%\documentclass[superscriptaddress,aps,prd,preprintnumbers,longbibliography,abbrv,twocolumn]{revtex4-1}
\documentclass[superscriptaddress,aps,prd,preprintnumbers,abbrv,nofootinbib,twocolumn]{revtex4-1}

%%%%%%%%%%%% packages 
\usepackage{amsfonts,amsmath,amssymb}
\usepackage{bm,tensor}
\usepackage{graphicx}
\usepackage{braket}
\usepackage{ifpdf}
\usepackage[utf8]{inputenc}
\usepackage{color}
\usepackage{xfrac}
%\usepackage{ulem}
\usepackage{comment}
% the color option for hyperref based on jcappub.sty
\usepackage[colorlinks=true
,urlcolor=blue
,anchorcolor=blue
,citecolor=blue
,filecolor=blue
,linkcolor=blue
,menucolor=blue
,linktocpage=true
,breaklinks=true
,pdfproducer=medialab
,pdfa=true
]{hyperref}
%\usepackage[inline]{showlabels}
\usepackage[normalem]{ulem} %to strike the words

%%%%%%%%%%%% new command 
\graphicspath{{./fig/}{./png/}}
\input macros
\newcommand{\red}[1]{\textcolor{red}{#1}}
\newcommand{\blue}[1]{\textcolor{blue}{#1}}
\newcommand{\SH}[1]{\textcolor[rgb]{0,0.6,0.1}{#1}}

\definecolor{darkgreen}{RGB}{0, 100, 0}

\newcommand{\AB}{\textcolor{red}}
\newcommand{\KK}{\textcolor{cyan}}
\newcommand{\KS}{\textcolor{blue}}
\newcommand{\ckm}[1]{{\color{green}#1}}
\newcommand{\HN}[1]{\textcolor{magenta}{#1}}

\newcommand{\df}{\text{d}}
\newcommand{\p}{\partial}
\newcommand{\Mpl}{{M_\text{pl}}}
\newcommand{\GeV}{\,{\rm GeV}}


\begin{document}

\title{Chiral magnetohydrodynamics with zero total chirality}

\date{\today}
\preprint{NORDITA-2023-014, RESCEU-5/23, KEK-TH-2504, MS-TP-23-13}

\author{Axel Brandenburg}
\affiliation{Nordita, KTH Royal Institute of Technology and Stockholm University, 10691 Stockholm, Sweden}
\affiliation{The Oskar Klein Centre, Department of Astronomy, Stockholm University, AlbaNova, SE-10691 Stockholm, Sweden}
\affiliation{School of Natural Sciences and Medicine, Ilia State University, 0194 Tbilisi, Georgia}
\affiliation{McWilliams Center for Cosmology and Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA}

\author{Kohei Kamada}
\affiliation{Research Center for the Early Universe (RESCEU), Graduate School of Science,\\ The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan}\

\author{Kyohei Mukaida}
\affiliation{KEK Theory Center, Tsukuba 305-0801, Japan}
\affiliation{Graduate University for Advanced Studies (Sokendai), Tsukuba 305-0801, Japan}

\author{Kai Schmitz}
\affiliation{Institute for Theoretical Physics, University of M\"unster, 48149 M\"unster, Germany}

\author{Jennifer Schober}
\affiliation{Institute of Physics, Laboratory of Astrophysics, \'Ecole Polytechnique F\'ed\'erale de Lausanne (EPFL), 1290 Sauverny, Switzerland}

\begin{abstract}
We study the evolution of magnetic fields coupled with chiral fermion
asymmetry in the framework of chiral magnetohydrodynamics with
zero initial total chirality.
The initial magnetic field has a turbulent spectrum peaking at a certain
characteristic scale and is fully helical with positive helicity.
The initial chiral chemical potential is spatially uniform and negative. 
We consider two opposite cases where the ratio of the length scale of
the chiral plasma instability (CPI) to the characteristic scale of the
turbulence is smaller and larger than unity.
These initial conditions might be realized in cosmological models such
as certain types of axion inflation. 
The magnetic field and chiral chemical potential evolve with inverse
cascading in such a way that the magnetic helicity and chirality cancel
each other at all times.
The CPI time scale is found to determine mainly the time when the magnetic
helicity spectrum attains negative values at high wave numbers.
The turnover time of the energy-carrying eddies, on the other hand,
determines the time when the peak of the spectrum starts to shift to
smaller wave numbers via an inverse cascade.
The onset of helicity decay is determined by the time when the chiral
magnetic effect becomes efficient at the peak of the initial magnetic
energy spectrum.
When spin flipping is important, the chiral chemical potential vanishes
and the magnetic helicity becomes constant, 
which leads to a faster
increase of the correlation length, as expected from magnetic helicity
conservation.
This also happens when the initial total chirality is imbalanced.
Our findings have important implications for baryogenesis after axion inflation.
\end{abstract}

\maketitle

\section{Introduction}

Relativistic plasmas are described by the evolution equations of
chiral magnetohydrodynamics (MHD) \cite{Son:2009tf,Neiman:2010zi,BFR12, BFR15, Pavlovic+17,
Roga_etal17, Bran_etal17, Schober+18, DelZanna+Bucciantini18}.
Chirality enters in two distinct ways: first, through a nonvanishing
chiral chemical potential, $\tilde{\mu}_5$, and second, through nonvanishing
magnetic helicity density, $\AAA\cdot\BB$, where $\BB=\nab\times\AAA$
is the magnetic field expressed in terms of the vector potential $\AAA$. 

It has been known for some time that fermion chirality can be transferred
into magnetic helicity and vice versa through the chiral anomaly~\cite{Adler:1969gk,Bell:1969ts}.
The transfer of fermion chirality to magnetic helicity occurs through
an instability \cite{JS97} known as the chiral plasma instability (CPI)
\cite{Akamatsu:2013pjd}.
This instability is the fastest at a specific wave number, whose value
depends on the chiral chemical potential.
The transfer from magnetic helicity to chiral chemical potential does
not involve any instability, but occurs just through a nonvanishing
nonlinear source term in the evolution equation for the chiral chemical
potential \cite{BFR12,Hirono+15, Schober+2020}.
These differences in the evolutions of the chiral chemical potential
and magnetic field can lead to nontrivial dynamics, which has triggered
a lot of research \cite{Manuel+15, Domcke+19b, Domcke+20}.
Since fermion chirality is tightly related to the baryon and lepton asymmetries
at high temperature in the early Universe,
their co-evolution with magnetic helicity in the context of cosmology
has also extensively studied \cite{Giovannini:1997eg, Giovannini:1997gp,
Dvornikov:2011ey, Tashiro:2012mf, Long:2013tha, Fujita:2016igl,
Kamada:2016eeb, Kamada:2016cnb, Pavlovic:2016gac}.

Previous investigations mostly assumed an initial imbalance between
fermion chirality and magnetic helicity.
This leads to a conversion of fermion chirality to a maximally helical
magnetic field \cite{BFR12}.
Also just spatial fluctuations can lead to magnetic field production
\cite{Schober+2022a,Schober+2022b}.
In many investigations, however, the initial fermion chirality is
nonvanishing while initial magnetic helicity is zero or vice versa.
Such chiral asymmetry, which can trigger the CPI, could be
generated \cite{GarciaBellido+99, Kamada:2018tcs, Domcke:2020quw}
in GUT baryogenesis in the early Universe~\cite{Yoshimura:1978ex,
Dimopoulos:1978kv, Toussaint:1978br, Weinberg:1979bt, Barr:1979ye}
or weak interactions in compact stars \cite{Ohnishi:2014uea,
Grabowska+15, Masada+18, Dvornikov+20, Matsumoto:2022lyb} (see also
Ref.~\cite{Kamada:2022nyt} and references therein).
However, numerical studies on other interesting initial conditions are
still lacking, where fermion chirality is exactly opposite to magnetic
helicity.
Such an initial condition is expected if the chiral symmetry in the fermion sector is only
broken through the topological density, $\partial_\mu J^\mu_5 = -
e^2 F_{\mu\nu} \tilde F^{\mu\nu} / (8 \pi^2 \hbar^2 c)$,
or the chiral anomaly~\cite{Adler:1969gk,Bell:1969ts}, 
with $J_5^\mu$ being
the chiral current and $e^2 F_{\mu\nu} \tilde F^{\mu\nu} / (8 \pi^2 \hbar^2 c)$
being the topological density.
Since the topological density can be written as a total derivative of
the magnetic helicity density, the sum of chiral asymmetry and magnetic
helicity vanishes when they are generated \cite{domcke2018}.

Configurations with vanishing total chirality are interesting not only
in the context of chiral MHD, but also in particle physics and cosmology.
At a high enough temperature realized in the early Universe, the
electron Yukawa interaction becomes inefficient for
$T \gtrsim 10^5\, \mathrm{GeV}$ \cite{Campbell:1992jd,Bodeker:2019ajh}.
There we find the conservation of the total chirality because of 
$\partial_\mu J^\mu_{e_R} = - g_Y^2 Y_{\mu\nu} \tilde Y^{\mu\nu} / (16 \pi^2 \hbar^2 c)$ 
with $J^\mu_{e_R}$ being the right-handed electron current and
$Y^{\mu\nu}$ being the field strength of the hypercharge gauge field
with gauge coupling $g_Y$.
For instance, in a certain class of axion inflation, configurations with
zero net chirality are generated during inflation~\cite{domcke2018},
which can be the origin of the observed baryon asymmetry of the
Universe \cite{Jimenez:2017cdr,Domcke+19, Domcke:2022kfs} and it
could explain the proposed intergalactic magnetic field; see, however,
Ref.~\cite{Kamada:2016cnb} for the baryon overproduction problem and
Ref.~\cite{Kamada:2020bmb} for the too large baryon isocurvature problem.
The main purpose of this paper is to perform a full numerical chiral MHD
simulation under the initial condition of vanishing total chirality and
provide a better understanding of the nonlinear dynamics in this case.

Before we begin our investigations, it is useful to recall the main
findings of earlier work where the total chirality was mostly different from zero.
Following the work of Ref.~\cite{Hirono+15}, who studied a system
consisting of the gauge field and the chiral chemical potential,
but without fluid velocity fields, and with the initial condition
$\aver{\AAA\cdot\BB} \not =0, \tilde{\mu}_5=0$, three stages can be identified:
(i) exponential decline of the magnetic helicity together with
an increase of $\tilde{\mu}_5$, followed by
(ii) a continued decrease of the typical peak wave number $\kp$,
while $\tilde{\mu}_5$ stays at its maximum value with $\aver{\AAA\cdot\BB}$
being essentially zero, and (iii) a phase when all the fermion chirality
$\tilde{\mu}_5$ gets transferred back to magnetic helicity.
As expected, owing to magnetic helicity conservation, and because the
magnetic field from the CPI is maximally helical,
the magnetic energy density $\aver{\BB^2}/2$ decays at late times such
that $\aver{\BB^2}\xiM\approx\const$, where $\xiM\equiv\kp^{-1}$ is the
magnetic correlation length.
In other words, both $\aver{\BB^2}$ and $\kp$ decay in the same fashion,
but, unlike the expected $t^{-2/3}$ scaling found previously for helical
turbulence \citep{Hat84,BM99,Kahn_etal13,BK17}, they find a $t^{-1/2}$
scaling both for $\aver{\BB^2}$ and $\kp$.
For sufficiently strong initial magnetic fields, the magnetic Reynolds
number can be much larger than unity and the eddy turnover scale much
longer than the estimated inverse peak momentum scale, if equipartition
between the magnetic fields and fluid velocity fields is established.
This suggests that the effect of the fluid velocity cannot be negligible
in general.

The earlier analytic study of Ref.~\cite{Hirono+15} was revisited using
direct numerical simulations of chiral MHD \cite{Schober+2020}.
At large magnetic Reynolds numbers, the authors found clear evidence
for a $t^{-2/3}$ scaling of both $\aver{\BB^2}$ and $\kp$ at late times.
They also found that the initial evolution is not exponential, as
suggested in Ref.~\cite{Hirono+15}, but linear in time.
However, they only considered the case where the initial fermion
chirality was zero.
When it is finite and balancing exactly the magnetic helicity, the
magnetic field decays in a way similar to the case of a strong, nonhelical
field~\cite{BKS23}, where the decay is governed by the conservation of the Hosking
integral \cite{Hosking+Schekochihin21, Hosking+Schekochihin22, Zhou+22}.
This integral describes the strength of magnetic helicity fluctuations
on different length scales and has the dimensions of $\cm^9\s^{-4}$,
which implies the scalings $\xiM\propto t^{4/9}$ and
$\aver{\BB^2}\propto t^{-10/9}$ \cite{Hosking+Schekochihin21}.
The general validity of the Hosking integral was further demonstrated by
applying a corresponding analysis to the decay of a nonhelical magnetic
field in neutron star crusts \cite{Bra23}, where the magnetic field
evolution is covered by the Hall effect \cite{GR92}.

Our goal here is to bridge the gap between the two extremes, where
the initial chirality is either only in the fermions or only in the
magnetic field, and to consider the intermediate case
where fermion chirality and magnetic helicity balance to zero, 
extending the study of the present authors \cite{BKS23}.
This is another case where the decay of $\aver{\BB^2}$ and $\kp$ are
described by a correspondingly adapted Hosking integral of the total
chirality.
In the following, we therefore refer to the Hosking integral with the
chiral chemical potential included as the ``adapted'' Hosking integral;
see Ref.~\cite{BKS23} for detail.

As mentioned, our findings on the evolution of the system with vanishing total chirality 
has a significant impact on the present baryon asymmetry of the Universe. 
Another goal of the present paper is then to clarify how the non-trivial
co-evolution of the magnetic field and fermion chirality affect the
model space of axion inflation consistent with the present Universe,
which has not been explored before.

We begin, by presenting the basic equations and the mathematical setup
of our simulations in \Sec{ChiralMHD}.
We then discuss the parameter dependence of characteristic time scales,
consider also the effect of spin flipping, and finally cases where the
perfectly vanishing chirality balance is relaxed in \Sec{Results}.
Applications to the early Universe are discussed in \Sec{Application}.
Conclusions are presented in \Sec{Conclusions}.

\section{Chiral magnetohydrodynamics}
\label{ChiralMHD}

\subsection{Chiral magnetic effect}

Using Lorentz-Heaviside units, the Amp\`{e}re-Maxwell equation 
for the QED-like model in the MHD limit (omitting the displacement current) reads
\begin{equation}
{\bm \nabla} \times {\bm B} = \frac{1}{c} {\bm J}. 
\end{equation}
The electric current ${\bm J}$ is the sum of the Ohmic current and 
the chiral magnetic effect (CME) \cite{Vilenkin80, Alekseev:1998ds, Fukushima:2008xe},
\begin{equation}
{\bm J} = \frac{\sigma}{c} \left(c{\bm E} + {\bm u} \times {\bm B}\right) + \frac{e^2}{2 \pi^2 \hbar^2 c} \tilde{\mu}_5 {\bm B}, 
\label{current}
\end{equation}
where we consider the case with $\tilde{\mu}_5 \ll (e^2/\hbar c) k_B T$.
By rewriting $c{\bm E} = - \partial {\bm A}/\partial t$ in the Weyl gauge,
$e^2/4\pi \hbar c\equiv \alpha$, \Eq{current} is rewritten as
\begin{equation}
\frac{\partial {\bm A}}{\partial t} = \frac{c^2}{\sigma} \left( \mu_5 {\bm B} - \nab\times{\bm B}\right) + {\bm u} \times {\bm B} ,
\label{dAdt}
\end{equation} 
where we defined \cite{Roga_etal17}
\begin{equation}
\mu_5 \equiv \frac{2\alpha}{\pi\hbar c}\tilde{\mu}_5.
\label{mu5-tilde}
\end{equation}
This expression agrees with Eq.~(32) of Ref.~\cite{Roga_etal17},
except for a factor of 2 resulting from our definition
$\tilde{\mu}_5=(\tilde{\mu}_{\rm R}-\tilde{\mu}_{\rm L})/2$ in
terms of the chemical potentials for right- and left-handed fermions
\cite{Kamada:2022nyt}.
The additional $4\pi$ factor in the numerator of the expression in
Ref.~\cite{Roga_etal17} is a consequence of their use of cgs units.

\subsection{Model description and basic equations}

We perform simulations in a cubic domain of size $L^3$ with side lengths
$L$ and triply-periodic boundary conditions.
The mass in the domain is therefore constant, so the mean density
$\meanrho$ is always equal to its initial value $\rho_0$
and put to unity in all cases.
The lowest wave number in the domain is $k_1=2\pi/L$.
Using $N^3$ mesh points, the largest wave number in the simulations is
the Nyquist wave number $k_{\rm Ny}=k_1 N/2$.

In the following, we set $c=1$, so $\JJ=\nab\times\BB$.
To include the effects of the cosmic expansion with scale factor
$a(t)\propto t^{1/2}$ in the radiation-dominated era, which we assume to
be a spatially flat Friedmann Universe, we use correspondingly scaled
quantities and conformal time, $\eta(t)=\int\dd t/a(t)$, in which the
evolution equations of MHD are the same as in the absence of expansion
\cite{BEO96}.
In order to obtain the physical quantities, we can simply normalize the
corresponding comoving quantities with the appropriate powers of the
scale factor $a$.
Furthermore, using $\lambda=3 \hbar (2 \alpha/ \pi \kB T)^2$ and including
spin flipping and spatial diffusion, our chiral anomaly equation is
\begin{equation}
\frac{\partial\mu_5}{\partial\eta} + {\bm \nabla}\cdot(\mu_5 {\bm u}) = \frac{\lambda}{\sigma}
\left({\bm J}-\mu_5 {\bm B} \right) \cdot {\bm B}+D_5\nabla^2\mu_5-\Gamma\mu_5,
\label{dmudt}
\end{equation}
where $D_5$ is an empirical diffusion coefficients for the chiral
chemical potential.
Here we used the relationship 
between the chiral chemical potential and the number density, 
\begin{equation}
n_5 \equiv n_R-n_L =  2 \times \frac{\tilde{\mu}_5}{6\hbar ^3} (k_B T)^2
=\frac{\pi \mu_5}{6 \alpha \hbar^2} (k_B T)^2,
\end{equation}
and used $J_5^\mu=(n_5,n_5 {\bm u}-D_5\nab n_5)$ for the chiral 4-current.

Owing to the chiral anomaly \citep{Adler:1969gk,Bell:1969ts},
the total chirality is conserved in the absence of spin flipping
interaction \citep{BFR12, Roga_etal17}.
It is then convenient to introduce the mean magnetic chirality
equivalent as
\begin{equation}
\aver{\mu_{\rm M}}\equiv\half\lambda\aver{\AAA\cdot\BB}, 
\label{muMdef}
\end{equation}
so that the conservation law derived from \Eqs{dAdt}{dmudt} can be
stated in the form
\begin{equation}
\mu_{\rm tot}=\aver{\mu_5}+\aver{\mu_{\rm M}}=\const.
\label{eq:mutot}
\end{equation}

We complement \Eqs{dAdt}{dmudt} by the momentum and continuity equations
\citep{Roga_etal17, Bran_etal17, BHKRS21}
\begin{align}
\frac{\DD\uu}{\DD\eta}&=\frac{2}{\rho}\nab\cdot\left(\rho\nu\SSSS\right)
-\frac{1}{4}\nab\ln\rho+\frac{\uu}{3}\left(\nab\cdot\uu
+\uu\cdot\nab\ln\rho\right)
\nonumber \\
&-\frac{\uu}{\rho}\left[\uu\cdot(\JJ\times\BB)+\eta \JJ^2\right]
+\frac{3}{4\rho}\JJ\times\BB,
\label{dudt} \\
\frac{\partial\ln\rho}{\partial\eta}
&=-\frac{4}{3}\left[\nab+ \left(\nab\ln\rho\right)\right] \cdot \uu
+\frac{1}{\rho}\left[\uu\cdot(\JJ\times\BB)+\eta \JJ^2\right]\!,
\nonumber
\end{align}
where $\DD/\DD\eta\equiv\partial/\partial\eta+\uu\cdot\nab$ is the
advective derivative, ${\sf S}_{ij}=(\partial_i u_j+\partial_j u_i)/2
-\delta_{ij}\nab\cdot\uu/3$ are the components of the rate-of-strain
tensor, $\nu$ is the viscosity, and $p$ is the
pressure, which is assumed to be proportional to the density, i.e.,
$p=\rho\cs^2$, with $\cs = 1/\sqrt{3}$ being the sound speed
for the ultrarelativistic fluid.

For all our simulations, we use the {\sc Pencil Code} \cite{JOSS},
where the relevant equations are readily implemented.
We use $N^3=1024^3$ mesh points for most of the runs, and $N^3=2048^3$
mesh points for one particular run.
In a small number of cases, we have included the slope-limited diffusion
(SLD) scheme of Ref.~\cite{Rem14,NewsletterSLD}.
In those cases, SLD acts in addition to the ordinary viscous and diffusive
processes stated in the equations above, but prevents the code from
crashing during an early more violent phase when the mesh resolution is
insufficient to dissipate the energy at high wave numbers.
At later times, however, this additional numerical device has little
effect.
Below, we demonstrate in one case that the solutions with and without
SLD yield the same result.

\subsection{Diagnostic quantities}

We introduce two characteristic times in our simulations, which are the
time scale of the CPI and the magnetic diffusion time,
\begin{equation}
\eta_{\rm CPI}=\sigma\mu_{50}^{-2}\quad\mbox{and}\quad
\eta_{\rm diff}=\sigma k_0^{-2},
\label{etaCPIdiff}
\end{equation}
respectively.
The ratio $(\eta_{\rm diff}/\eta_{\rm CPI})^{1/2}=|\mu_{50}|/k_0$
characterizes the degree of scale separation between the scales of
magnetic helicity and fermion chirality.
We also define the turnover time of the energy-carrying eddies,
which would determine the onset of the turbulence,
\begin{equation}
\eta_{\rm turb}=\left(\urms^{\max} k_0\right)^{-1}, 
\label{etaturb}
\end{equation}
where $\urms^{\max}$ is the maximum value (in time) of the rms velocity.

%FIG1
\begin{figure*}[t!]\begin{center}
\includegraphics[width=\textwidth]{AB}
\end{center}\caption[]{
Visualizations of $\AAA\cdot\BB\,\lambda/2$ (upper row) and $\mu_5$
(lower row) on the periphery of the computational domain for Run~O at
$\eta\approx4600$ (left) 15,000 (middle), and 46,000 (right).
}\label{AB}\end{figure*}

Next, we introduce several parameters with a dimension of velocity.
The nature of the CPI is characterized by the following parameters
\cite{Bran_etal17}
\begin{equation}
v_\lambda=|\mu_{50}|/(\meanrho\lambda)^{1/2}\quad\mbox{and}\quad
v_\mu=|\mu_{50}|/\sigma. 
\end{equation}
The former represents the ratio of the length scale of the magnetic field
at the saturation of the CPI to the CPI time scale,
while the latter represents the ratio of the length scale of the initial instability
to the CPI time scale.
The ratio $v_\lambda/v_\mu=\sigma/(\meanrho\lambda)^{1/2}$
characterizes the length of the $k^{-2}$ spectrum that develops if the
CPI operates without a strong pre-existing field \cite{Bran_etal17}.
In the unbalanced case, $\urms^{\max}$ is approximated by $v_\lambda$.
In the present case, however, it does not seem to play any role.
Instead, to compute $\urms^{\max}$, we approximate the velocity field by
the initial magnetic field such that $\Brms^2 \simeq {\bar \rho} \urms^2$.
Using \Eqs{muMdef}{eq:mutot}, we estimate
\begin{equation}
    \Brms^{(0)} \approx(k_0|\aver{\AAA\cdot\BB}|)^{1/2} \approx \left(\frac{2k_0|\mu_{50}|}{\lambda }\right)^{1/2}, 
\end{equation}
which thus defines a new quantity $\tilde{v}_\lambda$ as
\begin{equation} \label{tildevlam}
\tilde{v}_\lambda \equiv \left(\frac{2k_0|\mu_{50}|}{{\bar \rho} \lambda} \right)^{1/2}
\quad \left(\approx \frac{\Brms^{(0)} }{{\bar \rho}^{1/2}} \right). 
\end{equation}
A predictive estimate for the turnover time of the energy-carrying eddies
is thus
\begin{equation}
\eta_\lambda=(\tilde{v}_\lambda k_0)^{-1} = \left(\frac{{\bar \rho} \lambda}{2 k_0^3 |\mu_{50}|}\right)^{1/2},
\label{etalam}
\end{equation}
which is later used to predict the time when the inverse cascade sets in.

In this work, an important diagnostics is the magnetic energy spectrum, $\EM(k)$.
It is normalized such that $\int\EM(k)\,\dd k=\aver{\BB^2}/2\equiv\EEM$ 
where $\EEM$ is the magnetic energy density\footnote{
In terms of the mode function in the polarization basis,
${\bm A}({\bm x},t) \equiv \int d^3 k/(2\pi)^{3/2} \sum_{\lambda=\pm}
A_\lambda({\bm k},t) {\bm e}^\lambda ({\bm k}) e^{i {\bm k} {\bm x}}$,
$\EM$ is given as $\EM(k)= \sum_{s=\pm} (k^4/4\pi^2) |A_s(k)|^2$. 
We also have $\HM^s (k) = (k^3/2 \pi^2) |A_s(k)|^2$ and
$\HM (k) = (k^3/2 \pi^2) \sum_{s=\pm} s|A_s(k)|^2$.}.
The kinetic energy spectrum $\EK(k)$ is defined similarly, i.e.,
$\int\EK(k)\,\dd k=\aver{\rho\uu^2}/2\equiv\EEK$.
We also define the magnetic helicity spectrum $\HM(k)$, which is
normalized such that $\int \HM(k) \, \dd k = \aver{\AAA\cdot\BB}$.
In our simulations, $k|\HM(k)|/2$ approaches $\EM(k)$ near the maximum.
In fact, the spectra $\HM(k)$ and $\EM(k)$ satisfy the realizability
condition \cite{Mof78},
\EQ
k|\HM(k)|/2\le \EM(k).
\label{realizability}
\EN
When this inequality is saturated for specific wave numbers, we say that
the magnetic field is locally fully helical.

After some time, the magnetic helicity spectrum is characterized by two
subranges, one with positive and one with negative values of $\HM(k)$,
which are separated by the wave number $k_\pm(\eta)$, where the sign
changes.
In addition to the evolution of $k_\pm(\eta)$, we characterize the
spectrum and its evolution by the numbers  $k_{\rm I}(\eta)$ and
$k_{\rm II}(\eta)$, which are the wave numbers of the first positive
and second negative peak of $\HM(k)$.
The intermediate wave number $k_\pm(\eta)$ is often better determined
than $k_{\rm II}(\eta)$, especially at early times.

The wave number of the first peak of the spectrum is close to the
initial inverse correlation length,
\EQ
\xiM=\EEM^{-1}\int k^{-1}\EM(k)\,\dd k.
\EN
In fully helical turbulence, the value of $\xiM(\eta)$ tends to increase
with time in a power law fashion, $\xiM\propto\eta^q$, where $q=4/9$ in
our cases of balanced chirality \cite{BKS23}; see also Sec.~\ref{IIE}.
Note that in our setup the positive helicity modes always dominate
the energy density of the magnetic field, and hence approximately we have
$\xiM \simeq k_{\rm I}^{-1}$.

It is convenient to introduce the mean magnetic chirality for the positive
helicity modes for $k < k_\pm$ and the negative ones for $k>k_\pm$ as
\begin{align}
\label{muMp} \aver{\mu_{\rm M}^+} &= \frac{\lambda}{2} \int_0^{k_\pm} H_{\rm M}(k)\,\dd k, \\
\label{muMm} \aver{\mu_{\rm M}^-} &= -\frac{\lambda}{2} \int_{k_\pm}^\infty  H_{\rm M}(k)\,\dd k.
\end{align}
The conservation law takes then the form
\begin{equation}
\aver{\mu_5}+\aver{\mu_{\rm M}^+}-\aver{\mu_{\rm M}^-}=\mu_{\rm tot},
\end{equation}
where $\mu_{\rm tot}=\mu_{50}+\mu_{\rm M0}=\mu_{50}+\mu_{\rm M0}^+
-\mu_{\rm M0}^-$ is given by the initial values.

When we study the effect of spin flipping, we invoke a nonvanishing flipping
rate with
\EQ
\Gamma=
\left\{
\begin{aligned}
\Gamma_{\rm f0} & \quad\mbox{for}\quad 
\eta_{\rm flip}\leq\eta\leq\eta_{\rm off} \\
0\;\; & \quad\mbox{otherwise},\\
\end{aligned}
\right.
\EN
where $\eta_{\rm flip}$ denotes the time when spin flipping is turned
on, and in a few cases we allow for a finite value of $\eta_{\rm off}$,
which denotes the time when spin flipping is later turned off again.

\subsection{Initial conditions}

In our numerical experiments, the initial magnetic field is fully helical
with positive magnetic helicity and random phases.
The initial magnetic energy spectrum is a broken power law
\begin{equation}
\EM(k,\eta_0)\propto \begin{cases}
k^4 & \mbox{for}\;k<k_0,\\
k^{-5/3} & \mbox{for}\;k>k_0, 
\end{cases}
\end{equation}
where the initial peak is identified as $k_0 = k_{\rm I}(\eta_0)$.
The IR spectrum is motivated by causality constraints \cite{DC03},
while the UV spectrum is taken as a Kolmogorov-type spectrum.
The strength of the magnetic field is adjusted such that the initial
magnetic chirality obeys 
$\mu_{\rm M0}=-\mu_{50}$ such that $\mu_\mathrm{tot} =0$.
The chiral chemical potential is initially assumed to be uniformly
distributed in space.
Its initial value is always negative, i.e., $\mu_{50}<0$.
However, even for an initially uniform chiral chemical potential, there
is a specific length scale associated with the value of $\mu_5$ through
the wave number of the most unstable mode of the CPI, $k=|\mu_{50}|/2$. 
The initial velocity is assumed vanishing in all cases.

\subsection{Theoretical predictions} \label{IIE}

As was recently shown in Ref.~\cite{BKS23}, the present case of zero
total chirality, where the magnetic helicity is canceled by fermion
chirality, is remarkably similar to the case of ordinary MHD without
chemical potential and zero magnetic helicity.
In both cases, as already alluded to in the introduction,
one can define a correlation integral of the total chirality,
which is a quantity with dimensions $\cm^9\s^{-4}$
and is dubbed the (adapted) Hosking integral. 
The evolution of the system can be explained by the conservation 
of this quantity.
With self-similar evolution of the magnetic spectrum being assumed,
this yields the scalings $\xiM\propto\eta^{4/9}$ and
$\aver{\BB^2}\propto\eta^{-10/9}$ for typical length scales and the
magnetic energy density, respectively \cite{Hosking+Schekochihin21}.
Note that the conservation of the adapted Hosking integral suggests 
\begin{equation}
\xi_M^5 \langle \BB^2 \rangle^2 = \const, \quad\text{or} \quad k_{\rm I}^{-3} E_M(k_{\rm I})^2 = \const, \label{hoscons}
\end{equation}
if the magnetic energy density is 
dominated by the positive helicity mode, which is peaked at $k=k_{\rm I}$.
For the magnetic field with a IR spectrum $\propto k^4$, as motivated
from the causality constraints, the evolution of the magnetic field
exhibits inverse cascading.

The big difference between ordinary MHD without helicity on the one hand and chiral
MHD with helicity balanced by fermion chirality on the other hand is that in the latter,
both the magnetic helicity and the fermion chirality are decaying, 
which we shall call the anomalous chirality cancellation (ACC).
In the former, by contrast, the Hosking integral based just on the
ordinary magnetic helicity density is conserved.
In the latter, contrary to the naively expected exponential decay of
fermion chirality due to the CPI in chiral MHD, we actually have a much
slower power-law decay proportional to $\eta^{-2/3}$, since the magnetic
helicity is roughly estimated by $\xi_M \langle \BB^2 \rangle$, 
and likewise for $|\aver{\mu_5}|$ \cite{BKS23}. 
Here we have used the fact that the real space realizability condition
of magnetic helicity \citep{Kahn_etal13}, $|\HHM|\leq2\EEM\xiM$,
is nearly saturated.
Once this power law decay of the chirality starts,  
the CPI rate, $\aver{\mu_5}^2/\sigma$, decays faster than $\eta^{-1}$, 
which suggests that the CPI does not grow anymore. 
Hence the magnetic energy is always dominated by helicity modes with
the same sign as the initial ones, which, in our case, are positive
helicity modes.

The adapted Hosking integral makes sense only when the communication between 
the helicity and chirality through the CME becomes effective at the characteristic scale.
Therefore we expect that the scaling evolution discussed above starts 
at the time scale of the CME at the initial peak scale. 
With the evolution equation for the magnetic field, equivalent to Eq.~\eqref{dAdt}, 
\begin{equation}
\frac{\partial {\bm B}}{\partial \eta} = \frac{1}{\sigma} \left[\nabla^2 {\bm B}
+ {\bm \nabla} \times (\mu_5 {\bm B})\right]
+ \nab \times ({\bm u} \times{\bm B}) \label{BEOM}
\end{equation}
(where second term in the right-hand side represents the CME), 
we estimate $\eta_\mathrm{ACC}$ as the time when the following condition is satisfied:
\begin{equation}
\eta_\mathrm{ACC} \simeq \frac{\sigma}{\mu_5(\eta_\mathrm{ACC}) k_{\rm I}(\eta_\mathrm{ACC})}.
\label{etaacc}
\end{equation}
Note that from Eq.~\eqref{BEOM} we can also confirm that 
the magnetic field has an instability (the CPI) for one of the two
circular polarization modes with $k=|\mu_{50}|/2$ being the
most unstable mode.
The instability rate is roughly given as $\mu_{50}^2/\sigma$, 
which determines $\eta_\mathrm{CPI}$.

The evolution of the system is classified into two cases, 
determined by the comparison between $\eta_\lambda$ and $\eta_\mathrm{ACC}$
estimated by the initial conditions of $k_{\rm I}$ and $\mu_5$. 
Since ${\bar \rho}\simeq (\pi^2 g_*/30)T^4$, where $g_*$ is number
of the relativistic degrees of freedom, and $\sigma \simeq 10^2 T$ \cite{Baym:1997gq,Arnold:2000dr},
we have $\eta_\mathrm{ACC} < \eta_\lambda$ for $k_0 \ll |\mu_{50}|$ 
[more precisely, $k_0 \ll ({\bar \rho} \lambda/ 4 \sigma^2) |\mu_{50}|$,
which is independent of temperature], and vice versa.
For $k_0 \ll |\mu_{50}|$, we have the following estimate 
for the evolution of the system, 
\begin{enumerate}
    \item The system is frozen when $\eta<\eta_\mathrm{CPI}$. 
    \item The CPI starts to grow at $\eta \simeq \eta_\mathrm{CPI}$. 
    \item If the CPI does not sufficiently amplify the negative helicity modes, the chiral chemical potential starts to decay at $\eta=\eta_\mathrm{ACC} (>\eta_\mathrm{CPI})$ with
    \begin{equation} \label{etaACCcase1}
        \eta_\mathrm{ACC} \simeq \frac{\sigma}{|\mu_{50}| k_0}
    \end{equation}
    in a mild way.
    \item When $\eta \simeq \eta_\lambda (>\eta_\mathrm{ACC})$, the system starts to evolve
    according to the scaling law found in Ref.~\cite{BKS23}, 
    \begin{align}
        &k_{\rm I} \propto \eta^{-4/9}, \quad \EEM \propto \eta^{-10/9},\quad\mbox{and}\\
        &\aver{\mu_5}=-\aver{\mu_{\rm M}^+}+\aver{\mu_{\rm M}^-} \propto \eta^{-2/3}. 
    \end{align}
\end{enumerate}

For $k_0 \gg|\mu_{50}|$, on the other hand, we expect the following evolution of the system. 
\begin{enumerate}
    \item The system is frozen at $\eta<\eta_\lambda$. 
    \item The magnetic field evolve according the the inverse cascade
    at $\eta \simeq \eta_\lambda$
    in a similar way to the usual inverse cascade for nonchiral helical 
    magnetic field, 
    \begin{align}
        &k_{\rm I} \propto \eta^{-2/3}, \quad \EEM \propto \eta^{-2/3},\quad\mbox{and}\\
        &\aver{\mu_5}=-\aver{\mu_{\rm M}^+}+\aver{\mu_{\rm M}^-} = \mathrm{const}, 
    \end{align}
    since the CME is not effective at $k \simeq k_{\rm I}$ so that
    the magnetic helicity and chirality are individually conserved. 
    \item The CME becomes effective at $\eta \simeq \eta_\mathrm{ACC}$ $(>\eta_\lambda)$, 
    which is now evaluated as
    \begin{equation}
    \eta_\mathrm{ACC} \equiv \frac{\sigma^3}{|\mu_{50}|^3 k_0^3} \eta_\lambda^{-2}
    \simeq \frac{2\sigma^3}{{\bar \rho} \lambda \mu_{50}^2 }.
    \label{etaacck0>mu50}
    \end{equation}   
    Here we have used \Eq{etaacc} and
    $k_{\rm I}(\eta) = k_0 (\eta/\eta_\lambda)^{-2/3}$,
    as well as \Eq{tildevlam}.
    When $\eta>\eta_\mathrm{ACC}$ we have the inverse cascade with the conservation of the adapted Hosking integral, 
    \begin{align}
        & k_{\rm I} \propto \eta^{-4/9}, \quad \EEM \propto \eta^{-10/9},\quad\mbox{and}\\
        & \aver{\mu_5}=-\aver{\mu_{\rm M}^+}+\aver{\mu_{\rm M}^-} \propto \eta^{-2/3}. 
    \end{align}
    Note that in this case the CPI would not grow much due to the earlier
    onset of the chirality decay. 
\end{enumerate}
In \Tab{TAnalyticTimeScale}, we summarize the characteristic 
time scales relevant for the evolution of the system.

\begin{table*}[t]\caption{
Relevant time scales defined in this paper.
}\vspace{12pt}\centerline{\begin{tabular}{llll}
Time scale & Expression & Equation & Explanation \\
\hline
$\eta_{\rm CPI}$  & $\sigma\mu_{50}^{-2}$     & \Eq{etaCPIdiff} & time scale of the CPI \\
$\eta_{\rm diff}$ & $\sigma k_0^{-2}$         & \Eq{etaCPIdiff} & magnetic diffusion time \\
$\eta_{\rm turb}$ & $(\urms^{\max} k_0)^{-1}$ & \Eq{etaturb}    & turnover time of the energy-carrying eddies \\
$\eta_\lambda$    & $(\tilde{v}_\lambda k_0)^{-1} = [{\bar \rho} \lambda/(2 k_0^3 |\mu_{50}|)]^{1/2}$ & \Eq{etalam} & {\em predicted} turnover time of the energy-carrying eddies \\
$\eta_\mathrm{ACC}$ & $\sigma/[\mu_5(\eta_\mathrm{ACC}) k_{\rm I}(\eta_\mathrm{ACC})]$ & \Eq{etaacc} & onset time of the ACC \\
\label{TAnalyticTimeScale}\end{tabular}}\end{table*}

The features described above will be confirmed by direct numerical
simulations in the next section.
They can have important consequences for baryon production, as will
discussed at the end of the paper.

\section{Results}
\label{Results}

In this section, we show the results of the direct numerical simulation.
We first study the case with $k_0 \ll |\mu_{50}|$
until Sec.~\ref{sec:IIIG}. 
In Sec.~\ref{sec:IIIH} we study the case with $k_0 \ll \mu_{50}$.
Some of our observations will turn out to be consistent with the
theoretical prediction discussed in Sec.~\ref{IIE}.
We will also see some other features, which have not been addressed there.

\subsection{Visualization of magnetic and fermion chiralities} 

We begin by discussing the simulation of Ref.~\cite{BKS23}
with $k_0 \ll |\mu_{50}|$, which we refer to as Run~O.
In \Fig{AB}, we present visual impressions of magnetic and fermion
chiralities in Run O at different times.
We see that the turbulent structures gradually grow in size and the
extreme values away from zero decrease as time goes on.
Furthermore, $\mu_5$ and $\AAA\cdot\BB\,\lambda/2$ have predominantly
opposite signs, as expected.
Locally, however, there is no correspondence between the two fields.
This is because the vanishing total chirality is only a statistical
property.

\subsection{Evolution of characteristic scales}

As discussed in Ref.~\cite{BKS23}, it is important to allow for
sufficient scale separation between the smallest available wave number
$k_1\equiv2\pi/L$ and the initial wave number of the peak, $k_0$.
It is also important that there is enough separation between $k_0$
and the initial wave number of the CPI, $|\mu_{50}|/2$, 
to confirm distinct features of the evolution of the system.
Both, $k_0$ and $|\mu_{50}|/2$, in turn, must be much smaller than the
largest available wave number $k_{\rm Ny}=k_1N/2$.
Sufficient scale separation between $k_1$ and $k_0$ is particularly
important for obtaining the theoretically expected increase of
$\xiM\propto\eta^{4/9}$ along with the decay of
$\EEM\propto\eta^{-10/9}$, based on the conservation of the
Hosking integral adapted to the total chirality.
Indeed, in Run~O, an optimized balance between the two scale separation
requirements has been achieved.

With the start of the simulation, the helical random magnetic field, which is present
initially, drives turbulent motions through the Lorentz force.
Those motions are in approximate equipartition with the magnetic field at high
wave numbers; see \Fig{rspec_select_ub_1024a_mu10_k002o}, where we
compare kinetic and magnetic energy spectra at different times.
In this figure, we also mark the two scale separation ratios.
This observation supports the estimate of $\urms^\mathrm{max}$
to ${\tilde v}_\lambda$; see \Eq{tildevlam}.

%FIG2
\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{rspec_select_ub_1024a_mu10_k002o}
\end{center}\caption[]{
Magnetic energy (red lines) and kinetic energy (blue lines) spectra for
Run~O at times $\eta=460$, 4600, and 46,000.
The dotted lines denote the earliest outputted time $\eta=0.3$.
In the upper part, the two-sided arrows indicate the requirements
for scale separation at small and large $k$ to obtain the
$\kp\propto\eta^{-4/9}$ decay and to resolve $|\mu_{50}|$, respectively.
}\label{rspec_select_ub_1024a_mu10_k002o}\end{figure}

As already discussed in Ref.~\cite{BKS23}, even though there is vanishing
net chirality, $\aver{\mu_{\rm M}}+\aver{\mu_5}=0$, there is still some
degree of inverse cascading, just like in nonhelical magnetically dominated
turbulence \citep{BKT15,BK17}.
We see this clearly in \Fig{rspec_select_1024a_mu10_k002o}, where the position
of the magnetic peak, $k_{\rm I}(\eta)$, gradually moves to smaller values.
At the same time, the height of the peak decreases, following an
approximate power law $\propto k^\beta$, with $\beta=3/2$; see
\Fig{rspec_select_1024a_mu10_k002o}. 
This can be explained by the conservation of the Hosking integral
\cite{Hosking+Schekochihin21, Zhou+22}; see also Eq.~\eqref{hoscons}.
The exponent $\beta=3/2$ is characteristic of the fact that the net
chirality vanishes, even though near the peak itself the field is
locally fully helical, as we see from the proximity of 
$k|\HM(k)|/2$ and $\EEM(k)$; see \Eq{realizability}.

%FIG3
\begin{figure}[t]\begin{center}
\includegraphics[width=\columnwidth]{rspec_select_1024a_mu10_k002o}
\end{center}\caption[]{
Magnetic energy (solid lines) and normalized helicity spectra
$k\HM(k)/2$ (dotted lines with red and blue symbols for positive
and negative helicity spectra, respectively) for Run~O at
times $\eta=0.3$, $460$, 1500, 4600, 15,000, and 46,000.
The peaks $k_\mathrm{I}$ (peaks of the red curves) and $k_\mathrm{II}$
(peaks of the blue curves) evolve underneath the envelopes $\propto
k^{3/2}$ and $\propto k^{8/3}$, respectively.
}\label{rspec_select_1024a_mu10_k002o}\end{figure}

The newly injected magnetic helicity from the CPI leads to
a growth of the magnetic field at large wave numbers. 
It manifests itself mostly through the build-up of negative magnetic
helicity at high wave numbers.
At some point, we also see a gradual propagation of the
secondary peak $k_{\rm II}$ toward smaller $k$, 
which has not been addressed in Sec.~\ref{IIE}.
It lies underneath an envelope with an approximate $k^{8/3}$ slope; see
\Fig{rspec_select_1024a_mu10_k002o}.
At present, the exponent 8/3 is just empirical and there is no theory
for it.
It should be noted, however, that in other cases with a shorter inertial
range, we have found larger exponents.
Thus, the exponent could also be smaller when the inertial range is larger,
i.e., when there is more scale separation and $\aver{\mu_5}\xiM$ is larger.

Another characteristic wave number is $k_\pm$, where the sign of the
spectral magnetic helicity changes.
It is used in the definitions of $\aver{\mu_{\rm M}^+}$ and
$\aver{\mu_{\rm M}^-}$ in \Eqs{muMp}{muMm}.

%FIG4
\begin{figure}\begin{center}
\includegraphics[width=\columnwidth]{pkmax_comp_1024a_mu10_k002o}
\end{center}\caption[]{
Comparison of $k_{\rm I}$ (red), $k_\pm$ (orange),
and $k_{\rm II}$ (blue) for Run~O.
The dashed-dotted line indicates $\eta^{-4/9}$.
The green dashed line shows $\aver{\mu_5}$ and the
green dotted line shows the rms value $\mu_5^{\rm rms}$.
}\label{pkmax_comp_1024a_mu10_k002o}\end{figure}

%FIG5
\begin{figure}\begin{center}
\includegraphics[width=\columnwidth]{pkmax_comp_1024a_mu10_rep}
\end{center}\caption[]{
Similarly to \Fig{pkmax_comp_1024a_mu10_k002o}, but for Run~I.
The red dashed-dotted line indicates here the $\eta^{-1/3}$ scaling,
which describes the $k_{\rm I}$ scaling better than the
$\eta^{-4/9}$ scaling indicated by the red dotted line.
The orange and red dots indicate the crossings of the extrapolated
tangents on which the times $\eta_\pm^\mathrm{(i)}$, $\eta_\mathrm{I}$,
and $\eta_\pm^\mathrm{(ii)}$ are based.
}\label{pkmax_comp_1024a_mu10_rep}\end{figure}

\begin{table*}\caption{
Summary of the runs discussed in this paper.
Except for Run~P, where $\lambda=500$, we have in all other cases $\lambda=2\times10^4$.
Runs~A and B below the last horizontal line have nonvanishing
net chirality and are discussed at the end of the paper.
The asterisk on the value of $k_1=0.01$ Run~J'' indicates that the
resolution is $N^3=2048^3$, so the Nyquist wave number is here the
same as for Run~J' with $k_1=0.02$.
}\vspace{12pt}\centerline{\begin{tabular}{lccccccccccccccccccc}
Run & $k_0$ & $-\mu_{50}$ & $\sigma^{-1}$ & SLD & $\eta_{\rm CPI}$ & $\eta_{\rm diff}$ & $\eta_{\mu_\mathrm{M}^+}$ & $q_{\rm I}$ & $\eta_{\rm flip}$ &
$\eta_\pm^\mathrm{(i)}$ & $\eta_\mathrm{I}$ & $\eta_\pm^\mathrm{(ii)}$ & $\eta_{\mu_\mathrm{M}^-}$ & $|\aver{\muM^-}|^{\max}$ &
$k_\pm/k_{\rm I}$ & $k_{\rm II}/k_{\rm I}$ & $v_\lambda$ & $\urms^{\max}$ & $k_1$ \\ % $\eta_{\max}$
\hline
VI   &  1 &160 & $5\times10^{-4}$ &  no &0.08& $2\times10^3$ &   3 & 1/3 & --- & 0.3&   45 &   14 &   7 & 43     & 1.4 & 2.0 & 1.13  & 0.095 & 0.2  \\%  5,300 \\ % 1024c_mu160_k02
V    &  1 & 80 & $5\times10^{-4}$ &  no &0.3 & $2\times10^3$ &   4 & 1/3 & --- & 0.5&   55 &   40 &   4 & 13     & 1.5 & 1.8 & 0.57  & 0.050 & 0.2  \\%  5,300 \\ % 1024c_mu80_k02
IV   &  1 & 50 & $5\times10^{-4}$ &  no &0.8 & $2\times10^3$ &   6 & 1/3 & --- & 1.6&   50 &   55 &  50 & 5.1    & 1.5 & 2.0 & 0.35  & 0.050 & 0.1  \\%  5,300 \\ % 1024c_mu50
III  &  1 & 30 & $5\times10^{-4}$ &  no &2.2 & $2\times10^3$ &  14 & 1/3 & --- & 1.7&   80 &  108 & 150 & 1.5    & 1.8 & 2.3 & 0.21  & 0.018 & 0.2  \\%  5,300 \\ % 1024c_mu30_k02b
II+  &  1 & 20 & $5\times10^{-4}$ &  no & 5  & $2\times10^3$ &  18 & 1/3 & --- &  6 &   75 &  160 & 200 & 0.46   & 1.8 & 3.1 & 0.141 & 0.031 & 0.1  \\%    550 \\ % 1024a_mu20_5em4
II   &  1 & 20 & $2\times10^{-4}$ &  no &12.5& $5\times10^3$ &  60 & 1/3 & --- &  9 &   75 &  120 & 200 & 0.009  & 4.4 & 6.7 & 0.141 & 0.032 & 0.1  \\%  5,300 \\ % 1024a_mu20
II$-$&  1 & 20 & $1\times10^{-4}$ &  no & 25 & $1\times10^4$ & 160 & 1/3 & --- & 14 &   75 &  140 & 200 & 0.003  & 6.7 & 10  & 0.141 & 0.031 & 0.1  \\%  1,100 \\ % 1024b_mu20_1em4
I    &  1 & 10 & $2\times10^{-4}$ &  no & 50 & $5\times10^3$ & 125 & 1/3 & --- & 30 &   80 &  160 & 250 & 0.009  & 6.7 & 9.6 & 0.071 & 0.030 & 0.05 \\% 36,000 \\ % 1024a_mu10_rep
O    &  1 & 10 & $2\times10^{-4}$ &  no & 50 & $5\times10^3$ & 125 & 4/9 & --- & 20 &   70 &  120 & 300 & 0.008  & 7.3 & 9.5 & 0.071 & 0.0123& 0.02 \\% 52,000 \\ % 1024a_mu10_k002o
O'   &  1 & 10 & $2\times10^{-4}$ & yes & 50 & $5\times10^3$ & 125 & 4/9 & --- & 20 &  110 &  120 & 400 & 0.015  & 6.4 & 8.4 & 0.071 & 0.0103& 0.02 \\% 34,000 \\ % 1024a_mu10_k002b
L    &  1 & 10 & $2\times10^{-4}$ & yes & 50 & $5\times10^3$ & 125 & 4/9 & --- &180 &  400 &  500 &1000 & 0.027  & 6.7 & 8.7 & 0.071 & 0.0079& 0.01 \\%147,000 \\ % 1024a_mu10_k001c
M    &  1 &  7 & $2\times10^{-4}$ & yes &102 & $5\times10^3$ & 165 & 1/3 & --- &260 &  220 &  800 & 800 & 0.006  & 5.8 & 7.2 & 0.049 & 0.0065 & 0.01 \\%  6,000 \\ % 1024a_mu7_k001a
N    &  1 &  5 & $2\times10^{-4}$ & yes &200 & $5\times10^3$ & 235 & 1/3 & --- &350 &  200 &  800 &1000 & 0.0015 & 6.3 & 7.8 & 0.035 & 0.0055 & 0.01 \\%  6,000 \\ % 1024a_mu5_k001a
N'   &  1 &  5 & $2\times10^{-4}$ & yes &200 & $5\times10^3$ & 200 & 4/9 & --- &--- &  800 & 3000 &15000& 0.00004& 5.5 & 2.4 & 0.035 & 0.0035& 0.005\\% 49,000 \\ % 1024a_mu5_k0005b
F    &  1 &  5 & $2\times10^{-4}$ & yes &200 & $5\times10^3$ & --- & 4/9 & 100 &--- &  250 & 9000 & --- & ---    & 30  & 30  & 0.035 & 0.0055& 0.01 \\% 23,000 \\ % 1024d_mu5_Gam1em2_lam2e4
J    &  1 &  5 & $5\times10^{-4}$ &  no & 80 & $2\times10^3$ &  71 & 4/9 & --- &230 &  300 &  500 & 700 & 0.0003 & 6.1 & 7.6 & 0.035 & 0.0068 & 0.01 \\%  6,800 \\ % 1024b_mu5_k001a
J''  &  1 &  5 & $5\times10^{-4}$ &  no & 80 & $2\times10^3$ &  71 & 4/9 & --- & 90 &  300 &  500 & 460 & 0.0006 & 6.1 & 7.6 & 0.035 & 0.0071 & 0.01*\\%  3,000 \\ % 2048a_mu5_k001b
J'   &  1 &  5 & $5\times10^{-4}$ &  no & 80 & $2\times10^3$ &  76 & 1/3 & --- & 95 &  120 &  500 & 460 & 0.0005 & 5.8 & 7.7 & 0.035 & 0.0070 & 0.02 \\%  2,000 \\ % 1024a_mu5_k002b
P    &  1 & 0.1& $2\times10^{-4}$ &  no &$5\times10^5$& $5\times10^3$ &$10^4$&3/5 & --- & -- &  160 &  --- & --- &$3\times10^{-9}$ & --- & --- & 0.001 &0.0070 & 0.02 \\% 1024a_mu01_k002o <<==
G    & 0.5& 10 & $2\times10^{-4}$ &  no & 50 & $2\times10^4$ & 200 & 1/3 & --- & 30 &  360 &  360 & 300 & 0.044  & 4.6 & 6.6 & 0.071 & 0.0188 & 0.05 \\%  7,600 \\ % 1024a_mu10_k05_1024proc2
H    & 0.2& 10 & $2\times10^{-4}$ &  no & 50 &$1.2\times10^5$& 375 & 1/3 & --- & 75 & 2000 & 2000 &3000 & 0.42   & 2.0 & 2.7 & 0.071 & 0.0074 & 0.02 \\% 20,000 \\ % 1024b_mu10_k02_1024proc
\hline
A    &  1 & 10 & $2\times10^{-4}$ &  no & 50 & $5\times10^3$ & --- & 4/9 & --- &  8 &  110 &  210 & --- & ---    & 7.3 & 9.5 & 0.071 & 0.0109 & 0.02 \\% 52,000 \\ % 1024a_mu10_k002o_less
B    &  1 & 10 & $2\times10^{-4}$ &  no & 50 & $5\times10^3$ & --- & 4/9 & --- & 20 &   90 &  120 & 250 & ---    & 7.8 & 8.9 & 0.071 & 0.0137 & 0.02 \\% 52,000 \\ % 1024a_mu10_k002o_more
\label{Ttimescales}\end{tabular}}\end{table*}
% scr/axel/chiral_fluids/turbulent_decay/idl

In \Fig{pkmax_comp_1024a_mu10_k002o}, we plot the evolution of the
characteristic wave numbers $k_{\rm I}$, $k_\pm$, and $k_{\rm II}$.
We clearly see the $k_{\rm I}\propto\eta^{-4/9}$ decay predicted by
the conservation of the Hosking integral adapted to the total chirality
\cite{BKS23}.
It emerges after a time $\eta_{\rm I}$, which is expected to be close to
$\eta_\lambda$ (and also $\eta_\mathrm{turb}$); see Eq.~\eqref{etalam}.
In Run~O we find $\eta_{\rm I}\approx100$.

The evolution of $k_\pm$ and $k_{\rm II}$ can be seen more clearly
when the Nyquist wave number is larger.
We therefore discuss in \Fig{pkmax_comp_1024a_mu10_rep} another run, also
with $N=1024^3$ mesh points, but now with $k_1=0.05$ (instead of 0.02),
so $k_{\rm Ny}=25.6$, which is now a little over five times larger than
$|\mu_{\rm 50}|/2=5$.
In \Tab{Ttimescales}, this run is referred to as Run~I, which differs
from the previously discussed Run~O mainly in the value of $k_1$.
It also has a shallower scaling of the correlation length,
$\xiM\propto k_\mathrm{I}^{-1} \propto \eta^{1/3}$, which seems to be an artifact caused by
insufficient scale separation, i.e., the value of $k_1$ is not
sufficiently small.
Empirically, we find that if $k_0/k_1 \gg 20$, there is an
inverse cascade with $\xiM \propto k_\mathrm{I}^{-1} \propto \eta^{4/9}$.
The parameters $\eta_\mathrm{I}$, $\eta_\pm^\mathrm{(i)}$, and
$\eta_\pm^\mathrm{(ii)}$, listed in \Tab{Ttimescales}, are discussed
below.
We also give here the values of $v_\mu$ and $v_\lambda/v_\mu$, as well
as the number of mesh points and the length of the run, $\eta_{\max}$,
which can be useful in assessing the reliability of the numerical results.
Run~O' is similar to Run~O, except that here, SLD has been added.
The two runs are virtually indistinguishable.

The evolution of the peaks of the spectrum can be summarized as follows. 
(i) After the start of the run, the CPI induces a growth of the negative
helicity modes at the secondary peak $k_\mathrm{II}$, which stays constant
until $\eta=\eta_\pm^\mathrm{(i)}$, and then starts to decrease with time in a
power law fashion, $k_\mathrm{II} \propto\eta^{-q_{\rm IIa}}$ with
$q_{\rm IIa}\approx1$ in all cases.
(ii) The original large-scale spectrum is unchanged until some time
$\eta=\eta_{\rm I}$ and starts to decrease via an inverse cascade with
$k_\mathrm{I}(\eta) \propto\eta^{-q_{\rm I}}$, where $q_{\rm I}$ is
expected to be equal to the exponent $q=4/9$ found in Ref.~\cite{BKS23}.
(iii) At time $\eta=\eta_\pm^\mathrm{(ii)}$, the decay of
the secondary peak becomes slower with a smaller index,
$k_\mathrm{II} \propto\eta^{-q_{\rm IIb}}$, with
$q_{\rm IIb}<q_{\rm IIa}\approx1$.
Those parameters are summarized in \Tab{Ttimescales}.

The plot of characteristic wave numbers $k_{\rm I}$, $k_\pm$, and
$k_{\rm II}$ in \Fig{pkmax_comp_1024a_mu10_rep} shows three distinct
times $\eta_\pm^{\rm(i)}\la\eta_{\rm I}\la\eta_\pm^{\rm(ii)}$, where
$k_\pm$ begins to decrease first rapidly, at $\eta=\eta_\pm^{\rm(i)}$,
and later, at $\eta=\eta_\pm^{\rm(ii)}$, more slowly, approximately like
$\eta^{-4/9}$, i.e., $q_{\rm IIb}\approx q=4/9$.
The decay of $k_{\rm II}$ closely follows that of $k_\pm$.
The decay of $k_{\rm I}$, on the other hand, does not show the rapid
decay phase that we see in $k_\pm$ and $k_{\rm II}$, but turns directly
into the approximate $\eta^{-4/9}$ decay at $\eta=\eta_\pm^{\rm(i)}$.

\subsection{Onset of inverse cascading}

It is of interest to vary the separation between $|\mu_{50}|/2$ and $k_0$
to see the dependence of the relevant characteristic times on these
wavenumbers.
We have performed simulations for different values and consider runs
where we change $k_{\rm I}$ and keep $k_{50}$ fixed, and others where
we change $k_{50}$ and keep $k_0$ fixed.
It both cases, of course, since we want to satisfy
$\aver{\mu_5}+\aver{\mu_{\rm M}}=\const$, we need to adjust the
amplitude of the initial magnetic field correspondingly.
The results are summarized in \Tab{Ttimescales} and plotted
in \Figs{peta1_vs_mu50}{peta2_vs_vlam}.

%FIG6
%/home/brandenb/tex/mhd/kohei/idl
\begin{figure}\begin{center}
\includegraphics[width=\columnwidth]{peta1_vs_mu50}
\end{center}\caption[]{
Dependence of $\eta_\pm^{\rm(i)}$, $\eta_{\rm I}$, and
$\eta_\pm^{\rm(ii)}$ on $\eta_{\rm CPI}$.
$\eta_\pm^{\rm(i)}$ shows an approximate $\eta_{\rm CPI}^{2/3}$
dependence along two branches that are separated by a factor of about 6.
$\eta_{\rm I}$ and $\eta_\pm^{\rm(ii)}$ are essentially
independent of $\eta_{\rm CPI}$.
The inset shows that $\eta_\pm^{\rm(i)}$ scales inverse
quadratically with $|\mu_{50}|$.
}\label{peta1_vs_mu50}\end{figure}

%FIG7
\begin{figure}\begin{center}
\includegraphics[width=\columnwidth]{peta2_vs_vlam}
\end{center}\caption[]{
Dependence of $\eta_{\rm I}\approx\eta_\pm^\mathrm{(ii)}$ and
$\eta_\pm^\mathrm{(i)}$ on (a) $\eta_{\rm turb}$ and
(b) $\eta_\lambda$, as well as (c) the dependence of
$\eta_{\rm turb}$ on $\eta_\lambda$.
}\label{peta2_vs_vlam}\end{figure}

One may presume that $\eta_\pm^{\mathrm{(i)}}$ is roughly estimated by
$\eta_\mathrm{CPI}$ since the grow of negative helicity modes becomes
effective at that time.
We see, however, that, while $\eta_\pm^{\rm(i)}$ decreases
quadratically with increasing $|\mu_{50}|$, the dependence on
$\eta_{\rm CPI}=\sigma\mu_{50}^{-2}$ is shallower than linear and follows
approximately an $\eta_{\rm CPI}^{2/3}$ scaling; see \Fig{peta1_vs_mu50}.
Thus, $k_{\rm II}$ starts to decline more rapidly when $|\mu_{50}|$ is
large, although it is unclear why this exponent is here $\approx2/3$.
On the other hand, we see that the five data points with $k_1=0.01$
(Runs~L, M, N, J, and J'' with smaller $|\mu_{50}|$) lie on another
$\eta_{\rm CPI}^{2/3}$ line that is shifted upward by a factor of about
6 relative to the runs with larger $k_1$.
The reason for this is that for large values of $\eta_{\rm CPI}$, it
became necessary to decrease the value of $k_1$.
This decreased the Nyquist wave number since $N$
remained unchanged, which can cause artifacts in the values of $k_\pm$.
Small values of $k_1$ also facilitates the $\eta^{4/9}$
scaling of $\xiM$ and related length scales; see the comparison between
Runs~N and N' in \Tab{Ttimescales}.
This shows that $\eta_\pm^{\rm(i)}$ is currently very sensitive to
these restrictions which will be alleviated in future with larger
computational power.
Nevertheless, there is clearly a trend for an uprise in the dependence
of $\eta_\pm^{\rm(i)}$ on $\eta_{\rm CPI}$ for large values.

Next, we examine the dependence of $\eta_\mathrm{I}$ and
$\eta_\pm^\mathrm{(ii)}$ on $k_0$ and $\mu_{50}$.
Figure~\ref{peta1_vs_mu50} shows that
the time $\eta_{\rm I}$ of the onset of the decline of $k_{\rm I}$
does not strongly depend on the value of $\mu_{50}$.
Likewise, the time $\eta_\pm^\mathrm{(ii)}$ when the decay of $k_{\rm II}$ slows
down, does not strongly depend on $\mu_{50}$.
Again, however, there is an upward shift of data points for the four runs,
for which $k_1=0.01$.
As discussed in Sec.~\ref{IIE}, we expect that $\eta_\mathrm{I}$
is close to $\eta_\mathrm{turb}$ and $\eta_\lambda$. 
The upper two panels of Fig.~\ref{peta2_vs_vlam} show the dependence 
of $\eta_\pm^{\rm(i)}$, $\eta_{\rm I}$, and $\eta_\pm^{\rm(ii)}$ on $\eta_\mathrm{turb}$ and $\eta_\lambda$,
respectively.
From these plots, we estimate that
\begin{equation}
\eta_{\rm I}\approx1.4\, \eta_{\rm turb} \approx 2.2\, \eta_\lambda.
\label{etaIscaling}
\end{equation}
In the lowest panel of Fig.~\ref{peta2_vs_vlam}, we also show the relation
between $\eta_{\rm turb}$ and $\eta_\lambda$, i.e.,
\begin{equation}
    \eta_\mathrm{turb} \approx  1.6\, \eta_\lambda, 
\end{equation}
which shows the validity of the estimate of $\urms^\mathrm{max}$ 
in terms of ${\tilde v}_\lambda$. 
Equation~\eqref{etaIscaling} is useful for estimating the properties of
magnetic field strength and coherence length at later times.
Therefore, we conclude that the numerical results support,
at least for a moderate scale separation, $1 < |\mu_{50}|/k_0 \lesssim {\cal O}(10)$,
the theoretical estimate for the evolution of the characteristic scales
given in Sec.~\ref{IIE} with a more accurate determination 
of the time of the onset of the scaling evolution, \Eq{etaIscaling}.

%FIG8
\begin{figure}\begin{center}
\includegraphics[width=\columnwidth]{phmax_comp}
\end{center}\caption[]{
Comparison of $\aver{\mu_{\rm M}^\pm}$ for Run~II (red lines),
Run~J (blue lines), and Run~G (orange lines).
The times $\eta_{\rm ACC}$ are marked by the correspondingly
colored filled symbol at the crossing points of the extrapolated
$\eta^{-2/3}$ decay law with the initially constant values,
indicated by dotted line.
The $\eta^{-4/3}$ decay law $\aver{\muM^-}$ is shown as the
dashed-dotted line.
}\label{phmax_comp}\end{figure}

\subsection{Evolution of $\aver{\mu_5}$ and $\aver{\mu_{\rm M}^\pm}$}
\label{subsec:evolution}

We now discuss how the chirality of the system evolves.
Using \Eqs{muMp}{muMm}, we divide the magnetic helicity into
$\aver{\mu_{\rm M}^+}$ and $\aver{\mu_{\rm M}^-}$.
The typical evolution of $\langle \mu_5 \rangle$ and
$\langle \muM^{\pm} \rangle$ is as follows.
(i) $\aver{\mu_5}$ and $\aver{\muM^{+}}$ stay constant until the time
$\eta=\eta_{\mu_\mathrm{M}^+}$, when the ACC commences
exhibiting a power law decay.
(ii) $\aver{\muM^{-}}$ grows until the time
$\eta = \eta_{\mu_\mathrm{M}^-}$ and then decays.

As discussed in Sec.~\ref{IIE}, the decay of $\aver{\mu_5}$ and
$\aver{\muM^{+}}$ due to the ACC is expected to be like $\eta^{-2/3}$.
In \Fig{phmax_comp}, we have overplotted the asymptotic $\eta^{-2/3}$
decay laws of magnetic helicity with results of some of the representative
numerical runs (Runs~II, J, and G), which clearly shows that the numerical
results support the theoretical prediction.

The decay of $\langle \muM^{-} \rangle$ is faster than that of
$\aver{\mu_5}$ and $\langle \muM^{+} \rangle$ and follows an approximate
$\eta^{-4/3}$ law, resulting in a decay of the ratio $\langle \muM^{-}
\rangle/\langle \muM^{+} \rangle\propto\eta^{-2/3}$.
Therefore, unless $\langle \muM^{-} \rangle$ becomes comparable to
$\langle \muM^{+} \rangle$ when the grow stops, a complete cancellation
between $\langle \muM^{-} \rangle$ and $\langle \muM^{+} \rangle$
never occurs.

The production of $\aver{\muM^-}$ is expected to be a result of the CPI.
We now address the question of how much $\aver{\muM^-}$ is being
produced and what its maximum value depends on.
\FFig{phmax_comp} shows that $\aver{\muM^-}$ is generally rather small,
and at least for $\mu_{50}/k_0 \lesssim 20$ there is always a strong
imbalance between $|\aver{\muM^+}|$ and $|\aver{\muM^-}|$, which never
enters a phase with a near-complete cancellation.

To see whether this is related to the value of the conductivity, we
compare simulations with different values of $\sigma$.
It turns out that runs with smaller magnetic diffusivity
($\sigma^{-1}=10^{-4}$) result in an even larger imbalance, while those
with a larger diffusivity ($\sigma^{-1}=5\times10^{-4}$) have a smaller
imbalance; compare Runs~II$+$, II, and II$-$ in \Tab{Ttimescales}.

Before closing this section, let us comment on another trend
in the numerical runs we conducted regarding the absence
of a near-complete cancellation between
$\langle \muM^{-} \rangle$ and $\langle \muM^{+} \rangle$.
For Runs~III--VI, the ratio $|\aver{\muM^-}|^{\max}/|\mu_{50}|$ becomes rather large.
This could be due to the very large scale separation of $k_0$ and $|\mu_{50}|$. 
This suggests a possibility that the CPI completes the cancellation 
between the magnetic helicity and chirality immediately. 
However, the positive and negative helicity modes are 
distributed at separate length scales with the negative ones sitting 
at higher length scales and the latter receives a stronger magnetic diffusion. 
Therefore we expect the cancellation not to be complete and that the two helicity modes decay 
with a power-law decay, not an exponential one, 
though the scaling index can be different from $-2/3$. 
In order to investigate the evolution of the system in such extreme cases, 
$|\mu_{50}|/k_0 \gg {\cal O}(10)$, 
we need to have a sufficiently large box size to realize the corresponding 
scale separation. 
The detailed study is left for future study.

\subsection{Onset of ACC}

In \Fig{pkpm_vs_eta45b}, we show the dependence of
$\eta_{\mu_\mathrm{M}^+}$ and $\eta_{\mu_\mathrm{M}^-}$ on
$\eta_\mathrm{ACC} = \sigma/|\mu_{50}k_0|$
(for the case $k_0 \ll |\mu_{50}|$; see \Eq{etaACCcase1}).
It turned out that $\eta_{\mu_\mathrm{M}^+}$ increases with
$\eta_\mathrm{ACC} = \sigma/|\mu_{50}k_0|$ such that
\begin{equation}
\eta_{\mu_\mathrm{M}^+}\approx0.2\,\eta_{\rm ACC} = 0.2\, \sigma/|\mu_{50}k_0|
\label{eta_ACC}
\end{equation}
provides a good description to the data, which supports the discussion
in Sec.~\ref{IIE}.
Furthermore, $\eta_{\mu_\mathrm{M}^-}$ shows an approximately
linear dependence on $\eta_{\rm ACC}$.
This is reasonable because the CPI becomes ineffective when the ACC
onsets such that $\aver{\mu_\mathrm{M}^-}$ is no longer amplified by
the CPI after that.

%FIG9
\begin{figure}\begin{center}
\includegraphics[width=\columnwidth]{pkpm_vs_eta45b}
\end{center}\caption[]{
Dependence of $\eta_{\mu_\mathrm{M}^+}$ and $\eta_{\mu_\mathrm{M}^-}$
on (a) $\sigma/|\mu_{50}k_0|$ as well as the geometric means of
$\eta_{\rm CPI}$ and (b) either $\eta_{\rm turb}$ or (c) $\eta_\lambda$.
}\label{pkpm_vs_eta45b}\end{figure}

\subsection{The scale ratios $k_\pm/k_{\rm I}$ and $k_{\rm II}/k_{\rm I}$}

We also mention another observation in the case with $k_0 \ll |\mu_{50}|$.
At late times, the scale ratios $k_\pm/k_{\rm I}$ and $k_{\rm II}/k_{\rm I}$
reach values that are approximately constant in time.
It is about $10$ in the case of Run~O, i.e., equal to the initial scale
separation, $|\mu_{50}|/k_0=10$.
One might have expected the scale ratios to increase with $|\mu_{50}|/k_0$.
However, in all other cases, this ratio is smaller.
Some of this might also be caused by one of the two scale separation
constraints not being well enough obeyed, although the counter-intuitive
trend remains surprising.

%FIG10
\begin{figure}\begin{center}
\includegraphics[width=\columnwidth]{pkpm_vs_tratio}
\end{center}\caption[]{
Dependence of $k_\pm/k_{\rm I}$ and $k_{\rm II}/k_{\rm I}$
on $\eta_{\rm CPI}$, showing an $\eta_{\rm CPI}^{-0.4}$ behavior.
on $\eta_{\rm diff}$, showing a $\propto\eta_{\rm diff}^{1/2}$ behavior.
}\label{pkpm_vs_tratio}\end{figure}

In \Fig{pkpm_vs_tratio}, we show the ratios
$k_\pm/k_{\rm I}$ and $k_{\rm II}/k_{\rm I}$ versus
$|\mu_{50}|/k_0=(\eta_{\rm diff}/\eta_{\rm CPI})^{1/2}$.
The two insets give separately the dependencies on
$1/\eta_{\rm CPI}$, showing an $\eta_{\rm CPI}^{-0.4}$ behavior, and
on $\eta_{\rm diff}$, with a $\propto\eta_{\rm diff}^{1/2}$ behavior.
We see that $k_\pm/k_{\rm I}$ and $k_{\rm II}/k_{\rm I}$ decrease both
with $1/\eta_{\rm CPI}$ and with $\eta_{\rm diff}$, giving a combined
dependence on just the ratio $|\mu_{50}|/k_0$.
Thus, we see that, somewhat unexpectedly, large $|\mu_{50}|$ and small
$k_0$ tend to be detrimental to producing large scale ratios.

\subsection{Effect of chirality-flipping} \label{sec:IIIG}

The simulations discussed so far had $\Gamma=0$ and they resulted in
a final state where $\aver{\mu_5}$ and $\aver{\mu_{\rm M}}$ vanish at
late times.
As discussed in the introduction, spin flipping could prematurely
lead to a vanishing $\aver{\mu_5}$, which would imply that the decay
of $\aver{\mu_{\rm M}}$ would slow down and level off at a value away
from zero.
To study this quantitatively, we show in \Fig{p_comp2} the evolution of
$\aver{\mu_5}$, $\aver{\mu_{\rm M}}$, and $\aver{\mu_5}+\aver{\mu_{\rm M}}$
for Run~F with $\eta_{\rm flip}=100$ and $\Gamma=10^{-2}$ either for
the rest of the run or only until $\eta_{\rm off}=10^3$ (Run~F in
\Tab{Ttimescales}).

First, we study a case where spin flipping acts permanently (after $\eta=\eta_{\rm flip}$),
which is shown in \Fig{p_comp2} as solid lines.
We see that $|\aver{\mu_5}|$ begins to decrease rapidly to zero
after $\eta_{\rm flip}=100$.
This slows down the decay of $\aver{\mu_{\rm M}}$, which then declines
at a much smaller rate.
Qualitatively similar behaviors are also seen for smaller values of
$\Gamma$.
In all cases, we see that $\aver{\mu_5}+\aver{\mu_{\rm M}}$ evolves away
from zero.
This is because the total chirality is then no longer conserved.
The decay of $\aver{\mu_{\rm M}}$ is understood by magnetic diffusion. 
Thus we expect the decrease to slow down for a larger scale separation
between the magnetic diffusion scale and $k_{\rm I}$.

Next, it is also of interest to study a case 
where spin flipping acts 
only for a certain time interval and is then turned off again
at $\eta=\eta_{\rm off}$.
This case is shown in \Fig{p_comp2} as dashed lines.
We see that, when $\Gamma=0$ after $\eta_{\rm off}=10^3$, the sum
$\aver{\mu_5}+\aver{\mu_{\rm M}}$ is strictly constant and away from zero. 
This is in contrast to the case with permanently nonvanishing $\Gamma$,
where the sum continues to decrease slowly.
The constancy of the total chirality leads to the behavior
that $\aver{\mu_{\rm M}}$ stops to decline rapidly at a larger value.
Furthermore, during that time, some of the magnetic helicity 
decays due to the magnetic diffusion and is temporarily
converted back into fermion chirality through the total chirality conservation;
see the small increase of $\aver{\mu_5}$
with a positive maximum at $\eta\approx4000$ in \Fig{p_comp2}.
Later, however, this excess fermion chirality gets converted back into
magnetic fields, which explains the slight uprise of $\aver{\mu_{\rm M}}$
near the end of the simulation.
Indeed, this process is similar to the one seen in
Refs.~\cite{Hirono+15,Schober+2020}.
This is natural because after the decay of $\aver{\mu_5}$ the setup
becomes very similar to the ones in these studies.

%FIG11
\begin{figure}\begin{center}
\includegraphics[width=\columnwidth]{p_comp2}
\end{center}\caption[]{
Evolution of $\aver{\mu_{\rm M}}$ (red), $\aver{\mu_5}$ (blue), and their
sum (black) for Run~F with $\eta_{\rm flip}=100$ and $\Gamma=10^{-2}$
either for the rest of the run or only until $\eta_{\rm off}=10^3$.
}\label{p_comp2}\end{figure}

%FIG12
\begin{figure}\begin{center}
\includegraphics[width=\columnwidth]{pkmax_comp_1024d_mu5_Gam1em2_lam2e4}
\end{center}\caption[]{
$k_\pm^\mathrm{(i)}$ and $k_{\rm I}$ for Run~F with spin flipping,
$\eta_{\rm flip}=100$ and $\Gamma=10^{-2}$ for the rest of the run.
As in \Fig{pkmax_comp_1024a_mu10_k002o}, the green dashed line shows
$\aver{\mu_5}$ and the green dotted line shows $\mu_5^{\rm rms}$.
}\label{pkmax_comp_1024d_mu5_Gam1em2_lam2e4}\end{figure}

In \Fig{pkmax_comp_1024d_mu5_Gam1em2_lam2e4}, we show
$\eta_\pm^\mathrm{(i)}$ and $\eta_{\rm I}$ in the presence of
spin flipping.
The results suggests that the $\eta^{-4/9}$ decay changes into
the faster $\eta^{-2/3}$ decay.
Spin flipping brings $\aver{\mu_5}$ close to zero.
This process stops or slows down the decline of magnetic helicity, which
therefore remains positive.
At late times, $\aver{\mu_5}$, which was originally negative, now becomes
positive and settles at a value of around $\aver{\mu_5}\approx k_1$.
This is because at later times the positive chirality induced due
to the helicity decay by the magnetic diffusion through the chiral
anomaly is balanced by the erasure of the chirality through the
CME~\cite{Hirono+15,Schober+2020}, similar to the baryon asymmetry through
the magnetic helicity decay much before the electroweak phase transition
\cite{Giovannini:1997eg, Giovannini:1997gp, Fujita:2016igl}.

The sign of the final value of $\aver{\mu_5}$ is determined by the
magnetic helicity after the decay of $\aver{\mu_5}$ due to the onset of
spin flipping.
In the cases presented above, the sign of the magnetic helicity at the
time of the onset of spin flipping was positive and thus the chiral
chemical potential at later times was also positive.
If the initial magnetic field is weaker and the total chirality being
negative (see App.~\ref{Imbalanced}), the sign of the final value of
$\aver{\mu_5}$ can stay negative.

Our runs show that spin flipping can lead to a significant increase of the fraction
of the magnetic helicity that can be preserved in spite of the fact that
the system has vanishing total chirality.
This also reduces the total energy density dissipation of the system.
In the absence of spin flipping, both magnetic helicity and chiral chemical
potential would approach zero, so there would be no magnetic helicity available
for successful baryogenesis.
In the real Universe, however, spin flipping due to the electron Yukawa interaction,
which really violates the (total) chirality conservation, 
inevitably acts at $T\lesssim 10^2\,$TeV \cite{Campbell:1992jd,Bodeker:2019ajh}, 
and hence magnetic helicity survives more or less at the electroweak phase transition. 

In \Fig{pkmax_comp_1024d_mu5_Gam1em2_lam2e4}, we see an interval
between the onset of spin flipping, $\eta=\eta_\mathrm{flip} = 10^2$,
and the onset of the $\eta^{-2/3}$ scaling evolution of
$k_{\rm I}$, $\eta \sim 6 \times 10^3$, which
marks the real onset of the evolution with
(pure) magnetic helicity conservation.
For a rough estimate of the magnetic field evolution, however, 
we shall practically use $\eta_\mathrm{flip}$ as the 
switching time between the adapted Hosking integral conservation 
and the (pure) magnetic helicity conservation.

\subsection{Cases with initially small $|\mu_{50}|/k_0$} \label{sec:IIIH}

In all the cases considered so far, we assumed $|\mu_{50}|/k_0 > 1$.
We now consider the opposite case and discuss runs with $\mu_{50}=-0.1$,
keeping still $k_0=1$, so $|\mu_{50}|/k_0=0.1$ (Runs~P, Q and R), and
also a run with $\mu_{50}=-0.5$ and $k_0=1$ (Run~S).
To prevent the magnetic field from being too weak, while still preserving
vanishing total chirality, we now decrease the value of $\lambda$
and choose $\lambda=500$, 50, and 5 for Runs~ P (and S), Q, and R,
respectively.
All the runs end at $\eta\sim 10^4$.
The parameters of these runs are summarized in \Tab{Tbeta}.

\begin{table}[t]\caption{
Empirical values of $\beta$ for cases with $|\mu_{50}|<k_0$.
For a given value of $\lambda$, the values of $\vAz$ followed
from the requirement that the total chirality vanishes.
The resulting maximum rms velocity $\urms^{\max}$ is listed for
completeness.
}\vspace{12pt}\centerline{\begin{tabular}{cccccccccc}
Run & $-\mu_{50}$ & $\lambda$ & $v_\mu$ & $\tilde{v}_\lambda$ & $\vAz$ & $\urms^{\max}$ & SLD & $\beta$ & $\eta_{\rm I}$ \\
\hline
P & 0.1 &500 & $2\times10^{-5}$ & 0.014 & 0.026 & 0.008 &  no & 0.33 & 160 \\% 1024a_mu01_k002o
Q & 0.1 & 50 & $2\times10^{-5}$ & 0.045 & 0.081 & 0.028 &  no & 0.15 &  50 \\% 1024a_mu01_k002r
R & 0.1 &  5 & $2\times10^{-5}$ & 0.141 & 0.257 & 0.076 & yes & 0.05 &  27 \\% 1024a_mu01_k002q2
S & 0.5 &500 & $       10^{-4}$ & 0.032 & 0.057 & 0.019 &  no & 0.33 &  70 \\% 1024a_mu05_k002o
\label{Tbeta}\end{tabular}}\end{table}

\begin{table}[t]\caption{
Possible combinations of $q=2/(\beta+3)$, $p=2(1-q)$, and $r=p-q$
in the range $0\leq\beta\leq1/3$.
}\vspace{12pt}\centerline{\begin{tabular}{llll}
$\qquad\beta$ & $\qquad q$ & $\qquad p$ & $\qquad r$ \\
\hline
$3/2=1.50$        & $4/9\approx0.44$   & $10/9\approx1.11$ & $2/3\approx0.67$  \\
$1/3\approx0.33$  & $3/5=0.60$         & $4/5=0.80$         & $1/5=0.20$         \\
$0.15          $  & $    0.63$         & $    0.73$         & $0.10    $         \\
$0.05          $  & $    0.66$         & $    0.69$         & $0.03    $         \\
$0             $  & $2/3=0.67$         & $2/3=0.67$         & $0       $         \\
\label{Tpqr}\end{tabular}}\end{table}

%FIG13
\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{penerg_RunsPR}
\end{center}\caption[]{
Time dependence of $\EEM$ (black), $\xiM^{-1}$ (orange), $\HHM$ (red),
and $-2\aver{\mu_5}/\lambda$ (blue), for Runs~P (a) and R (b).
}\label{penerg_RunsPR}\end{figure}

%FIG14
\begin{figure}[t]\begin{center}
\includegraphics[width=\columnwidth]{pEMxi_pq_runPQR}
\end{center}\caption[]{
$pq$ diagram for Runs~P (red symbols), Q (orange symbols), and
R (blue symbols) at times $t=700$, 1000, 1500, 2200, 3200, 4600, 6800,
$10^4$, $1.5\times10^4$, $1.5\times10^4$, $2.2\times10^4$, and
$3.2\times10^4$, corresponding to symbols of increasing size.
The solid line denotes the scale-invariance line $p=2(1-q)$, the dashed
line the empirical $\beta=1/3$ line, and the dashed-dotted
line is the resulting $r=1/5$ line for the magnetic helicity decay.
We also show cases with stronger magnetic field strength for
Runs~Q and R, where the solutions evolve along $\beta\approx0.15$
and $\beta\approx0.05$, respectively.
Toward the end of the runs, the finite size effects of the domain
begin to affect the solution.
The dotted line denotes the $\beta=0$ line for magnetic helicity
conservation and is shown for comparison.
}\label{pEMxi_pq_runPQR}\end{figure}

%FIG17 --> FIG15
\begin{figure}\begin{center}
\includegraphics[width=\columnwidth]{pkmax_comp_1024a_mu05_k002o}
\end{center}\caption[]{
Comparison of $k_{\rm I}$ (red), $k_\pm$ (orange),
and $k_{\rm II}$ (blue) for Run~S with $\mu_{50}=-0.5$.
The dashed-dotted line indicates $\eta^{-4/9}$.
The green dashed line shows $\aver{\mu_5}$ and the
green dotted line shows the rms value $\mu_5^{\rm rms}$.
}\label{pkmax_comp_1024a_mu05_k002o}\end{figure}

Smaller values of $\lambda$ correspond to to larger magnetic fields.
We see that this also leads to a gradual decrease of 
the scaling index of the envelope of the magnetic energy spectrum,
$\beta$, toward zero.
For a given value of $\beta$, we expect that the scaling indices 
$q$, $p$, and $r$, which are those of the
evolution of the magnetic coherence length, energy density, and helicity, respectively
($\xiM\propto\eta^q$, $\EEM\propto\eta^{-p}$, and $\HHM\propto\eta^{-r}$),  
are given as $q=2/(\beta+3)$, $p=2(1-q)$, and $r=p-q$. 
In \Fig{penerg_RunsPR}(a), we see that for Run~P the exponents in
agree reasonably well with those expected for $\beta=1/3$.
In \Fig{penerg_RunsPR}(b), we also show the results for Run~R, where
$\lambda$ is a hundred times smaller and the magnetic field ten times
stronger.
Now the value of $\beta$ is very small (about 0.05), corresponding
to $q=0.66$, $p=0.69$, and $r=0.03$.

In \Tab{Tpqr}, we list several combinations of the expected scaling
indices $q$, $p$, and $r$ for $0\leq \beta \leq 3/2$.
Interestingly, in the range $0\leq\beta\leq1/3$, the values of $q$
and $p$ do not vary much in this range, especially compared to the case 
for the evolution with the (adapted) Hosking integral conservation, $\beta=3/2$,
so if they do not agree with
those from the simulations, the discrepancy cannot easily be resolved
by changing the value of $\beta$ within reasonable limits.
Note that $\beta=0$ is expected if the evolution is governed by
(pure) magnetic helicity conservation.

In the corresponding $pq$ diagram \Fig{pEMxi_pq_runPQR}, we see that Run~P
approaches the scale-invariance line $p=2(1-q)$ along the line $r=1/5$.
At the intersection, we have $q=3/5$ and $p=4/5$.
However, for Runs~Q and R with stronger magnetic field
strengths, here measured in terms of the initial Alfv\'en speed
$\vAz =\tilde{v}_\lambda=B_0/\sqrt{\rho_0}$, the solution approaches the
$\beta=0$ line, which suggests better conservation of magnetic helicity.
Note that near the end of those runs, the data points may not be
reliable because of the finite size of the domain.
In addition, because of the stronger magnetic field, the Alfv\'en time
is shorter and therefore $k_{\rm I}$ reaches $k_1$ more quickly.
In any case, it is likely that for small $|\mu_{50}|/k_0$ we see an
intermediate stage of the evolution of the system where the magnetic
helicity and chirality are temporarily conserved individually as
discussed in Sec.~\ref{IIE}.
This is supported by the fact that the theoretically predicted time
of the onset of ACC comes much later than the end of the run; see
\Eq{etaacck0>mu50}.

For all the simulations where initially $|\mu_{50}|<k_0$,
we find that $\mu_5$ decays more slowly than $k_{\rm I}$; see
\Fig{pkmax_comp_1024a_mu05_k002o} for Run~S, as an example,
where we see the crossing of $\mu_5$ and $k_{\rm I}$.
The same is also seen for $|\mu_{50}|=0.1$, but then the crossing
of $|\mu_5|$ and $k_{\rm I}$ is less prominent.
Again, these observations suggest that the magnetic
helicity-conserving phase is an intermediate one before the solution
resumes the decay governed by the adapted Hosking integral, as discussed
in Sec.~\ref{IIE}.

%FIG16
\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{rspec_select_HoskM_1024a_mu01_k002o}
\end{center}\caption[]{
(a) Magnetic energy and (b) total helicity variance spectra at $t=31$ (dashed),
$100$ (solid), $316$ (dotted), $10^3$ (blue), $3.16\times10^3$ (green),
$10^4$ (orange), and $3.16\times10^4$ (red) for Run~P.
In (a), note that the $\EM(k,t)$ evolve underneath the envelope $k^{1/3}$,
and the upward arrow indicates the sense of time.
In (b), the slopes $k^4$ and $k^{-4}$ have been indicated
and the inset compares $\Sp(2\mu_5/\lambda)$ (solid) with
$\Sp(h_{\rm tot})$ (dotted) at the last time.
}\label{rspec_select_HoskM_1024a_mu01_k002o}\end{figure}

%FIG17
\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{psaff_1024a_mu01_k002o}
\end{center}\caption[]{
(a) ${\cal I}_H(R,t)$ versus $R$ for different times $t_*$ (indicated by
the same colors/line styles, and
(b) ${\cal I}_H(R,t)$ versus $t$ (normalized) for $R=\xiM(t_*)$
marked by the four colors for Run~P.
In (a), the four colored symbols indicate the positions of
$k_0\xiM(t_*)$, and in (b), the time dependencies are plotted
for those $R=\xiM(t_*)$.
}\label{psaff_1024a_mu01_k002o}\end{figure}

The time evolution of the magnetic energy and helicity spectra for Run~P
are given in \Fig{rspec_select_HoskM_1024a_mu01_k002o}(a).
We can see that a negative magnetic helicity part of the spectrum still
emerges, again only at large wave numbers, although now much later.
This means that $|\aver{\muM^-}|$ is induced by the CPI, but it stays
extremely small.
Furthermore, $|\aver{\muM^+}|$ does not decay much during the time of the
run; see also \Fig{penerg_RunsPR}(b).
This can easily be understood by the fact that
$\eta_{\rm CPI}$ becomes very large in this run
because of the small value of $|\mu_{50}|$, which enters the CPI time
inverse quadratically; see Eq.~\eqref{etaCPIdiff}.
Many other features of the magnetic field evolution remain superficially
similar to the limit of large $|\mu_{50}|/k_0$.
One still sees inverse cascading of positive magnetic helicity.

In \Fig{rspec_select_HoskM_1024a_mu01_k002o}(b), we show 
the total helicity variance spectrum $\Sp(h_{\rm tot})$.
We clearly see that the spectral slope changes from $k^2$
to $k^4$ as time goes on.
Again, this suggests that the adapted Hosking integral, as defined in
Ref.~\cite{BKS23}, is not conserved.

Looking at the scaling of the magnetic helicity correlation function
in \Fig{psaff_1024a_mu01_k002o}, we see that there is a plateau
only for very small values of $R$, which are generally smaller
than $\xiM$ (marked by colored symbols).
This shows that the conservation of the Hosking integral is now
superseded by the conservation of the magnetic helicity.
This can be understood by the fact that the CME is inefficient at $k_{\rm
I}$ during the time of the run [see Eq.~\eqref{etaacck0>mu50}], and
hence the magnetic helicity and fermion chirality would be individually
relatively well conserved quantities.

Regarding the conserved quantity for runs in the limit of small
$|\mu_{50}|/k_0$, we can say that, in spite of vanishing total chirality,
the Hosking integral is here not conserved, because the magnetic energy
now peaks at scales where the CME is not effective during the time of
the run and the magnetic helicity is conserved.
As a result, the net chirality is no longer random, but systematically
of positive sign.
The subinertial range of the magnetic helicity variance begins to be
dominated by a $k^4$ spectrum, which suggests that the Hosking integral
in the expansion $\Sp(h_{\rm tot})=I_{\rm H} k^2/2\pi^2+O(k^4)$ is now
subdominant.

To summarize, these runs are consistent with the theoretical prediction
in Sec.~\ref{IIE}, although a moderate violation of helicity
conservation has been seen for Run~P.
Note that Run~P has a larger value of $\lambda$, which makes
the theoretically predicted $\eta_\mathrm{ACC}$ smaller [see
\Eq{etaacck0>mu50}], so that an earlier transition to the evolution
with adapted Hosking integral conservation is expected.
For an analytic estimate of the evolution of the system in the next section, 
we shall use the theoretical prediction discussed in Sec.~\ref{IIE}. 
Namely, the system is frozen until $\eta=\eta_\lambda$ and then evolves
with the usual inverse cascade for the helical magnetic field as an
intermediate stage until $\eta=\eta_\mathrm{ACC}$.
Then, it starts to evolve with a decay law determined by the conservation
of the adapted Hosking integral.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Application to the early Universe}
\label{Application}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{From QED to the Standard Model}

Now we investigate the impact of our findings in the previous sections
on the cosmology of the early Universe, especially, baryogenesis.
Up to here, we focused on a QED-like theory.
Thus, we first would like to clarify
its relation to the dynamics in the early Universe.
The Standard Model (SM) of particle physics involves the right-handed
leptons $e_{Rf}$, the left-handed lepton doublets $\ell_{Lf}$, the
right-handed up- and down-type quarks, $u_{Rf}$ and $d_{Rf}$, and the
left-handed quark doublets $q_{Lf}$ with the flavor index running through
$f=1,2,3$, alongside the scalar Higgs doublet $\Phi$, which are in total
$16$ species.
On top of this, we have gauge interactions of
$\mathrm{U}(1)_Y \times \mathrm{SU}(2)_L \times \mathrm{SU}(3)_C$.
It is not obvious why we can reduce this complicated system to chiral
MHD based on a QED-like theory like the one introduced in Sec.~\ref{ChiralMHD}.

What we are interested in here is the slow dynamics at long wave lengths
compared to interactions among particles.
The key idea for the reduction is to assume the equilibration of fast
interactions and to keep only the slow variables.
The hypermagnetic field of $\mathrm{U}(1)_Y$ with a correlation length
much larger than the mean free path of the particles stands out as a slow
variable because the magnetic flux cannot be cut thanks to the absence
of monopoles.
This feature does not hold for non-Abelian gauge fields because they are
charged under their own gauge group.
We also need the chiral chemical potential, since it is related to the
magnetic field via the anomaly equation.
Apart from these two fundamental building blocks, we can coarse-grain
the microscopic properties of all particle in the form of transport
coefficients such as the diffusion constant and the electric conductivity,
besides macroscopic quantities such as the pressure, energy density,
and velocity field. In this way, one may see that the system can be reduced to chiral MHD
as far as the slow and long-wave dynamics is concerned.

Still, one might wonder why we can just focus on one particular chiral
chemical potential, as in Eq.~\eqref{dAdt}, since we have $15$ chiral fermion species in the SM.
To illustrate this, let us focus on the temperature right above $10^5\,
\mathrm{GeV}$, where the electron Yukawa interaction is not efficient
compared to the cosmic expansion, but other interactions are fast enough.
In this case, the chiral chemical potential for the right-handed electron,
$\tilde{\mu}_{e}$, should be counted as a slow variable, as it is directly
related to the hypermagnetic field via the anomaly equation.
On the other hand, other chiral chemical potentials are subject to fast
SM interactions, which provides $11$ nontrivial constraints among them.
Recalling that the SM has four conserved charges, hypercharge $Y$ and
the flavored baryon-minus-lepton numbers $B/3 - L_f$ with $f = 1,2,3$,
one may immediately see that the remaining $15$ chemical potentials
can be expressed as a function of $\tilde{\mu}_e$ by solving $11 +
4$ constraints.
The chiral chemical potential $\tilde{\mu}_5$
originates from the generalized Ohm's law,
\begin{equation}
\bm{J}_Y = \sigma_Y \bm{E}_Y + \frac{2 \alpha_Y}{\pi}\tilde{\mu}_5\,\bm{B}_Y,
\end{equation}
where $\alpha_Y$ is now the $\mathrm{U}(1)_Y$ fine-structure constant
and $\sigma_Y$ the hyperelectric conductivity of the plasma.
In the following, we will work with the $\alpha_Y$ value around the electroweak
scale, $\alpha_Y \simeq 0.01$, and neglect its renormalization group running when
considering the dynamics of the hypermagnetic field at high energies.
Also, note that, in this section, we set $\hbar = c = k_B = 1$, and all
quantities are physical rather than comoving, unless explicitly stated otherwise.
For the SM $\mathrm{U}(1)_Y$, at $T \sim 10^{5\cdots6} \mathrm{GeV}$, one may express this $\tilde{\mu}_5$ as a summation
of chiral chemical potentials for the SM fermions as \cite{JS97,
Domcke:2022kfs}
\begin{equation}
\label{eq:mu5Y}
\tilde{\mu}_5 = \sum_{i=1}^{15} \epsilon_i\,g_i\,y_i^2\,\frac{\tilde{\mu}_i}{2} = \frac{711}{481}\,\frac{\tilde{\mu}_e}{2} \,,
\end{equation}
where $i$ runs over all SM fermions, $\epsilon_i = \pm$ for right-
and left-handed fermions, respectively, $g_i$ counts internal degrees
of freedom, and $y_i$ is the hypercharge of fermion species $i$. In
the second equality, we inserted the solution of the $15$ constraint
equations mentioned above. We now see that, up to the $\mathcal{O}(1)$
coefficient of $711/481$, one chiral chemical potential suffices to
describe the system.
In higher temperature regimes, we will have additionally more slow
variables that enter the expression of $\tilde{\mu}_5$, but it is
still written as a linear combination of their chemical potentials with
$\mathcal{O}(1)$ coefficients.
It still holds that the evolution of the system is described by chiral
MHD as discussed in Sec.~\ref{ChiralMHD} with $\tilde{\mu}_5$ being
evaluated accordingly.

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Baryogenesis}

After these general remarks, let us now turn to the implications of our
analysis for the generation of the baryon asymmetry of the Universe.
We are primarily interested in the scenario of baryogenesis from decaying
hypermagnetic helicity~\cite{Giovannini:1997eg,Giovannini:1997gp,Fujita:2016igl,Kamada:2016cnb,Kamada:2016eeb},
which assumes the presence of a strongly helical hypermagnetic field
during the radiation-dominated era in the early Universe.
This scenario is based on the observation that the helicity stored in the
hypermagnetic field decays at the time of the electroweak phase transition,
not because of some exotic helicity-violating interactions, but simply
because hypermagnetic helicity is converted to magnetic helicity.
This decay of hypermagnetic helicity then sources a baryon asymmetry
via the chiral anomaly of the baryon-number current.

One possibility to generate the helical hypermagnetic field required for baryogenesis consists of axion inflation featuring a Chern--Simons coupling to $\mathrm{U}(1)_Y$.
Such a model leads to the nonperturbative production of hypermagnetic gauge fields in combination with charge asymmetries for the 15 chiral SM fermion species~\cite{Domcke+19, Domcke:2022kfs},
%
\begin{equation}
\label{eq:ninibar}
n_i - \bar{n}_i = \frac{1}{6}\,g_i\,\tilde{\mu}_i\,T^2 = - \epsilon_i\,g_i\,y_i^2\,\frac{\alpha_Y}{2\pi}h_Y + \cdots \,,
\end{equation}
%
where the ellipsis represents all other SM contributions, which, however,
can safely be neglected during inflation.%
%
\footnote{The top-quark Yukawa interaction would be a
possible exception; see the discussion in footnote~$5$
of Ref.~\cite{Domcke:2022kfs} for more details.}
%
Furthermore, $h_Y$ in Eq.~\eqref{eq:ninibar} is the physical helicity
density, which we define in terms of the comoving vector potential
$\bm{A}_{Y,\rm com}$, comoving hypermagnetic field $\bm{B}_{Y,\rm com}$,
and scale factor $a$,
%
\begin{equation}
h_Y = \frac{1}{a^3}\, \langle \bm{A}_{Y,\rm com} \cdot \bm{B}_{Y,\rm com} \rangle \,, 
\end{equation}
%
where the angle brackets now stand for a double average including the spatial average and the quantum mechanical expectation value during inflation. From Eq.~\eqref{eq:ninibar}, we can read off the fermion chemical potentials at the end of inflation in terms of the helicity density at the end of inflation.
Inserting this result into Eq.~\eqref{eq:mu5Y}, we obtain the chiral
chemical potential at the end of inflation,
%
\begin{equation}
\label{eq:mu5ini}
\frac{\tilde{\mu}_5}{T} = -\frac{c_5}{2}\,6\,\chi \,, \qquad c_5 = \frac{95}{18} \,,
\end{equation}
%
where the dimensionless yield parameter $\chi$ quantifies the amount of $CP$ violation during axion inflation~\cite{Domcke:2022kfs},
%
\begin{equation}
\label{eq:chi}
\chi = \frac{\alpha_Y}{2\pi}\frac{h_Y}{T^3} \,.
\end{equation}
%
Here, we assume instantaneous reheating.
The same coefficient $c_5$ was found in Ref.~\cite{Domcke:2022kfs}; in total, the expression for $\tilde{\mu}_5$ in Eq.~\eqref{eq:mu5ini} is, however, smaller than the one in Ref.~\cite{Domcke:2022kfs} by a factor of $\sfrac{1}{2}$ because, in the present paper, we include a factor of $\sfrac{1}{2}$ in Eq.~\eqref{eq:mu5Y}.

The fermion asymmetries generated during axion inflation are consistent with the chiral anomalies of the respective fermion currents. In fact, it is straightforward to generalize the conversion law in Eq.~\eqref{eq:mutot} to the early Universe. To see this, let us rewrite Eq.~\eqref{eq:mu5ini} as follows,
%
\begin{equation}
\mu_5 + \frac{3\,c_5}{2} \left(\frac{2\alpha_Y}{\pi}\right)^2\frac{1}{2a^3T^2}\, \langle \bm{A}_{Y,\rm com} \cdot \bm{B}_{Y,\rm com} \rangle = 0 \,,
\end{equation}
%
where we used $\mu_5 = \left(2\alpha_Y/\pi\right)\tilde{\mu}_5$. Then, introducing
%
\begin{equation}
\lambda_Y = 3\left(\frac{2\alpha_Y}{\pi aT}\right)^2 \,,
\end{equation}
%
we obtain the relation
%
\begin{equation} \label{conseqSM}
\mu_5 + \frac{c_5}{2}\,\mu_{\rm M}^Y = 0 \,,
\end{equation}
%
where 
%
\begin{equation}
\mu_{\rm M}^Y = \frac{1}{2a}\lambda_Y \langle \bm{A}_{Y,\rm com} \cdot \bm{B}_{Y,\rm com} \rangle \,.
\end{equation}

As the temperature in the early Universe decreases, more and more SM interactions reach chemical equilibrium. This includes the SM Yukawa interactions, which violate parity and hence render the coefficient $c_5$ in Eq.~\eqref{eq:mu5ini} a time-dependent quantity~\cite{Domcke:2022kfs}. During axion inflation, $c_5$ assumes its maximal value, $c_5 = 95/18 \simeq 5.3$, before it then decreases down to $c_5 = 711/481 \simeq 1.5$ at temperatures of a few 100 TeV [see Eq.~\eqref{eq:mu5Y}]. This change in $c_5$ is reflected in a changing value of the chiral chemical potential $\mu_5$, which is always given by $\mu_5 = -c_5/2\,\mu_{\rm M}^Y$ according to Eq.~\eqref{conseqSM}, with $\mu_{\rm M}^Y$ remaining constant until the onset of ACC or electroweak phase transition. At $T \lesssim 10^5\,\textrm{GeV}$, $c_5$ and hence $\mu_5$ vanish because all SM interactions have reached chemical equilibrium.

The $CP$ asymmetry parameter $\chi$ in Eq.~\eqref{eq:chi} controls the
outcome of baryogenesis from helicity decay.
That is, if no CPI or ACC takes place before the onset of spin flipping,
the decay of hypermagnetic helicity around the electroweak phase transition
results in a present-day baryon asymmetry (quantified in terms of the baryon-to-photon ratio) that is fully controlled by
$\chi$ \cite{Domcke:2022kfs},
%
\begin{equation}
\label{eq:etaold}
\eta_B^0 \equiv \frac{n_B^0}{n_\gamma^0} \simeq 0.15\,c_B^{\rm dec} \chi \,,
\end{equation}
%
where $n_\gamma = 2\,\zeta(3)T^3/\pi^2$ and the superscript $0$ indicates that 
a quantity is evaluated at the present time.
%
Here, the coefficient $c_B^{\rm dec}$ has a theoretical uncertainty of possibly two orders of magnitude~\cite{Kamada:2016cnb}.
In the following, we will work with the representative value
$c_B^{\rm dec} = 0.05$~\cite{Kamada:2020bmb,Domcke:2022kfs}, which implies that $\chi$ values of the order
of $\chi \sim 10^{-7}$ are necessary to reproduce the observed
baryon asymmetry, $\eta_B^{\rm obs} \simeq 6.1 \times 10^{-10}$~\cite{Planck:2018vyg,ParticleDataGroup:2022pth}.
Meanwhile, the parameter $\chi$ also allows us to evaluate the ratio of $k_0$ and $\mu_5$ at the end of axion inflation.
Specifically, if we estimate the comoving peak wave number $k_0$ in terms of the comoving wave number
that enters the Hubble horizon at the end of reheating,
$k_{\rm rh} = a_{\rm rh} H_{\rm rh}$~\cite{Domcke:2022kfs}, we find
%
\begin{align}
\label{eq:mu5krh}
\frac{\left|\mu_5\right|}{k_{\rm rh}/a_{\rm rh}} & = \frac{6\,\alpha_Y c_5\,\chi}{\pi}\frac{T_{\rm rh}}{H_{\rm rh}} = \frac{6\,\alpha_Y c_5\,\chi}{\pi}\frac{M_*}{T_{\rm rh}} \notag \\
& \sim 10^{-4}\:\bigg(\frac{\chi}{10^{-7}}\bigg) \left(\frac{10^{14} \mathrm{GeV}}{T_{\rm rh}}\right) \,,
\end{align}
%
where $M_* = \left(90/\pi^2/g_*\right)^{1/2}M_{\rm Pl} \simeq 7.1 \times 10^{17}\,\textrm{GeV}$
is the reduced Planck mass, $M_{\rm Pl} \simeq 2.4 \times 10^{18}\,\textrm{GeV}$,
rescaled by the effective number of relativistic degrees of freedom in
the Standard Model plasma, $g_* = 427/4$.
Axion inflation typically results in small values of the $\chi$
parameter (e.g., $\chi \sim 10^{-7}$; see above) and large values of the
reheating temperature (e.g., $T_{\rm rh} \sim 10^{14}\,\textrm{GeV}$;
see Ref.~\cite{Domcke:2022kfs}), which puts us in the parametric regime
where $\left|\mu_5\right| \ll k_{\rm rh}/a_{\rm rh}$ at the end of
axion inflation.
Moreover, smaller values of $T_{\rm rh}$ typically result in
smaller values of $\chi$, following the scaling relation $\chi \propto (T_{\rm rh}/M_*)^3$~\cite{Domcke:2022kfs}, which means that the opposite hierarchy,
$\left|\mu_5\right| \gg k_{\rm rh}/a_{\rm rh}$, cannot simply be obtained
by considering a smaller reheating temperature.

For $\left|\mu_5\right| \ll k_{\rm rh}/a_{\rm rh}$, we can estimate the time when the ACC sets in after axion inflation based on Eq.~\eqref{etaacck0>mu50},
%
\begin{equation}
\eta_{\rm ACC} \sim \left[\frac{1}{a^3}\frac{2\,\sigma_Y^3}{\bar{\rho}\,\lambda_Y \mu_5^2}\right]_{\rm rh} \,,
\end{equation}
%
where the factor of $a^{-3}$ follows from the mass dimension of the factor $\sigma_Y^3/(\bar{\rho}\mu_5^2)$.
%
Next, we write the hyperelectric conductivity $\sigma_Y$ as $\sigma_Y = c_{\sigma_Y} T$, with $c_{\sigma_Y} \sim 100$~\cite{Baym:1997gq,Arnold:2000dr}, the average radiation energy density $\bar{\rho}$ as $\bar{\rho} = c_{\bar{\rho}}\,T^4$, with $c_{\bar{\rho}} = \pi^2 g_*/30$, and the parameter $\lambda_Y$ as $\lambda_Y = c_{\lambda_Y}/(aT)^2$ with $c_{\lambda_Y} = 12\,\alpha_Y^2/\pi^2$.
%
With these definitions, we find
%
\begin{equation}
\label{eq:etaACC}
\eta_{\rm ACC} \sim \frac{c_{\rm ACC}}{\chi^2}\frac{T_{\rm rh}}{M_*}\frac{1}{k_{\rm rh}} \,
\end{equation}
%
with the coefficient $c_{\rm ACC}$ being given as follows,
%
\begin{equation}
c_{\rm ACC} = \frac{2\,c_{\sigma_Y}^3}{c_{\bar{\rho}}\,c_{\lambda_Y}}\left(\frac{\pi}{6\,\alpha_Yc_5}\right)^2 \sim 10^{11} \,.
\end{equation}

The time $\eta_{\rm ACC}$ marks the onset of the anomalous chirality cancellation and needs to be compared to the time $\eta_{\rm sf} = 1/k_{\rm sf}$ when spin flipping for left- and right-handed electrons becomes efficient, where $k_{\rm sf} = a_{\rm sf} H_{\rm sf}$ is the comoving horizon scale at $\eta = \eta_{\rm sf}$. Using Eq.~\eqref{eq:etaACC}, together with $c_{\rm ACC} = 10^{11}$ and $T_{\rm sf} = 10^5\,\textrm{GeV}$ for the electron Yukawa interaction in the SM~\cite{Campbell:1992jd,Bodeker:2019ajh}, we obtain
%
\begin{equation}
\label{eq:etaACCsf}
\frac{\eta_{\rm ACC}}{\eta_{\rm sf}} \sim \frac{c_{\rm ACC}}{\chi^2}\frac{T_{\rm sf}}{M_*} \sim \frac{0.01}{\chi^2} \,.
\end{equation}
Therefore, in order to have ACC before the onset of spin flipping,
we require very large values of the $CP$ asymmetry parameter,
$\chi \gtrsim 0.1$.
However, for such large $\chi$ values, the naive estimate of the baryon
asymmetry according to Eq.~\eqref{eq:etaold} is hopelessly too large.
Furthermore, the reduction of the magnetic helicity because of ACC in
the time window between $\eta_{\rm ACC}$ and $\eta_{\rm sf}$ will not
be enough to counteract this overproduction of baryon number.
To see this explicitly, one may use the ratio in Eq.~\eqref{eq:etaACCsf}
to introduce a dilution factor, $\Delta = (\eta_{\rm ACC}/\eta_{\rm sf})^{2/3}$,
that multiplies the naive baryon asymmetry in Eq.~\eqref{eq:etaold}
whenever ACC should occur before the onset of spin flipping,
\begin{equation}
\label{eq:etanew}
\eta_B^0 \simeq \textrm{min}\left\{1,\Delta\right\} \times 0.15\,c_B^{\rm dec} \chi \,.
\end{equation}
This formula suggests that at extremely large $\chi$ values, $\chi \ggg 1$,
it appears to be possible to reach $\eta_B^0 \simeq \eta_B^{\rm obs}$.
In realistic models of axion inflation, it should, however, be extremely
difficult, if not impossible, to realize such large $\chi$ values.
Moreover, no such solution would be self-consistent, as we would always
violate our initial assumption that $\left|\mu_5\right| \ll k_{\rm rh}/a_{\rm rh}$;
see the relation in Eq.~\eqref{eq:mu5krh}.

Finally, for completeness, we mention that any scenario that does
allow for the hierarchy $\left|\mu_5\right| \gg k_{\rm rh}/a_{\rm rh}$
in one way or another would be interesting as it might display
similar dynamics as our Runs~III--VI, which we commented on at the end of
Sec.~\ref{subsec:evolution}.
The decay law for the magnetic helicity may then be different from
the $\eta^{-2/3}$ behavior that we typically find for ACC, which may
be relevant for the outcome of baryogenesis.
As already stated in Section~\ref{subsec:evolution}, we leave a more
detailed study of this more exceptional case for future work.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Conclusions}
\label{Conclusions}

We have performed numerical simulations of chiral MHD 
with zero initial total chirality for a range of parameters to
determine the dependence of characteristic time and scale ratios,
which are well explained by the analytical estimate in Sec.~\ref{IIE}. 
Namely, they are consistent with the scaling evolution, $\xi_\mathrm{M} \propto \eta^{4/9}, \,
{\cal E}_\mathrm{M} \propto \eta^{-10/9}$, and $\aver{\mu_5} \propto \eta^{-2/3}$, 
derived from the conservation of the adapted Hosking integral~\cite{BKS23}, 
and also the time scale of the onset of this scaling evolution, $\eta_\mathrm{ACC}$; see Eqs.~\eqref{etaACCcase1} and \eqref{etaacck0>mu50}.
Our numerical simulations also assess the possibility of artifacts resulting from insufficient
scale separation.
A particularly important constraint is a sufficiently large size of the
computational domain (small $k_1$), which is needed to obtain the expected
$\eta^{4/9}$ scaling of the correlation length.
When this constraint is not obeyed, the scaling is closer to $\eta^{1/3}$.
The second constraint of a sufficiently large Nyquist wave number is
important to obtain the correct values of the scale ratio of the positive
and negative magnetic helicity peaks, i.e., $k_{\rm II}/k_{\rm I}$.
Somewhat surprisingly, this ratio scales inversely with the initial scale
separation between the scale of the magnetic field and the CPI scale.
Increased values of $\aver{\mu_{\rm M}^-}$, which characterize the
strength of the CPI, are obtained when $\sigma$ is small or $|\mu_{50}|$
is large and therefore the coupling to the CPI is more efficient.

In the absence of spin flipping, even the slightest initial imbalance
will amplify as the magnetic energy decays; see \App{Imbalanced}.
On long time scales, this eventually leads to a fully helical state,
although simulations of this are at present unable to demonstrate this
conclusively owing to the finite size of the computational domain.
Spin flipping is another mechanism that can produce an imbalance between
magnetic helicity and fermion chirality.
In any case, however, the finally available magnetic energy and helicity
densities are always limited by the finiteness of the initial total
chirality imbalance.
For $\eta < \eta_\mathrm{ACC}$, when the chiral magnetic effect is not effective 
at the peak scale, magnetic helicity conservation governs
the decay of magnetic energy and the Hosking integral does not play a role.

We also discussed the implications of our findings for the 
generation of the baryon asymmetry of the Universe, in particular,
the scenario of baryogenesis from helicity decay.
The final baryon asymmetry in this scenario is controlled by
a dimensionless yield parameter $\chi$ that quantifies the helicity
density produced in the very early Universe, for instance, during a
stage of axion inflation.
In previous work, it was shown how the observed baryon asymmetry
can be generated from helicity decay at the time of the electroweak
phase transition for a specific $\chi$ value, $\chi_0 \sim 10^{-7}$; see Eq.~\eqref{eq:etaold}
and Ref.~\cite{Domcke:2022kfs}.
The situation at larger $\chi$ values, however, remained unclear.
At $\chi \gg 1$, one may have anticipated either (A) the overproduction
of baryon number or (B) catastrophic helicity erasure by the chiral
plasma instability and consequently no baryon asymmetry at all.

Thanks to the analysis in this paper, we now understand that, instead
of helicity erasure owing to the CPI, we should rather be concerned
about the possible effect of ACC on the primordial hypermagnetic helicity.
However, for realistic $\chi$ values, $10^{-7} \lesssim \chi \ll 0.1$, even ACC
will not become efficient before the onset of spin flipping
triggered by the electron Yukawa interaction. We therefore conclude that, at large $\chi$ values, neither the CPI
nor ACC occurs. Instead, we encounter the problem of baryon-number overproduction that also naively follows from
Eq.~\eqref{eq:etaold} in the large-$\chi$ limit. This leaves us with one viable solution for
baryogenesis from the decay of hypermagnetic helicity, $\chi_0 \sim 10^{-7}$, which had already
been identified in Ref.~\cite{Domcke:2022kfs}. Our analysis in the present paper confirms the validity 
of this solution and demonstrates that it is indeed unique.

\vspace{2mm}
{\bf Data availability}---The source code used for the
simulations of this study, the {\sc Pencil Code},
is freely available from Ref.~\cite{JOSS}.
The simulation setups and the corresponding data
are freely available from Ref.~\cite{SMZeroTotalChirality}.

\medskip
\begin{acknowledgments}
We thank V.~Domcke for fruitful discussions and J.~Warnecke for his work
on the implementation of SLD, which is used in some of the simulations.
Support through the
grant 2019-04234 from the Swedish Research Council (Vetenskapsr{\aa}det) (AB),
Grant-in-Aid for Scientific Research No.\ (C) JP19K03842 from the JSPS KAKENHI (KK),
MEXT Leading Initiative for Excellent Young Researchers  No.\ JPMXS0320200430 (KM),
Grant-in-Aid for Young Scientists No.\ JP22K14044
from the JSPS KAKENHI (KM),
and the grant 185863 from the Swiss National Science Foundation (JS)
are gratefully acknowledged.
We acknowledge the allocation of computing resources provided by the
Swedish National Infrastructure for Computing (SNIC)
at the PDC Center for High Performance Computing Stockholm and Link\"oping.
\end{acknowledgments}

\appendix

\section{Behavior in imbalanced chirality decay}
\label{Imbalanced}

In \Sec{Conclusions}, we emphasized that even the slightest initial
imbalance between magnetic helicity and fermion chirality will amplify
as the magnetic energy decays.
It is therefore important to remember that the dynamics discussed in
this paper is specific to the case of balanced chirality, which is
arguably also the most generic case.
We know that the decay of magnetic energy and the increase of
the correlation length follow a different behavior in the completely
imbalanced case compared to the unbalanced one.
We now discuss the behavior for the mildly imbalanced case.
Here, we show that there is a tendency for the system to approach the
behavior of a completely imbalanced one.

%FIG18
\begin{figure}[t]\begin{center}
\includegraphics[width=\columnwidth]{rspec_select_1024a_mu10_k002o_less}
\end{center}\caption[]{
Magnetic energy (solid lines) and normalized helicity spectra
$k\HM(k)/2$ (dotted lines with red and blue symbols for positive
and negative helicity spectra, respectively) for Run~A at
times $\eta=32$, 100, $320$, 1000, 3200, 15,000, 22,000, and 32,000.
}\label{rspec_select_1024a_mu10_k002o_less}\end{figure}

We discuss two runs, Run~A where the initial $\aver{\muM}$ is enhanced
by 20\% compared with $|\aver{\mu_5}|$, and Run~B where it is decreased
by 20\%.
Apart from that, the runs are the same as Run~O, i.e., the run discussed
in Ref.~\cite{BKS23}.

%FIG19
\begin{figure}[t]\begin{center}
\includegraphics[width=\columnwidth]{rspec_select_1024a_mu10_k002o_more}
\end{center}\caption[]{
Same as \Fig{rspec_select_1024a_mu10_k002o}, but for Run~B
at times $\eta=32$, 100, $320$, 1000, 3200, 10,000, and 32,000.
}\label{rspec_select_1024a_mu10_k002o_more}\end{figure}

%FIG20
\begin{figure}[t]\begin{center}
\includegraphics[width=\columnwidth]{penerg_comp}
\end{center}\caption[]{
Time dependence of $\EEM$ (black), $\xiM^{-1}$ (orange), $\HHM$ (red),
and $-2\aver{\mu_5}/\lambda$ (blue), for (a) Run~A with smaller and (b) Run~B
with larger magnetic helicity than in the balanced case.
Dashed lines indicate negative values; at late times
$-2\aver{\mu_5}/\lambda$ changes sign in (a), and
$\HHM$ changes sign in (b).
In (a) the dotted line denotes the $\eta^{-1/2}\log(\eta/\eta_{\log})$ scaling
of Ref.~\cite{Schober+2020} with $\eta_{\log}=3$.
}\label{penerg_comp}\end{figure}

In Run~A, where the magnetic helicity is weaker than in Run~O, the CPI
becomes dominant and overcompensates the magnetic helicity.
The net chirality is then negative.
Eventually, the sign of the magnetic helicity changes and all the
remaining fermion chirality is converted to magnetic fields with negative
helicity; see \Fig{rspec_select_1024a_mu10_k002o_less}, where we show
the magnetic energy $\EEM(k,t)$ and the normalized magnetic helicity
spectra $k\HM(k)/2$ for Run~A at times $\eta=32$, $320$, 1000, 3200,
10,000, and 32,000.
We see that $k|\HM(k)|/2$ approaches $\EEM(k)$ near the maximum.
In view of the spectral realizability condition, \Eq{realizability},
this means that the magnetic field is fully helical.
Away from the maxima, the inequality is no longer saturated, but this
is a typical effect in all turbulent flows where the current helicity
spectrum shows a Kolmogorov-type spectrum, making the magnetic helicity
spectrum therefore steeper than what could still be allowed by the
spectral realizability condition \cite{BS05b}.

On the other hand, when the fermion chirality is weak (Run~B),
the usual inverse magnetic cascade quickly gets established;
see \Fig{rspec_select_1024a_mu10_k002o_more}.
In either case, the fermion chirality gets ultimately
converted into magnetic helicity.
It just takes a little longer than when the magnetic helicity is
initially weak.
At the end, however, the usual inverse cascade for a fully helical
magnetic field commences.
The sign of magnetic helicity can be positive or negative, depending
on the sign of the initial total chirality.

To illustrate how the decay laws change when magnetic helicity and
fermion chirality no longer balance, we plot in \Fig{penerg_comp} the
time dependencies of $\EEM$, $\xiM$, $\HHM$, and $-2\aver{\mu_5}/\lambda$,
for (a) Run~A with 20\% smaller and (b) Run~B with 20\% larger magnetic
helicity than in the balanced case.
In both cases, we see a tendency of the decays of $\EEM$ and $\xiM^{-1}$
to slow down while those of $\HHM$ and $-2\aver{\mu_5}/\lambda$ follow
separate evolutions.
Especially in the case of Run~B, where the magnetic helicity
dominates of the fermion chirality, we see a tendency
toward a $\EEM\propto\xi^{-1}\propto t^{-2/3}$ as well as
$\mathcal{H}_\mathrm{M}=\const$ evolution, as expected from
magnetic helicity conservation.

\bibliography{ref}{}
%\bibliographystyle{apsrev4-1}
\end{document}
