# Decay law of magnetic turbulence with helicity balanced by chiral fermions

Axel Brandenburg  Nordita, KTH Royal Institute of Technology and Stockholm
University, 10691 Stockholm, Sweden  The Oskar Klein Centre, Department of
Astronomy, Stockholm University, AlbaNova, 10691 Stockholm, Sweden  School of
Natural Sciences and Medicine, Ilia State University, 0194 Tbilisi, Georgia
McWilliams Center for Cosmology and Department of Physics, Carnegie Mellon
University, Pittsburgh, Pennsylvania 15213, USA  Kohei Kamada  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
Jennifer Schober  Institute of Physics, Laboratory of Astrophysics, École
Polytechnique Fédérale de Lausanne (EPFL), 1290 Sauverny, Switzerland

(Received 1 February 2023; accepted 4 April 2023; published 11 May 2023)

###### Abstract

In plasmas composed of massless electrically charged fermions, chirality can
be interchanged with magnetic helicity while preserving the total chirality
through the quantum chiral anomaly. The decay of turbulent energy in plasmas
such as those in the early Universe and compact stars is usually controlled by
certain conservation laws. In the case of zero total chirality, when the
magnetic helicity density balances with the appropriately scaled chiral
chemical potential to zero, the total chirality no longer determines the
decay. We propose that in such a case, an adaptation to the Hosking integral,
which is conserved in nonhelical magnetically dominated turbulence, controls
the decay in turbulence with helicity balanced by chiral fermions. We show,
using a high resolution numerical simulation, that this is indeed the case.
The magnetic energy density decays and the correlation length increases with
time just like in nonhelical turbulence with vanishing chiral chemical
potential. But here, the magnetic helicity density is nearly maximum and shows
a scaling with time $t$ proportional to $t^{-2/3}$. This is unrelated to the
$t^{-2/3}$ decay of magnetic energy in fully helical magnetic turbulence. The
modulus of the chiral chemical potential decays in the same fashion. This is
much slower than the exponential decay previously expected in theories of
asymmetric baryon production from the hypermagnetic helicity decay after axion
inflation.

††preprint: NORDITA-2023-001, RESCEU-1/23

Magnetic helicity characterizes the knottedness of magnetic field lines and
plays important roles in cosmological, astrophysical, and laboratory plasmas.
Since the early work of Woltjer of 1958 Woltjer (1958), we know that the
magnetic helicity is an invariant of the ideal magnetohydrodynamic (MHD)
equations. Even in the non-ideal case of finite conductivity, it is
asymptotically conserved in the limit of large magnetic Reynolds numbers
(Taylor, 1974). This is because, unlike the magnetic energy dissipation, which
is finite at large magnetic Reynolds numbers, the magnetic helicity
dissipation converges to zero in that limit (Brandenburg and Subramanian,
2005). The magnetic helicity controls the decay of magnetic fields in closed
or periodic domains, provided the magnetic helicity is finite. However, even
when the net magnetic helicity over the whole volume vanishes, there can still
be random fluctuations of magnetic helicity. In this case, the conservation of
magnetic helicity still plays an important role, but only in smaller
subvolumes, as was shown recently (Hosking and Schekochihin, 2021). The
conserved quantity in that case is what is now known as the Hosking integral
(Schekochihin, 2022; Zhou  _et al._ , 2022), which characterizes magnetic
helicity fluctuations in smaller subvolumes (Hosking and Schekochihin, 2021).

At relativistic energies, the chirality of fermions combines with the helicity
of the magnetic field to a total chirality that is strictly conserved in a
periodic or closed domain – even for finite magnetic diffusivity (Boyarsky
_et al._ , 2012; Rogachevskii  _et al._ , 2017) which is a consequence of the
chiral anomaly (Adler, 1969; Bell and Jackiw, 1969). This can have a number of
consequences. There is an instability that can amplify a helical magnetic
field Joyce and Shaposhnikov (1997). It is now often referred to as the chiral
plasma instability (CPI) Akamatsu and Yamamoto (2013) and it causes the chiral
chemical potential carrying the chirality of the fermions to decay such that
the total chirality remains unchanged (Kamada, 2018; Domcke  _et al._ , 2022;
Co  _et al._ , 2022). Conversely, if a helical magnetic field decays, the
chiral chemical potential can increase (Hirono  _et al._ , 2015; Schober  _et
al._ , 2020a). Finally, when the chiral chemical potential balances the
magnetic helicity to produce vanishing total chirality of the system, which is
realized in, e.g., cosmological MHD after axion inflation (Domcke and Mukaida,
2018; Domcke  _et al._ , 2019, 2023), the magnetic field can only decay. It
has been thought that the decay is triggered by the CPI and that it would be
therefore exponential (Domcke and Mukaida, 2018; Domcke  _et al._ , 2019). In
this Letter, however, we show that this decay occurs only in a power-law
fashion. This has consequences for explaining the baryon asymmetry of the
Universe Giovannini and Shaposhnikov (1998a, b); Kamada and Long (2016) and
for theories of primordial magnetic fields, which will open up a new direction
for early Universe cosmology model building. The purpose here is to show that
the decay of the magnetic field in chiral MHD is governed—similarly to
nonhelical MHD—by a conserved quantity that we call the adapted Hosking
integral. While the model adopted here is based on quantum electrodynamics,
the extension to the realistic cosmological models based on the standard model
of particle physics is straightforward; see, e.g., Refs. Kamada  _et al._
(2023); Domcke  _et al._ (2022).

The Hosking integral $I_{\rm H}$ is defined as the asymptotic limit of the
relevant magnetic helicity density correlation integral, ${\cal I}_{\rm
H}(R)$, for scales $R$ which are large compared to the correlation length of
the turbulence, $\xi_{\rm M}$, but small compared to the system size $L$. The
function ${\cal I}_{\rm H}(R)$ is given by

${\cal I}_{\rm H}(R)=\int_{V_{R}}\langle
h(\bm{x})h(\bm{x}+\bm{r})\rangle\,{\rm d}{}^{3}r,$ (1)

where $V_{R}$ is the volume of a ball of radius $R$ and, in MHD,
$h=\bm{A}\cdot\bm{B}$ is the magnetic helicity density with $\bm{A}$ being the
magnetic vector potential, so $\bm{B}={\bm{\nabla}}\times\bm{A}$. Here, angle
brackets denote averages over the volume $L^{3}$.

For relativistic chiral plasmas, on the other hand, we now amend the magnetic
helicity density with a contribution from the chiral chemical potential
$\mu_{5}$. We work here with the scaled chiral chemical potential
$\mu_{5}\to\mu_{5}^{\prime}=(4\alpha/\hbar c)\,\mu_{5}$, where $\alpha$ is the
fine structure constant, $\hbar$ is the reduced Planck constant, and $c$ is
the speed of light. Our rescaled $\mu_{5}^{\prime}$ has the dimension of a
wave number. From now on, we drop the prime and only work with the rescaled
chiral chemical potential. We also define the quantity $\lambda=3\hbar
c\,(8\alpha/k_{\rm B}T)^{2}$, where $k_{\rm B}$ is the Boltzmann constant and
$T$ is the temperature. We define the total helicity density $h_{\rm
tot}\equiv\bm{A}\cdot\bm{B}+2\mu_{5}/\lambda$ and replace $h\to h_{\rm tot}$
when defining the adapted Hosking integral.

Similarly to earlier studies of non-relativistic chiral plasmas ($\mu_{5}\to
0$) with a helical magnetic field, the case of a finite net chirality,
$\langle h_{\rm tot}\rangle\neq 0$, is governed by the conservation law for
$\langle h_{\rm tot}\rangle$. Of course, when $\langle h_{\rm tot}\rangle=0$,
it is still conserved, but it can then no longer determine the dynamics of the
system. This is when we expect, instead, $I_{\rm H}$ to control the dynamics
of the decay. As before, we define $I_{\rm H}={\cal I}_{\rm H}(R_{*})$ for
values of $R_{*}$ for which ${\cal I}_{\rm H}(R)$ shows a plateau. In the
following, we focus on this case using numerical simulations to compute the
decay properties of a turbulent magnetic field and the conservation properties
of $I_{\rm H}$ using the total helicity in a relativistic plasma.

Now setting $c=1$, the evolution equations for $\bm{A}$ and $\mu_{5}$ are
(Rogachevskii  _et al._ , 2017)

$\displaystyle\frac{\partial\bm{A}}{\partial t}$ $\displaystyle=$
$\displaystyle\eta(\mu_{5}\bm{B}-\bm{J})+\bm{u}\times\bm{B},\quad\bm{J}={\bm{\nabla}}\times\bm{B},$
(2) $\displaystyle\frac{\partial\mu_{5}}{\partial t}$ $\displaystyle=$
$\displaystyle-\frac{2}{\lambda}\eta(\mu_{5}\bm{B}-\bm{J})\cdot\bm{B}-{\bm{\nabla}}\cdot(\mu_{5}\bm{u})+D_{5}\nabla^{2}\mu_{5},\;\;$
(3)

where $\eta$ is the magnetic diffusivity, $D_{5}$ is the diffusion coefficient
of $\mu_{5}$, spin flipping is neglected here, and $\bm{u}$ is the velocity,
which is governed by the compressible Navier-Stokes equations (Rogachevskii
_et al._ , 2017; Brandenburg  _et al._ , 2017a, 2021)

$\displaystyle\frac{{\rm D}{}\bm{u}}{{\rm D}{}t}$ $\displaystyle=$
$\displaystyle\frac{2}{\rho}{\bm{\nabla}}\cdot\left(\rho\nu\mbox{\boldmath${\sf
S}$}{}\right)-\frac{1}{4}{\bm{\nabla}}\ln\rho+\frac{\bm{u}}{3}\left({\bm{\nabla}}\cdot\bm{u}+\bm{u}\cdot{\bm{\nabla}}\ln\rho\right)$
(4) $\displaystyle-$
$\displaystyle\frac{\bm{u}}{\rho}\left[\bm{u}\cdot(\bm{J}\times\bm{B})+\eta\bm{J}^{2}\right]+\frac{3}{4\rho}\bm{J}\times\bm{B},$
$\displaystyle\frac{\partial\ln\rho}{\partial t}$ $\displaystyle=$
$\displaystyle-\frac{4}{3}\left({\bm{\nabla}}\cdot\bm{u}+\bm{u}\cdot{\bm{\nabla}}\ln\rho\right)+\frac{1}{\rho}\left[\bm{u}\cdot(\bm{J}\times\bm{B})+\eta\bm{J}^{2}\right],$

where ${\sf
S}_{ij}=(\partial_{i}u_{j}+\partial_{j}u_{i})/2-\delta_{ij}{\bm{\nabla}}\cdot\bm{u}/3$
are the components of the rate-of-strain tensor, $\nu$ is the kinematic
viscosity, $\rho$ is the density (which includes the rest mass density), and
the ultrarelativistic equation of state for the pressure $p=\rho/3$ has been
employed. We assume uniform $\nu$, $\eta$, and $D_{5}$ such that
$\nu=\eta=D_{5}$. Our use of Eqs. (4) and (LABEL:dlnrhodt) compared to the
nonrelativistic counterpart only affects the kinetic energy and not the
magnetic field evolution; see Ref. Brandenburg  _et al._ (2017b) for
comparisons in another context.

We define spectra of a quantity $h(\bm{x})$ as $\mbox{\rm
Sp}(h)=\oint_{4\pi}|\tilde{h}|^{2}\,k^{2}{\rm d}{}\Omega_{k}/(2\pi L)^{3}$,
where a tilde denotes the quantity in Fourier space and $\Omega_{k}$ is the
solid angle in Fourier space, so that $\int\mbox{\rm Sp}(h)\,{\rm
d}{}k=\langle h^{2}\rangle$. Here, $k\equiv|\bm{k}|$. The magnetic energy
spectrum is $E_{\rm M}(k,t)\equiv\mbox{\rm Sp}(\bm{B})/2$ and $\int E_{\rm
M}\,{\rm d}{}k=\langle\bm{B}^{2}\rangle/2$ is the mean magnetic energy
density. The mean magnetic helicity density is ${\cal H}_{\rm
M}=\langle\bm{A}\cdot\bm{B}\rangle$, the magnetic helicity spectrum is $H_{\rm
M}(k,t)$ with $\int H_{\rm M}\,{\rm d}{}k={\cal H}_{\rm M}$, and $\xi_{\rm
M}={\cal E}_{\rm M}^{-1}\int k^{-1}E_{\rm M}\,{\rm d}{}k$ is the correlation
length.

For an initially uniform $\mu_{5}\equiv\mu_{50}$, Eq. (2) has exponentially
growing solutions proportional to $e^{{\rm i}\bm{k}\cdot\bm{x}+\gamma_{5}t}$,
when $k<\mu_{50}$. The maximum growth rate is $\gamma_{5}=\mu_{50}^{2}\eta/4$
for $k=k_{5}\equiv\mu_{50}/2$ (Rogachevskii  _et al._ , 2017; Brandenburg  _et
al._ , 2017a). As an initial condition for $\bm{A}$, we consider a Gaussian
distributed random field with a magnetic energy spectrum that is a broken
power law with $E_{\rm M}(k,t)\propto k^{4}$ for $k<k_{0}$, motivated by
causality constraints Durrer and Caprini (2003), and a Kolmogorov-type
spectrum, $E_{\rm M}(k,t)\propto k^{-5/3}$, for $k>k_{0}$, which may be
expected if there is a turbulent forward cascade. By setting $k_{0}=1$ for the
spectral peak, we fix the units of velocity and length. The unit of time is
then $(k_{0})^{-1}$. We initially set $\rho=\rho_{0}=1$, which then also fixes
the units of energy.

We solve the governing equations using the Pencil Code (Pencil Code
Collaboration  _et al._ , 2021), where the equations are already implemented
(Schober  _et al._ , 2018, 2020b). We consider a cubic domain of size $L^{3}$,
so the smallest wave number is $k_{1}=2\pi/L$. The largest wave number is
$k_{\rm Ny}=k_{1}N/2$, where $N$ is the number of mesh points in one
direction. In choosing our parameters, it is important to observe that
$k_{1}\ll k_{0}\ll k_{5}\ll k_{\rm Ny}$. Here, we choose $k_{1}=0.02$,
$k_{0}=1$, $k_{5}=5$, and $k_{\rm Ny}=10.24$, using $N=1024$ mesh points in
each of the three directions. This means that $|\mu_{50}|=10$, which is
virtually the same as $k_{\rm Ny}$. However, experiments with other choices,
keeping $N=1024$, showed that ours yields an acceptable compromise that still
allows us to keep $k_{1}$ small enough. We choose the sign of $\mu_{5}$ to be
negative, and adjust the amplitude of the magnetic field such that $2{\cal
E}_{\rm M}\xi_{\rm M}={\cal H}_{\rm M}=-2\mu_{50}/\lambda$. Using
$\eta=2\times 10^{-4}$ and $\lambda=2\times 10^{4}$, we have, following Ref.
(Brandenburg  _et al._ , 2017b),
$v_{\lambda}\equiv\mu/\sqrt{\rho_{0}\lambda}\approx 0.07$ and
$v_{\mu}\equiv\mu\eta=0.002$, so $v_{\lambda}/v_{\mu}\approx 35\gg 1$,
corresponding to what is called regime I.

In Fig. 1(a), we present magnetic energy spectra at different times. We
clearly see an inverse cascade where the spectral magnetic energy increases
with time for $k\ll k_{0}$ (indicated by the upward arrow), but decays for
$k\gg k_{0}$. As time goes on, the peak of the spectrum moves to smaller wave
numbers with $k_{\rm peak}\approx\xi_{\rm M}^{-1}$, where $\xi_{\rm M}$
increases approximately like a power law, $\xi_{\rm M}\propto t^{q}$, while
the energy density decreases, also approximately like a power law with ${\cal
E}_{\rm M}\propto t^{-p}$. The spectral peak always evolves underneath an
envelope $\propto k^{3/2}$, which implies that $\max[E_{\rm M}(k,t)]=\xi_{\rm
M}(t)^{-\beta}$ with $\beta=3/2$, indicated by the upper dashed-dotted line in
Fig. 1(a).

Figure 1:  (a) Magnetic energy and (b) total helicity variance spectra at
$t=31$ (dashed), $100$ (solid), $316$ (dotted), $10^{3}$ (blue), $3.16\times
10^{3}$ (green), $10^{4}$ (orange), and $3.16\times 10^{4}$ (red). In (a),
note that $\mbox{\rm Sp}(\bm{B})$ evolves underneath the envelope $k^{3/2}$,
and the upward arrow indicates the sense of time. For orientation, the slopes
$k^{-5/2}$ and $k^{-4}$ have been indicated in what is expected to correspond
to the inertial ranges in (a) and (b), respectively. In (a), the inset shows
$(k/2)\,H_{\rm M}(k)$ at the last time with positive (negative) values in red
(blue), and in (b), the inset compares $\mbox{\rm Sp}(2\mu_{5}/\lambda)$
(solid) with $\mbox{\rm Sp}(h_{\rm tot})$ (dotted) at the last time.  Figure
2:  $I_{\rm H}(t)$ normalized by its initial value. The inset shows ${\cal
I}_{\rm H}(R,t)$ versus $R$ at different times $t$: solid lines correspond to
$t=70$, 200, 700, 2000, 7000, and 20,000, which are also marked by selected
colored symbols in the graph of $I_{\rm H}(t)$. The adapted Hosking integral
is evaluated as $I_{\rm H}(t)={\cal I}_{\rm H}(R_{*},t)$. The vertical dashed-
dotted line marks the value $k_{0}R_{*}=100$ where the curves show a plateau.
The slopes $\propto R^{2}$ and $\propto R^{3}$ are also marked by dashed-
dotted lines.

To compute ${\cal I}_{\rm H}$ (and thereby $I_{\rm H}$), we employ a spectral
technique by computing the total helicity variance spectrum $\mbox{\rm
Sp}(h_{\rm tot})$; see Fig. 1(b). Compared to the inverse cascade seen in
$\mbox{\rm Sp}(\bm{B})$, here we see the conservation of the large-scale total
helicity variance spectrum $\propto k^{2}$. We thus obtain

${\cal I}_{H}(R,t)=L^{-3}\int w(\bm{k},R)\,\mbox{\rm Sp}(h_{\rm tot})\,{\rm
d}{}^{3}\bm{k}/(2\pi)^{3}.$ (6)

We choose $w(k,R)=(4\pi R^{3}/3)[6j_{1}(kR)/kR]^{2}$ as weight function (Zhou
_et al._ , 2022) with $j_{n}$ being spherical Bessel functions.

In Fig. 2, we plot the adapted Hosking integral $I_{\rm H}(t)$, normalized by
its initial value. It is evaluated as $I_{\rm H}(t)={\cal I}_{\rm H}(R_{*},t)$
with $k_{0}R_{*}=100$, where ${\cal I}_{\rm H}(R,t)$ is shown in the inset at
different times as functions of $R$. Note that $I_{\rm H}(t)$ is essentially
flat and shows only toward the end a slight decline $\propto t^{-0.12}$, which
is similar to what has been seen for other simulations at that resolution;
see, e.g., Ref. Brandenburg (2023). Thus, the adapted Hosking integral appears
to be well conserved – even better so than the Hosking integral in ordinary
MHD, studied in Refs. (Hosking and Schekochihin, 2021; Zhou  _et al._ , 2022).
There is not even the slight uprise $I_{\rm H}(t)$ reported in Ref. (Zhou  _et
al._ , 2022), which was there argued to be due to strong non-Gaussian
contributions to the field that emerged during the nonlinear evolution of the
system. Note also that for $R\ll R_{*}$, we see ${\cal I}_{\rm H}(R,t)\propto
R^{2}$, which is shallower than the expected cubic scaling. This might change
at larger resolution, although an intermediate range $\propto R^{2}$ is also
seen in Fig. 4(d) of Ref. (Zhou  _et al._ , 2022), before cubic scaling
emerged for $R/2\pi<10^{-4}$.

Figure 3:  Time dependence of ${\cal E}_{\rm M}$ (black), $\xi_{\rm M}$
(orange), ${\cal H}_{\rm M}$ (blue), and $-2\langle\mu_{5}\rangle/\lambda$
(red). The inset confirms that $2{\cal E}_{\rm M}\xi_{\rm M}/{\cal H}_{\rm
M}\approx 1$ during the whole time.

As in the case of nonrelativistic MHD ($\mu_{5}\to 0$), the dimensions of
${\cal I}_{\rm H}$ and $I_{\rm H}$ are $\,{\rm cm}^{9}\,{\rm s}^{-4}$. This
implies that in $\xi_{\rm M}\propto t^{q}$, the value of the exponent is
$q=4/9$, if the conservation of ${\cal I}_{\rm H}$ determines the time
evolution of the magnetic field around the characteristic scale. Next,
assuming selfsimilarity, the magnetic spectra can be collapsed on top of each
other by plotting them versus $k\xi_{\rm M}(t)$ and compensating the decline
in the height by $\xi_{\rm M}^{\beta}$ to yield the universal function
$\phi(k\xi_{\rm M})=\xi_{\rm M}^{\beta}E_{\rm M}(k\xi_{\rm M})$; see Appendix
B of Ref. Zhou  _et al._ (2022) and Refs. Brandenburg and Kahniashvili (2017);
Brandenburg (2023) for examples in other contexts. Using also the invariance
of the spectrum under rescaling (Olesen, 1997),
$\bm{x}\to\bm{x}^{\prime}=\bm{x}\ell$ and $t\to t^{\prime}=t\ell^{1/q}$, and
since the dimension of $E_{\rm M}(k,t)$ is $\,{\rm cm}^{3}\,{\rm s}^{-2}$, we
have $E_{\rm M}(k\ell^{-1},t\ell^{1/q})=\ell^{3-2/q+\beta}[\xi_{\rm
M}\ell]^{-\beta}\phi(k\xi_{\rm M})$, and therefore $\beta=2/q-3=3/2$, which
agrees with Fig. 1(a). Finally, for ${\cal E}_{\rm M}\propto t^{-p}$, we find
with ${\cal E}_{\rm M}(t)=\int E_{\rm M}\,{\rm d}{}k\propto t^{-(\beta+1)q}$
the line $p=2(1-q)$, which is also known as the self-similarity line
(Brandenburg and Kahniashvili, 2017; Zhou  _et al._ , 2022). With $q=4/9$, we
thus obtain $p=10/9$. This is completely analogous to the MHD case with zero
magnetic helicity111See also the discussion in Ref. Uchida  _et al._ (2022)
for weak magnetic field with zero magnetic helicity, where selfsimilarity is
not assumed.; see also Table 2 of Ref. Brandenburg (2023). Thus, the
cancellation of finite magnetic helicity by fermion chirality with ${\cal
H}_{\rm M}(t)=-2\langle\mu_{5}\rangle(t)/\lambda\neq 0$ has the same effect as
that of zero magnetic helicity.

To understand the decay of magnetic helicity density in the present
simulations, it is important to remember that the real space realizability
condition of magnetic helicity (Kahniashvili  _et al._ , 2013) is always valid
and implies $|{\cal H}_{\rm M}|\leq 2{\cal E}_{\rm M}\xi_{\rm M}$. Assuming
the inequality to be saturated, we find the scaling $|{\cal H}_{\rm
M}|\propto|\langle\mu_{5}\rangle|\propto t^{-r}$ with $r=p-q=2/3$. This is
well obeyed, as is shown in Fig. 3. In the inset, we show that $2{\cal E}_{\rm
M}\xi_{\rm M}/{\cal H}_{\rm M}\approx 1$ at early times and about 1.1 at late
times. It is thus fairly constant, therefore confirming the validity of our
underlying assumption. On top of this evolution of the chiral asymmetry, the
growth rate of the CPI, $\gamma_{5}\propto\langle\mu_{5}\rangle^{2}\propto
t^{-4/3}$, decays more rapidly than $t^{-1}$, which causes it to grow less
efficiently so as not to spoil the scaling properties of the system.

To characterize the scaling expected from the conservation of the adapted
Hosking integral further, in Fig. 4 we plot the $pq$ diagram of the
instantaneous scaling exponents $p(t)=-{\rm d}{}\ln{\cal E}_{\rm M}/{\rm
d}{}\ln t$ versus $q(t)={\rm d}{}\ln\xi_{\rm M}/{\rm d}{}\ln t$. The solution
converges to a point close to the crossing point between the $\beta=3/2$ line
and the scale-invariance line $p=2(1-q)$. The approach to the point
$(p,q)=(10/9,\,4/9)$ does not occur predominantly along the $\beta=3/2$ line,
as in nonhelical standard MHD, but is now closer to the $r=2/3$ line, where
$p=q+r$. In the unbalanced case, where the net chirality is non-vanishing,
however, the decay is solely governed by $\langle h_{\rm tot}\rangle={\rm
const}{}$ Brandenburg  _et al._ (2023).

Figure 4:  $pq$ diagram for times $t=700$, 1000, 1500, 2200, 3200, 4600, 6800,
$10^{4}$, $1.5\times 10^{4}$, $1.5\times 10^{4}$, $2.2\times 10^{4}$, and
$3.2\times 10^{4}$, corresponding to symbols of increasing size. The solid
line denotes the scale-invariance line $p=2(1-q)$, the dashed line the
$\beta=3/2$ line for adapted Hosking scaling, and the dashed-dotted line is
the new $r=2/3$ line that does not have any correspondence in standard MHD.

In conclusion, we have presented evidence that, in the balanced case of zero
total chirality, the Hosking integral, when adapted to include the chiral
chemical potential, is approximately conserved around the characteristic
scale. This implies decay properties for magnetic energy and correlation
length that are unchanged relative to nonhelical MHD, but here with ${\cal
H}_{\rm M}+2\langle\mu_{5}\rangle/\lambda=0$ (instead of ${\cal H}_{\rm
M}=0$). This yields the scaling $|{\cal H}_{\rm
M}|\propto|\langle\mu_{5}\rangle|\propto t^{-2/3}$, along with the familiar
scalings ${\cal E}_{\rm M}\propto t^{-10/9}$ and $\xi_{\rm M}\propto t^{4/9}$
that also apply to the case with ${\cal H}_{\rm M}=0$. These scalings have
consequences for understanding the properties of the chiral magnetic effect in
the early Universe (Kamada, 2018; Del Zanna and Bucciantini, 2018; Domcke and
Mukaida, 2018; Domcke  _et al._ , 2019, 2023) and young neutron stars (Masada
_et al._ , 2018; Dvornikov  _et al._ , 2020). Our work has significant impact
on the baryon asymmetry of the Universe from hypermagnetic helicity decay
after axion inflation. It also exposes a rather unexpected application of the
general idea behind the recently developed Hosking integral, raising therefore
the hope that there may be other ones yet to be discovered.

###### Acknowledgements.

We thank Valerie Domcke and Kai Schmitz for useful comments on the manuscript
and Kyohei Mukaida for fruitful discussions. Support through Grant No.
2019-04234 from the Swedish Research Council (Vetenskapsrådet) (AB), Grant-in-
Aid for Scientific Research No. (C) JP19K03842 from the JSPS KAKENHI (KK), and
Grant No. 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 Linköping.

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