Can a nonstandard invisible pair mimic the Michel distribution?

We ask whether a measured Michel distribution, apparently in excellent agreement with the Standard Model interpretation of the \ell_i \to \ell_j \nu\bar\nu <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>→</mml:mo> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mi>ν</mml:mi> <mml:mover> <mml:mi>ν</mml:mi> <mml:mo accent="true">‾</mml:mo> </mml:mover> </mml:mrow> </mml:math> decay, could instead arise from a different invisible sector. Within a general low-energy effective field theory, we analyze lepton decays \ell_i \to \ell_j + X\bar X <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>→</mml:mo> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mi>j</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>X</mml:mi> <mml:mover> <mml:mi>X</mml:mi> <mml:mo accent="true">‾</mml:mo> </mml:mover> </mml:mrow> </mml:math> for electrically neutral, color-singlet, mutually conjugate invisible pairs X\bar X <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mover> <mml:mi>X</mml:mi> <mml:mo accent="true">‾</mml:mo> </mml:mover> </mml:mrow> </mml:math> of spin up to 2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mn>2</mml:mn> </mml:math> , allowing (pseudo)scalar, (axial)vector, and antisymmetric tensor interactions in the lepton current, focusing on the massless limit relevant for exact degeneracies. We formulate a criterion for indistinguishability based on the full set of measurable differential distributions. Under these assumptions, besides the obvious spin 1/2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mi>/</mml:mi> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> case, there is a unique nontrivial solution: a massless complex scalar pair coupled through a purely left-handed vector current exactly reproduces the standard Michel pattern, including its extensions to daughter-lepton polarization and radiative channels. All other cases studied here are distinguishable, in particular because higher-spin invisible pairs produce additional kinematic prefactors. These results isolate the only nonstandard and nontrivial invisible sector that can remain hidden in Michel-type lepton decay measurements. Phenomenologically, a subpercent branching fraction into this channel could bias the conventional muon-lifetime extraction of G_F <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>G</mml:mi> <mml:mi>F</mml:mi> </mml:msub> </mml:math> upward by enough for the corrected value to agree with the CKM-unitarity determination, while increasing the discrepancy with the electroweak fit.