>±</g:mo> </g:msup> </g:math> can exchange their energies by hard Bhabha scattering, causing muonium to fake an antimuonium decay signal. We compute the rate for this background and find that, while negligible for MACE, it will become larger than the signal for conversion probabilities less than <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msup> <i:mn>10</i:mn> <i:mrow> <i:mo>−</i:mo> <i:mn>18</i:mn> </i:mrow> </i:msup> </i:math> .