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\title{Supplemental Material to ``Decay law of magnetic turbulence with helicity balanced by chiral fermions''}

\author{by Axel Brandenburg, Kohei Kamada, \& Jennifer Schober}
\date{\today}
\begin{document}
\maketitle

Following Zhou+22 [6], the function ${\cal I}_{\rm H}(t,R)$
can be recast as a weighted integral
\begin{equation}
{\cal I}_{\rm H}(t,R)=\frac{1}{V}\int\frac{\dd^3k}{(2\pi)^3}\ w_R(\kk)\ h^*(\kk)h(\kk).
\label{eqn:Iv_general}
\end{equation}
The weight function $w_R(\kk)$ depends on the shape of $V_R$.
For a cubic region with length $2R$, we have
\begin{equation}
w_R(\kk)=w_{\rm cube}^{\rm BC}(\kk)\equiv8R^3\prod_{i=1}^3
j_0^2(k_iR);
\end{equation}
see \Fig{psaff_Box_cube_1024a_mu10_k002b}.
Here $j_0(x)=\sin x/x$.
For a spherical region with radius $R$,
\begin{equation}
w_R(\kk)=w_{\rm sph}^{\rm BC}(k)\equiv\frac{4\pi R^3}{3}
\left[\frac{6j_1(kR)}{kR}\right]^2,
\label{eq:wsph}
\end{equation}
which is the version shown in the main paper.

\begin{figure}[t!]\begin{center}
\includegraphics[width=.9\columnwidth]{psaff_Box_cube_1024a_mu10_k002b}
\end{center}\caption[]{
Box counting method with cubic volumes showing
(a) ${\cal I}_H(R,t)$ versus $R$ for different times (the normalized
values of $t\eta k_0^2$ are indicated by different colors), and
(b) ${\cal I}_H(R,t)$ versus $t$ (normalized) for two choices of $R$,
indicated by solid ($k_0R\approx1.5$) and dashed ($k_0R\approx6$) lines
in both panels.
The $t^{-0.16}$ scaling is indicated as the dashed-dotted curve for
comparison.
}\label{psaff_Box_cube_1024a_mu10_k002b}\end{figure}

\begin{figure}[t!]\begin{center}
\includegraphics[width=.9\columnwidth]{psaff_Cor_cube_1024a_mu10_k002b}
\end{center}\caption[]{
Similar to \Fig{psaff_Box_cube_1024a_mu10_k002b} using the
correlation-integral method for cubic volumes.
}\label{psaff_Cor_cube_1024a_mu10_k002b}\end{figure}

\begin{figure}[t!]\begin{center}
\includegraphics[width=.9\columnwidth]{psaff_Cor_sph1024a_mu10_k002b}
\end{center}\caption[]{
Similar to \Fig{psaff_Box_cube_1024a_mu10_k002b} using the
correlation-integral method for spherical volumes.
}\label{psaff_Cor_sph1024a_mu10_k002b}\end{figure}

Alternatively, by the correlation-integral (CI) method, we have
\begin{equation}
w_R(\kk)=w_{\rm cube}^{\rm CI}(\kk)\equiv8R^3\prod_{i=1}^3
j_0(k_i R),
\label{eqn:w_spc}
\end{equation}
for a cubic region $V_R$; see \Fig{psaff_Cor_cube_1024a_mu10_k002b}, and
\begin{equation}
w_R(\kk)=w_{\rm sph}^{\rm CI}(k)\equiv\frac{4\pi R^3}{3}
\frac{6j_1(kR)}{kR}.
\end{equation}
for a spherical region; \Fig{psaff_Cor_sph1024a_mu10_k002b}.

\end{document}
