Gauge dependence of helicity in an open magnetic flux tube ($H_m/L=\pi bC{R}_0^2$)

An open flux tube carries magnetic flux through its end caps. Its ordinary magnetic helicity can therefore change under a single-valued gauge transformation. For $\mathbf{B}=aR{\mathbf{e}}_{\varphi }+b{\mathbf{e}}_z$ inside $R<R_0$ and zero outside, a regular family of magnetic vector potentials is $A_{\varphi }=bR/2$, $A_z=C-a{R}^2/2$ inside and $A_{\varphi }=b{R}_0^2/(2R)$, $A_z=C-a{R}_0^2/2$ outside. Their tangential components agree at $R_0$. Inside, $\mathbf{A}\cdot \mathbf{B}=bC$, so the helicity per unit length is $\pi bC{R}_0^2$. Changing $C$ is the gauge transformation $\Lambda =\Delta Cz$, and changes this result whenever $b\ne 0$. The convention $A_z=0$ outside gives $C=a{R}_0^2/2$ and helicity per unit length $\pi ab{R}_0^4/2$, while $C=0$ gives zero. Without an end-cap or gauge convention the ordinary helicity is not unique.