We thank referees A and B for their comments. We respond to each of them
in turn. Our changes to the manuscript are marked in blue. Regarding
the justification for publication in PRL, we wish to recall that the
recently discovered Hosking integral governs the decay of non-helical MHD
turbulence. There is now a growing number of applications, for example to
neutron star crusts. Here, we study the Hosking integral in yet another
and completely different system, applicable to the early universe, which
includes the chiral chemical potential. The importance of our paper,
justifying the publication in PRL, lies in promoting the idea that the
Hosking integral has broad applications. Particularly concrete are those
to cosmological MHD after axion inflation, which can also be connected
with the origin of the observed baryon asymmetry of the Universe.

Response to Referee A

> 1. Why is this particular scenario important so as to justify
> publication in PRL? It may be argued that the evolution discussed in
> this paper is another routine calculation of MHD evolution along the
> lines of several other analyses in the literature. What makes this
> analysis special?

The idea that a Hosking-type integral can describe the decay of a fully
helical magnetic field with helicity that is canceled by fermion chirality
was a long shot, but it worked. The numerical calculation of chiral MHD
is standard and implemented in the Pencil Code, but the interpretation
of the results is not routine.

Such a situation would indeed be realized in the context of axion
inflation. Until now, the chiral chemical potential was assumed to
decay exponentially, but now we show that the decay is actually slower
and of power-law fashion. This will give a significant impact on our
understanding of the present baryon asymmetry of the Universe, which
is highly relevant for the particle physics and cosmology communities.
Therefore, our work is more than just routine calculation of numerical
MHD evolution.

We have now improved the presentation of the novel aspects of this
work both in the second and last paragraphs of this paper. We have also
restructured the abstract to make this point clearer.

> 2. In the Supplemental Material Section 2, there appears to be a
> puzzle related to Fig. 2 that needs clarification -- the authors find
> that with evolution $\mu_5$ goes to zero and they say that "This
> halts the decay of $H_M$.... but eventually it also decays.'' The
> puzzle is that once $\mu_5=0$, further evolution is just that of
> helical magnetic fields in a non-chiral plasma. We know $H_M$ is
> conserved in this situation. So why do the authors find helicity
> non-conservation for a non-chiral plasma?

In Section 2 of the Supplemental Material, we discussed two cases:
(i) where Gamma was different from 0 for the entire time after t=100, and
(ii), where it was different from 0 only for a limited interval. If spin
flipping is left on, i.e., case (i), the total chirality is no longer
conserved. Magnetic helicity \mu_M decays due to magnetic diffusion,
while \mu_5 stays vanishingly small due to the large spin-flipping
term. So there is no conflict.

> 3. Another puzzle is that the authors find (Section 1 of Supplemental
> Material) that the Hosking integral describes magnetic decay when
> $<h_tot>=0$, but not when $<h_tot> \ne 0$. However we know from
> previous work [4,5,6] that for non-chiral plasmas ($\mu_5=0$) $h_tot$
> is just the magnetic helicity and the Hosking integral correctly
> describes decay. So the statement that the Hosking integral does not
> give a correct description for $<h_tot>\ne 0$ cannot be completely
> correct. Perhaps the statement has some caveats that need to be
> spelled out.

There are two conserved quantities that potentially govern the decay: the
Hosking integral and the total chirality. The latter constraint is rather
strong and controls the decay once the total chirality is sufficiently
far away from zero. This is the case for the two models studied in the
Supplemental Material. The Hosking integral is then no longer dominant.

In previous works [4,5,6], the evolution of the system was studied in the
case $\mu_5=<h>=0$ so that also $<h_tot>=0$. In that case, the Hosking
integral was identified just for the magnetic helicity $h$, and was found
to determine the evolution of the magnetic field. So there is no puzzle.

> Some other minor comments that need to be addressed:
> 1) $k_0$ below (4) is not defined.

We have now moved the original remark about k_0 to the end of the
following paragraph after having defined k_0 as the peak of the spectrum.

> 2) In Fig. 1 caption, is it $Sp(B)$ or $Sp(B^2)$?

Yes, this was a typo and has now been corrected to Sp(B). One could
of course also ask about the spectrum of B^2, but this is quartic in
Fourier space and has been studied previously on another occasion;
see https://ui.adsabs.harvard.edu/abs/2020ApJ...892...80B

> 3) Below Eq. (5): "grow cubically for small $R$ and is flat for larger
> $R$" seems inconsistent with Fig. 2(a).

We think the current wording is correct. We recall that in this sentence
we talk about two different cases: non-helical standard MHD (where the
scaling is cubic for small R) and the novel case of helical chiral
(non-standard) MHD where helicity is balanced by fermion chirality
(where the scaling is cubic for large R). The scaling the referee A
mentioned is for the former, but not the latter (Fig. 2(a)).

> 4) In Fig. 3(a), $\mu_5$ label is in blue and $H_M$ label is in red,
> but the caption says $\mu_5$ is red and $H_M$ is blue. (Similarly in
> Fig. 1 of Supplemental Material.)

We have now corrected this.

> 5) In the last sentence of Supplemental Material Section 1, $E_M$
> should be $H_M$.

No, here we talk about the case where the system is no longer governed
by the adapted Hosking integral, but by the total chirality.

> 6) $\mu_M$ needs to be defined in Supplemental Material.

We have done this now and marked it in blue.

Response to Referee B

> Regarding the technical aspects, there are numbers of quantities that
> are not defined in the manuscript. For example, $I_H$ in the paragraph
> starting with "Similarly to earlier studies of ..." is not defined.
> Probably, it is related to $\mathcal{I}_H$ in Eq.(1), but how?

We note that two paragraphs above, we did write 

  The Hosking integral I_H is defined as the asymptotic limit of the
  relevant magnetic helicity density correlation integral, {\cal I}_H(R),
  for scales R large compared to the correlation length of the turbulence,
  xi_M, but small compared to the system size L.

In the present case, however, the relevant value of R tends to be less
than the correlation length, so we have now added the sentence:

  As before, we define I_H={\cal I}_H(R_*) for values of R_* for which
  {\cal I}_H(R) shows a plateau, which is here the case for values of
  R that are comparable to or less than xi_M.

> Similarly, the authors state that they set $k_0=1$ and $\rho_0=1$ for
> units of length and energy, but such quantities do not appear in
> Eqs.(2),(3),(4).

This is true, so we have now moved the sentences about k_0 and rho_0 by
2 paragraphs forward, just after k_0 was defined as the peak wave number.

> Finally, $Sp(B)$ appears in the paragraph starting with
> "In the following, we ...", but it is not fully specified (only
> its normalization is specified). Because Sp(*) appears in the
> discussion of Fig.1, readability of this manuscript is largely
> spoiled. Although they may be obvious for specialists, the manuscript
> is not written in a way understandable to non-specialists.

The specification of the normalization does already define the spectrum
uniquely. Nevertheless, we have now added the definition in the paragraph
following Eqs.(4).

> Regarding the physical significance, the main finding of this
> manuscript is scalings of various quantities out of chiral
> magnetohydrodynamics in Eqs.(2),(3),(4). But why such scalings are
> important? At the very end of this manuscript, it is only briefly
> stated that they have consequences for understanding the properties of
> the chiral magnetic effect in the early universe and young neutron
> stars. If so, this part should be elaborated on. Otherwise, this
> manuscript looks just a technical report of simulation results. Any
> physical significance or novel innovations cannot be found in this
> manuscript from the viewpoint of non-specialists. Therefore, I cannot
> recommend its publication in Physical Review Letters.

We have now restructured the abstract to bring out the importance of the
universality of the Hosking integral and its broad range of applications,
especially to the early Universe. We have also improved the wording in the
second paragraph of the paper to stress that the decay law of magnetic
turbulence with helicity balanced by chiral fermions is realized in,
e.g., cosmological MHD after axion inflation [18-20]. Previously, it has
been thought that the magnetic decay is triggered by the chiral plasma
instability and that it is therefore exponential. However, as is shown
in this Letter, the decay occurs actually more slowly in a power-law
fashion. This feature has a significant impact on our understanding of
the baryon asymmetry of the Universe.