Extending theories on muon-specific interactions

The proton radius puzzle, the discrepancy between the proton radius measured in muonic hydrogen and electronic hydrogen, has yet to be resolved. There are suggestions that beyond-the-standard-model physics could resolve both this puzzle and the muon anomalous magnetic moment discrepancy. Karshenboim et al. point out that simple, nonrenormalizable, models in this direction involving new vector bosons have serious problems when confronting high energy data. The prime example is radiative corrections to $W\to \mu \nu$ decay which exceed experimental bounds. We show how embedding the model in a larger and arguably renormalizable theory restores gauge invariance of the vector particle interactions and controls the high energy behavior of decay and scattering amplitudes. Thus, beyond-the-standard-model explanations of the proton radius puzzle can still be viable.

Figure 1

The illustrative process $\nu \overline{\nu }\to W^{+}{W}^{-}$.

Figure 2

$W\to \mu \nu \varphi$.

Figure 3

The parameter space necessary to satisfy experimental constraints. Solid lines refer to constraints on $C_V^{\mu }$. Dashed lines refer to constraints on $C_A^{\mu }$. The green band, outlined by solid lines, is the constraint on $C_V^{\mu }$ necessary to solve the proton radius problem ($\pm 2\sigma$). The shaded red region, above the line labeled BR $(W\to \mu \nu \varphi )\dots$, is the forbidden region of $C_V^{\mu }$ due to the constraint that the branching ratio for $W$ goes to $\mu \nu {\varphi }_V+\mu \nu {\varphi }_A$ must be less than 4% under the assumption that $C_A^{\mu }$ solves the muonic $g-2$ problem. The shaded orange region is the restricted region on $C_V^{\mu }$ due to energy splittings in muonic Mg and Si at $2\sigma$. The green band, outlined by dashed lines, is the constraint on $C_A^{\mu }$ necessary to solve the muonic $g-2$ problem ($\pm 2\sigma$) under the assumption $C_V^{\mu }$ solves the proton radius problem ($\pm 2\sigma$).

Figure 4

$Z\to {\mu }^{-}{\mu }^{+}\varphi$.

Figure 5

$W\to \mu \nu \varphi$.

Figure 6

The parameter space necessary to satisfy experimental constraints on the scalar coupling $C_S^{\mu }$. The green band, outlined by solid lines, is the constraint on $C_S^{\mu }$ necessary to solve the proton radius problem ($\pm 2\sigma$). The shaded red region, above the line labeled BR $(W\to \mu \nu \varphi )\dots$, is the forbidden region of $C_S^{\mu }$ due to the requirement that the branching ratio for $W$ goes to $\mu \nu {\varphi }_S+\mu \nu {\varphi }_P$ must be less than 4% under the assumption that $C_P^{\mu }$ solves the muonic $g-2$ problem. The shaded orange region is the restricted region on $C_S^{\mu }$ due to energy splittings in muonic Mg and Si at $2\sigma$. The dashed green line is the constraint on $C_P^{\mu }$ necessary to solve both the proton radius and muonic $g-2$ discrepancies (for clarity, only the central value is shown).