Nuclear-level effective theory of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>μ</mml:mi><mml:mo>→</mml:mo><mml:mi>e</mml:mi></mml:mrow></mml:math> conversion: Inelastic process

Mu2e and COMET will search for electrons produced via the neutrinoless conversion of stopped muons bound in 1s atomic orbits of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mmultiscripts><a:mi>Al</a:mi><a:mprescripts/><a:none/><a:mn>27</a:mn></a:mmultiscripts></a:math>, improving existing limits on charged lepton flavor violation (CLFV) by roughly four orders of magnitude. Conventionally, <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:mi>μ</b:mi><b:mo>→</b:mo><b:mi>e</b:mi></b:mrow></b:math> conversion experiments are optimized to detect electrons originating from transitions where the nucleus remains in the ground state, thereby maximizing the energy of the outgoing electron. Clearly, detection of a positive signal in forthcoming experiments would stimulate additional work—including subsequent conversion experiments using complementary nuclear targets—to further constrain the new physics responsible for CLFV. Here we argue that additional information can be extracted without the need for additional experiments, by considering inelastic conversion in <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mmultiscripts><c:mi>Al</c:mi><c:mprescripts/><c:none/><c:mn>27</c:mn></c:mmultiscripts></c:math>. Transitions to low-lying nuclear excited states can modify the near-endpoint spectrum of conversion electrons, with the ratio of the elastic and inelastic responses being sensitive to the underlying CLFV operator. We extend the nuclear effective theory of <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mi>μ</d:mi><d:mo>→</d:mo><d:mi>e</d:mi></d:mrow></d:math> conversion to the inelastic case, which adds five new response functions to the six that arise for the elastic process. We evaluate these nuclear response functions in <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mmultiscripts><e:mi>Al</e:mi><e:mprescripts/><e:none/><e:mn>27</e:mn></e:mmultiscripts></e:math> and calculate the resulting conversion-electron signal, taking into account the resolution anticipated in Mu2e/COMET. We find that <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mmultiscripts><f:mi>Al</f:mi><f:mprescripts/><f:none/><f:mn>27</f:mn></f:mmultiscripts></f:math> is an excellent target choice from the perspective of the new information that can be obtained from inelastic <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:mrow><g:mi>μ</g:mi><g:mo>→</g:mo><g:mi>e</g:mi></g:mrow></g:math> conversion.