Revisiting series expansions of neutrino oscillation and decay probabilities in matter

We present analytic expressions for three-flavor neutrino oscillations in presence of invisible neutrino decay and matter effects. Using the well-known Cayley-Hamilton formalism, the leading-order terms are derived for oscillation probabilities in all major channels assuming the neutrino mass eigenstate <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>ν</a:mi><a:mn>3</a:mn></a:msub></a:math> to decay. Our work extends and complements previous studies utilizing the Cayley-Hamilton theorem, providing the series expansions for <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>ν</c:mi><c:mi>e</c:mi></c:msub><c:mo stretchy="false">→</c:mo><c:msub><c:mi>ν</c:mi><c:mi>e</c:mi></c:msub></c:math>, <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:msub><f:mi>ν</f:mi><f:mi>e</f:mi></f:msub><f:mo stretchy="false">→</f:mo><f:msub><f:mi>ν</f:mi><f:mi>μ</f:mi></f:msub></f:math>, <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msub><i:mi>ν</i:mi><i:mi>e</i:mi></i:msub><i:mo stretchy="false">→</i:mo><i:msub><i:mi>ν</i:mi><i:mi>τ</i:mi></i:msub></i:math>, <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:msub><l:mi>ν</l:mi><l:mi>μ</l:mi></l:msub><l:mo stretchy="false">→</l:mo><l:msub><l:mi>ν</l:mi><l:mi>μ</l:mi></l:msub></l:math>, and <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:msub><o:mi>ν</o:mi><o:mi>μ</o:mi></o:msub><o:mo stretchy="false">→</o:mo><o:msub><o:mi>ν</o:mi><o:mi>τ</o:mi></o:msub></o:math>. The accuracy of the analytical formulas is investigated by comparing the results with numerically calculated probabilities. We also comment on the implications on unitarity violation.