Generating the cosmic microwave background power asymmetry with $g_{NL}$

We consider a higher order term in the $\delta N$ expansion for the CMB power asymmetry generated by a superhorizon isocurvature field fluctuation. The term can generate the asymmetry without requiring a large value of $f_{NL}$. Instead it produces a non-zero value of $g_{NL}$. A combination of constraints lead to an allowed region in $f_{NL}-g_{NL}$ space. To produce the asymmetry with this term without a large value of $f_{NL}$ we find that the isocurvature field needs to contribute less than the inflaton towards the power spectrum of the curvature perturbation.