Can a Nonstandard Invisible Pair Mimic the Michel Distribution?

Abstract

We ask whether a measured Michel distribution, apparently in excellent agreement with the Standard Model interpretation of the ${\ell }_i\to {\ell }_j\nu \bar{\nu }$ decay, could instead arise from a different invisible sector. Within a general low-energy effective field theory, we analyze lepton decays ${\ell }_i\to {\ell }_j+X\bar{X}$ for electrically neutral, color-singlet, mutually conjugate invisible pairs $X\bar{X}$ of spin up to $2$, 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 $1/2$ 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 $G_F$ upward by enough for the corrected value to agree with the CKM-unitarity determination, while increasing the discrepancy with the electroweak fit.

6 pages, 1 (new) figure (expanded phenomenological discussion on the possibility of bringing into agreement the low-energy determinations of $G_F$, from muon decay and Cabibbo unitarity, as well as the corresponding implications and feasibility)

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