# Axial Inverse Magnetic Catalysis

Yuanyuan Wang<EMAIL_ADDRESS>Shinya Matsuzaki<EMAIL_ADDRESS>Center for Theoretical Physics and College of Physics, Jilin University,
Changchun, 130012, China

###### Abstract

We propose a new inverse magnetic catalysis emerging around the chiral
crossover regime in the thermomagnetic QCD with 2 + 1 flavors at physical
point, that is, the inverse magnetic catalysis for $U(1)$ axial symmetry
(AIMC: axial inverse magnetic catalysis): difference of susceptibilities for
quark bilinears forming the $U(1)_{A}$ partner increases with respect to the
applied magnetic field at zero temperature, while decreases at higher
temperatures as well as its (pseudo) critical temperature. This phenomenon is
compared with the already discovered IMC for the chiral $SU(2)_{L}\times
SU(2)_{R}$ symmetry (CIMC: chiral IMC). We demonstrate the AIMC based on a
Nambu-Jona-Lasinio model with 2 + 1 quark flavors, where introduced anomalous
magnetic moments of the quarks play the essential role to drive both the CIMC
and AIMC. Our finding is shortly testable on lattices. Possible
phenomenological and cosmological implications are also briefly addressed.

††journal: Physics Letters B

##  1 Introduction

Violation of $U(1)$ axial symmetry plays a key role to address the QCD vacuum
characterized by quark condensates, as well as the chiral symmetry breaking.
In particular, it has been a longstanding issue how much the $U(1)_{A}$
breaking contributes to the quark condensate. This question could be related
to what is the major role for the origin of mass in the thermal history of the
universe.

The state-of-art-lattice simulations have so far clarified that in the hot QCD
with 2 + 1 (light up and down quarks and one heavy strange quark), the
$U(1)_{A}$ axial breaking tends to survive longer than the chiral $SU(2)$
breaking for the lightest two flavors as temperature grows [40, 2]. This
result is also supported from a rigorous argument based on QCD-inequality like
relations [3] and its generalized evidence based on the lattice QCD setup [4].
Moreover, a recent lattice study (with two lightest flavors) has shown a hint
that significantly dominant contributions from the $U(1)_{A}$ breaking are
left in the quark condensate, during the chiral phase transition (crossover)
[5]. All those may imply that in a view of the thermal history of the
universe, the main source for the origin of mass is supplied from the
$U(1)_{A}$ breaking.

However, it might be not the end of the story: even in early universe
including the QCD phase transition epoch, a strong enough magnetic field might
be present as a background field, which could be generated due to some
primordial (electroweak) phase transitions of strong first order [6, 7, 8, 9,
10, 11, 12]. Therefore, the QCD dynamics in the thermal history might be
thermomagnetic, which is called thermomagnetic QCD. In addition, heavy ion
collision experiments [13, 14, 15, 16, 17, 18, 19, 20] can create a strong
enough magnetic field in the thermal plasma of QCD, due to the relativistic
motion of the colliding nuclei and the smallness of the system. Thus, the
thermomagnetic QCD has nowadays opened a vast ballpark, non only with
cosmological, but also experimental interests, involving lattice simulations,
and chiral-effective model approaches.

Some outstanding results have already been reported from lattice studies on
the QCD thermodynamics in a strong external magnetic field [21, 22]. Of
particular importance related to the chiral symmetry breaking are the
reduction of the pseudo-critical temperature for the chiral crossover [23, 24,
25, 26, 27] and the inverse magnetic catalysis [28]. Both two imply a faster
effective restoration of the chiral symmetry in hot-magnetized early universe.
To our best knowledge, however, no definite analysis on the $U(1)_{A}$
breaking and effective restoration in the thermomagnetic QCD has been curried
out, or no implications of the inverse magnetic catalysis for the chiral
symmetry to the $U(1)_{A}$ symmetry has been argued.

In this paper, prior to lattice simulations in the near future, based on a
chiral effective model we propose a definite implication of the inverse
magnetic catalysis for the chiral symmetry, to the $U(1)_{A}$ breaking in
thermomagnetized QCD: that is the inverse magnetic catalysis (IMC) for the
$U(1)_{A}$ symmetry, which we call the axial IMC (AIMC), and in comparison, we
call the IMC for the chiral $SU(2)_{L}\times SU(2)_{R}$ symmetry CIMC. We
employ a Nambu–Jona-Lasinio (NJL) model with 2 + 1 flavors at the physical
point, in which we introduce a couple of anomalous magnetic moments (AMMs) of
quarks, as well as so-called Kobayashi-Maskawa-‘t Hooft determinant term [29,
30, 31, 32] mimicked as the instanton-induced $U(1)_{A}$ anomaly in the
underlying QCD.

We first look into the viable parameter space to realize the CIMC, which is
mapped on the AMM parameters. Then we evaluate the difference of
susceptibilities for the $U(1)_{A}$ partner, such as $\pi$ and $\delta$
mesons, denoted as $\chi_{\pi}$ and $\chi_{\delta}$ as a function of the
applied constant magnetic field and temperature. In terms of the axial
susceptibility, $\chi_{\pi-\delta}\equiv\chi_{\pi}-\chi_{\delta}$, the AIMC is
dictated by observing that at zero temperature, $\chi_{\pi-\delta}$ becomes
larger and larger, as the magnetic field strength increases, and the pseudo-
critical temperature of $\chi_{\pi-\delta}$, $T_{pc}^{A}$, gets smaller as the
magnetic field gets stronger. Here the $T_{pc}^{A}$ is defined as the
inflection point of the temperature evolution of $\chi_{\pi-\delta}$.

In light of the CIMC, the two-flavor NJL model with AMMs of quarks has so far
been discussed [33, 34, 35, 36, 37, 38, 39]. There, in terms of the symmetry
argument, realization of the CIMC can be understood by emergence of
destructive interference in the light quark condensates between two explicit
chiral-breaking sources, where one comes from the current quark mass, while
the other from the AMMs of quarks coupled to a strong magnetic field. No
explicit work on the chiral phase transition has been done with 2 + 1 flavors
coupled with the AMMs in the framework of NJL.

We will shed the first light on the 2 + 1 flavor case, and clarify the fully
viable parameter space to realize the desired CIMC at high temperatures, and
MC at zero temperature. We find that at any temperature including zero
temperature, the AMMs of $u$ and $d$ quarks contribute to the $u$ and $d$
quark condensates, or the constituent-$u$ and $d$ quark masses, destructively
against effects from the current quark mass and the $U(1)_{A}$ anomaly, while
the AMM of strange quark acts as destructive interference at lower
temperature, and constructive one at higher temperatures: namely, the AMM of
strange quark tends to cease realization of MC and CIMC, at zero temperature
and higher temperatures, respectively. This latter feature is a new finding
characteristic to the 2 + 1 flavor model.

Since the AMM interactions break not merely the chiral $SU(2)_{L}\times
SU(2)_{R}$ symmetry, but the $U(1)_{A}$ symmetry, similar destructive
interference is expected to happen in the $U(1)_{A}$ sector, i.e.,
$\chi_{\pi-\delta}$. Such coincidental correlation between the chiral and
$U(1)_{A}$ symmetries may also be deduced in a context of the QCD-inequality
argument [3, 40], and a recent lattice study detailed on the Dirac spectrum
[42, 41]. The latter, in particular, works through an operator identity-like
relation between the Dirac spectrum, the chiral condensate, and the $U(1)_{A}$
susceptibility, and should hold even including external fields, such as a
constant magnetic field 111 We thank Akio Tomiya for comments on the validity
of identities with the Dirac spectrum. . Still, however, it would be
nontrivial in a sense of framework of chiral effective models to monitor such
chira- $U(1)_{A}$ axial coherence in the 2 + 1 flavor case. Based on the NJL
model with AMMs, we will explicitly clarify how efficiently the destructive
interference arises in the $U(1)_{A}$ susceptibility $\chi_{\pi-\delta}$,
coherently in the chiral condensate.

We find that the AIMC can take place in the chiral crossover regime where the
CIMC is present, which depends on the size of AMM, and the pseudo-critical
temperature $T_{pc}^{A}$ drops as increasing the magnetic field strength.
Lattice simulations in the near future will give a decisive conclusion on
existence of the AIMC and the competition for the survival between the chiral
and $U(1)_{A}$ breaking at higher temperatures.

This AIMC may provide a hint to reveal whether in a sense of early
thermomagnetic universe, the remnant of the $U(1)_{A}$ breaking in the origin
of mass might be comparable with what the chiral breaking leaves, in contrast
to the pure-thermal QCD in which the former might highly dominate [5].

Other phenomenological and cosmological implications deduced from the AIMC are
also briefly addressed.

##  2 NJL model with quark AMM: preliminaries

To demonstrate the presence of AIMC, we work on an NJL model with 2 + 1
flavors with AMMs of quarks in a constant magnetic field. The model is
described by the following Lagrangian:

$\displaystyle\mathcal{L}=\sum_{f=u,d,s}\bar{\psi}_{f}\left(i\gamma^{\mu}D_{\mu}^{(f)}-\hat{m}_{0f}+\kappa_{f}q_{f}F_{\mu\nu}\sigma^{\mu\nu}\right)\psi_{f}$
$\displaystyle+G\left\\{(\bar{\psi}\lambda^{a}\psi)^{2}+\left(\bar{\psi}i\gamma^{5}\lambda^{a}\psi\right)^{2}\right\\}$
(1)
$\displaystyle-K\left[\operatorname{det}\bar{\psi}\left(1+\gamma_{5}\right)\psi+\operatorname{det}\bar{\psi}\left(1-\gamma_{5}\right)\psi\right].$

Here $\psi_{f}=(u,d,s)^{T}$ denotes the three-flavor quark field forming the
$SU(3)$ triplet; the current-quark mass matrix $\hat{m}_{0f}$ is diagonal in
the flavor space: $\hat{m}_{0f}={\rm diag}(m_{0u},m_{0d},m_{0s})$, where the
present 2 + 1 flavor setup leads to $m_{0u}=m_{od}\equiv m_{0}$; $\lambda^{a}$
($a=0,\cdots,8$) are the Gell-Mann matrices in the flavor space with
$\lambda^{0}=\sqrt{2/3}$ ${\rm diag}(1,1,1)$; the covariant derivative
$D^{(f)}_{\mu}=\partial_{\mu}-iq_{f}A_{\mu}$ contains the coupling between the
quark and the external magnetic field with the electromagnetic charge matrix
$q_{f}=e\,{\rm diag}\\{2/3,-1/3,-1/3\\}$, where the magnetic field is applied
along the $z$ direction and embedded in the electromagnetic gauge field
$A_{\mu}$ as $A_{\mu}=(0,0,Bx,0)$;
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
electromagnetic field strength; $G$ and $K$ are the four-fermion coupling
constant and the determinant (six-fermion) coupling constant, respectively;
$\sigma^{\mu\nu}=\frac{i}{2}\left[\gamma^{\mu},\gamma^{\nu}\right]$. The
$\kappa_{f}$ is the AMM coupling, dependent on quark flavors.

Under the chiral $U(3)_{L}\times U(3)_{R}$ transformation: $\psi\to
U\cdot\psi$ with $U=\exp[-i\gamma_{5}\sum_{a=0}^{8}(\lambda^{a}/2)\theta^{a}]$
and the chiral phases $\theta^{a}$, the four-fermion interaction term is
$U(3)_{L}\times U(3)_{R}$ invariant. The mass term explicitly breaks the
chiral $U(3)_{L}\times U(3)_{R}$ symmetry. The determinant term, called
Kobayashi-Maskawa-‘t Hooft (KMT) determinant [29, 30, 31, 32], is induced from
the instanton coupled to quarks in the underlying QCD, and preserves the
chiral $SU(3)_{L}\times SU(3)_{R}$ invariance (associated with the chiral
phases labeled as $a=1,\cdots,8$), but breaks the $U(1)_{A}$ (corresponding to
$a=0$) symmetry. The AMM term with the coupling $\kappa_{f}$ explicitly breaks
not only the chiral $SU(3)_{L}\times SU(3)_{R}$ symmetry, but also the
$U(1)_{A}$ axial symmetry.

In addition to the quark mass terms, the KMT determinant term, and the AMM
term, the chiral $U(3)_{L}\times U(3)_{R}$ symmetry is spontaneously broken by
the nonperturbative dynamics of the present NJL. To monitor the spontaneous
breaking, we simply employ the mean-field approximation, and construct the
thermodynamics potential in the presence of a constant magnetic field. The
thermodynamic potential is then given as a function of thermally averaged
quark condensates $\langle\bar{u}u\rangle,\langle\bar{d}d\rangle$, and
$\langle\bar{s}s\rangle$ with
$\langle\bar{f}f\rangle=-iN_{c}\operatorname{tr}\int\frac{d^{4}p}{(2\pi)^{4}}S^{f}(p)$,
where $S^{f}(p)$ stands for the full propagator of $f$-quark and $N_{c}$ is
the number of colors, to be fixed to three. With respect to those quark
condensates, the thermodynamic potential is minimized at the nontrivial vacuum
with the stationary condition, which leads to the coupled gap equations in
terms of the constituent quark masses $M_{f}=(M_{u},M_{d},M_{s})$:

$\displaystyle M_{f}=m_{0f}+\sigma_{f}\,,$ (2)

where

$\displaystyle\sigma_{u}$
$\displaystyle=4iG\langle\bar{u}u\rangle-2K\langle\bar{d}d\rangle\langle\bar{s}s\rangle\,,$
$\displaystyle\sigma_{d}$
$\displaystyle=4iG\langle\bar{d}d\rangle-2K\langle\bar{u}u\rangle\langle\bar{s}s\rangle\,,$
$\displaystyle\sigma_{s}$
$\displaystyle=4iG\langle\bar{s}s\rangle-2K\langle\bar{u}u\rangle\langle\bar{d}d\rangle\,.$
(3)

In evaluating the thermally averaged quark condensates, we apply the imaginary
time formalism, and make the following replacement:

$\displaystyle p_{0}$ $\displaystyle\rightarrow i\omega_{n}=i(2n+1)\pi T\,,$
$\displaystyle\int\frac{d^{4}p}{(2\pi)^{4}}$ $\displaystyle\rightarrow
iT\sum_{n=-\infty}^{\infty}\int\frac{d^{3}p}{(2\pi)^{3}},$ (4)

with the Matsubara frequency $\omega_{n}$.

The $f$-quark propagator $S^{f}(p)$ is available in the literature [38],
which, including the AMM term, takes the form

$\displaystyle S^{f}(p)$
$\displaystyle=ie^{-\frac{\vec{p}_{\perp}^{2}}{\left|q_{f}B\right|}}\sum_{n=0}^{\infty}\frac{D_{n}\left(q_{f}B,p\right)F_{n}\left(q_{f}B,p\right)}{A_{n}\left(q_{f}B,p\right)}\,,$
(5)

where $p_{\perp}=(p_{1},p_{2})$, and 222 Here $L_{n}^{\alpha}$ denotes the
generalized Laguerre polynomials.

(6) $\displaystyle D_{n}\left(q_{f}B,p\right)$
$\displaystyle=\left(p^{0}\gamma^{0}-p^{3}\gamma^{3}+M_{f}+\kappa_{f}q_{f}B\sigma^{12}\right)$
$\displaystyle\times\left[\left(1+i\gamma^{1}\gamma^{2}\operatorname{sign}\left(q_{f}B\right)\right)L_{n}\left(\frac{2\vec{p}_{\perp}^{2}}{\left|q_{f}B\right|}\right)-\left(1-i\gamma^{1}\gamma^{2}\operatorname{sign}\left(q_{f}B\right)\right)L_{n-1}\left(\frac{2\vec{p}_{\perp}^{2}}{\left|q_{f}B\right|}\right)\right]$
$\displaystyle+4\left(p^{1}\gamma^{1}+p^{2}\gamma^{2}\right)L_{n-1}^{1}\left(\frac{2\vec{p}_{\perp}^{2}}{\left|q_{f}B\right|}\right)\,,$
$\displaystyle F_{n}\left(q_{f}B,p\right)$
$\displaystyle=\left(\kappa_{f}q_{f}B-p^{0}\gamma^{3}\gamma^{5}+p^{3}\gamma^{0}\gamma^{5}\right)^{2}-M_{f}^{2}-2n\left|q_{f}B\right|\,,$
(7) $\displaystyle A_{n}\left(q_{f}B,p\right)$
$\displaystyle=\left[\left(\kappa_{f}q_{f}B+\sqrt{\left(p^{0}\right)^{2}-\left(p^{3}\right)^{2}}\right)^{2}-M_{f}^{2}-2n\left|q_{f}B\right|\right]$
(8)
$\displaystyle\times\left[\left(\kappa_{f}q_{f}B-\sqrt{\left(p^{0}\right)^{2}-\left(p^{3}\right)^{2}}\right)^{2}-M_{f}^{2}-2n\left|q_{f}B\right|\right]\,.$

To regularize the intrinsic divergence terms arising in the momentum
integration along the magnetic field direction ($p_{3}$), we adopt a smooth
cutoff scheme with the cutoff function,

$\displaystyle
f_{\Lambda}(p_{3},n)=\frac{\Lambda^{10}}{\Lambda^{10}+(p^{2}_{3}+2n\lvert
q_{f}B\rvert)^{2}}\,,$ (9)

where $\Lambda$ is the cutoff parameter.

The AMM term breaks the full $U(3)_{L}\times U(3)_{R}$ symmetry as noted
above, and could actually be dynamically developed [43, 44, 45, 46, 47, 33,
48] as another chiral order parameter, simultaneously with the quark
condensates. In the present analysis, we assume the dynamical AMM to dominate
over the perturbative one generated at the leading order of QED, so that the
AMM can be related to the dynamical mass part of the constituent quark mass as
333 A similar procedure has been applied in Ref. [38] for the 2-flavor NJL
case, with $v_{u}=v_{d}$ assumed.

$\displaystyle\kappa_{f}=v_{f}\cdot\sigma_{f}\,.$ (10)

The flavor-dependent coefficient $v_{f}$ is assumed to be flavor-universal for
$u$ and $d$ quarks, i.e., $v_{u}=v_{d}\equiv v$, but is taken to be different
for $s$ quark: $v_{s}\neq v$. This would be a reasonable setup because the AMM
parameter $\kappa_{f}$ is defined exclusively out of the overall
electromagnetic charge, as seen from Eq.(1), and QCD (or NJL dynamics) is
potentially flavor-universal, and the discrepancy only arises from the chiral-
explicit breaking sources, which can be fully incorporated in the dynamical
mass part $\sigma_{f}$.

In closing this preliminary section, we introduce five input parameters fixed
at vacuum with $eB=T=0$: $m_{0}=0.0055$ GeV, $m_{0s}=0.1407$ GeV,
$\Lambda=0.6023$ GeV, $G\Lambda^{2}=1.835$, and $K\Lambda^{5}=12.36$, which
are quoted from the literature [49] where the physical pion mass is used to
fix those parameters in part.

##  3 CIMC and “phase” diagram

With the preliminary setup provided in the previous section, we first evaluate
the constituent quark mass for $u$ and $d$ quarks as a function of temperature
$T$, given the strength of the applied magnetic field and the AMM parameters
$v$ and $v_{s}$. See Fig. 1. We have taken $\sqrt{eB}\sim(0.5-0.6)$ GeV for a
wide range of $T$, $(0.05-0.25)$ GeV, where the strength of magnetic field is
strong enough in light of lattice QCD: the constant magnetic field applied on
lattice QCD has the minimal size fixed as $|eB|_{\rm min}\propto T^{2}$ [28],
and one can check that $\sqrt{eB}|_{\rm min}<T_{\rm highest}\sim 0.25$ GeV
$<\sqrt{{\rm eB}}|_{\rm applied}\sim(0.5-0.6)$ GeV. In the figure, we have
plotted the averaged mass for $u$ and $d$ quarks, $M=\frac{M_{u}+M_{d}}{2}$,
which is a reasonable chiral order parameter in the presence of a couple of
isospin breaking sources, and taken the AMM parameters $v=vs=1.4$, as a
reference point, a priori clarified below. We see that for small $T$, $M$
increases as the magnetic field gets larger, while at higher $T$, it turns to
decrease with the magnetic field. This confirms the CIMC phenomenon, and is a
successful result generalized from the two-flavor NJL with the AMM [33, 34,
35, 36, 37, 38, 39], to the 2 + 1 flavor case including the KMT-determinant
$U(1)_{A}$-anomaly contribution ($K$ terms in Eq.(3)). The CIMC is
successfully realized essentially due to a moderately large AMM for $u$ and
$d$ quarks which constructively contributes to $M$ with the thermal
corrections, to make $M$ dropped faster.

Figure 1: Plot of the averaged constituent mass for $u$ and $d$ quarks, $M$,
versus temperature evolution, with $eB$ varied and the AMM parameters $v$ and
$v_{s}$ being fixed to a reference point (included in the allowed regime of
the “phase” diagram in Fig. 2). Figure 2: The “phase” diagram on $(v,v_{s})$
plane classified by the CMC and CIMC features at $T=0$ and higher temperatures
(HT). The three phases filled by shaded lines, corresponding to [1], [2] and
[3] in the text, are ruled out because of failure of realizing the magnetic
features on the chiral symmetry breaking, reported from lattice QCD. Only the
centered regime of green-leave shape is allowed. All the four phases merge at
the tetra-critical point, TCP, denoted as a blob.

Scanning over the AMM parameter space $(v,v_{s})$ with the size of $eB$ varied
in an appropriately strong enough range (as noted above), we examine the MC
and IMC features at $T=0$ and at higher $T$. Thus, the “phase” diagram is
drawn on the $(v,v_{s})$ space, as depicted in Fig. 2. The diagram turns out
to be divided into four “phases”, where the model realizes

  * [1

] CIMC for any $T$ including $T=0$, due to larger AMMs for $u$ and $d$ quarks;

  * [2

] CMC for any $T$ including $T=0$, due to smaller AMM for $u$ and $d$ quarks ;

  * [3

] CIMC at $T=0$, and CMC at higher temperatures, due to significant AMM for
strange quark;

  * [4

] CMC at $T=0$, and CIMC at higher temperatures, due to moderate AMMs for 2 +
1 quarks.

The first three “phases” [1], [2], and [3] are excluded, because they do not
reproduce the lattice results on the CMC and CIMC. Thus, only the “phase” [4]
survives, which corresponds to the “Allowed” regime in Fig. 2. Once may notice
that there is a critical point at which all four “phases” merge on the
diagram, that is, the tetra-critical point (TCP), which has been also
specified in Fig. 2.

Of particular interest is to note the “phase” [3], where the AMM of strange
quark acts like a destructive interference against realization of the CMC at
$T=0$ and CIMC at higher temperatures. This is in contrast to the role of the
AMM for $u$ and $d$ quarks. It is operative in the light-quark constituent
mass $(M)$, and contributes destructively against the current quark mass and
the $U(1)_{A}$ anomaly at any temperature including zero temperature, so that
when it is too large, the CIMC is driven even at $T=0$ (“phase” 1), while the
CMC shows up at any temperature, when the light-quark AMM is too small
(“phase” 2 ). This feature can also be observed by viewing the plot along the
$v$ axis at $v_{s}=0$, in Fig. 2.

With the “phase” diagram taken into account, below we will discuss the $T$\-
and $eB$\- dependence on the axial susceptibility.

##  4 Axial susceptibility: $\chi_{\pi-\delta}$, AMC and AIMC

The axial susceptibility is constructed from difference of two
susceptibilities related to the $U(1)_{A}$ partners. In the present NJL model
with 2 + 1 flavors, we have two candidates of the $U(1)_{A}$ partners, which
are, in terms of meson names, $(\sigma,\eta)$ and $(\pi,\delta)$. Those cases
should be just alternatives each other, and exhibit the same axial property,
which would be so even at finite $T$ and $eB$. In the present analysis, we
take $\pi$ and $\delta$ meson channels, and investigation of the other partner
is to be pursued in another publication.

We start with evaluating the $\pi$ channel. The $\pi$ meson susceptibility
$\chi_{\pi}$ is defined as

$\chi_{\pi}=\int_{T}d^{4}x\left[\langle(\bar{u}(0)i\gamma_{5}u(0))(\bar{u}(x)i\gamma_{5}u(x))\rangle_{\rm
conn}+\langle(\bar{d}(0)i\gamma_{5}d(0))(\bar{d}(x)i\gamma_{5}d(x))\rangle_{\rm
conn}\right]\,,$ (11)

with $\langle\cdot\cdot\cdot\rangle_{\rm conn}$ being the connected part of
the correlation function. Here the spacetime integral with the subscript
symbol $T$ means $\int_{0}^{\beta=1/T}d\tau\int d^{3}x$, reflecting the
currently employed imaginary time formalism. Following the literature [50],
the explicit formula for $\chi_{\pi}$ in the present NJL model reads

$\chi_{\pi}=\frac{\Pi_{\pi}\left(0,0\right)}{1-\left[2G-K\left\langle\bar{s}s\right\rangle\right]\Pi_{\pi}\left(0,0\right))},$
(12)

with $\Pi_{\pi}(0,0)\equiv\Pi_{\pi}(\omega,\vec{p})$ being the polarization
(correlation) function for the $\pi$ channel. Presently, we focus only on the
neutral meson channel, so that $\Pi_{\pi}(0,0)$ is evaluated as

$\displaystyle\Pi_{\pi}\left(0,0\right)=-i\cdot\sum_{f=u,d}\left(\int\frac{d^{4}p}{(2\pi)^{4}}\operatorname{tr}\left[i\gamma_{5}S^{f}(p)i\gamma_{5}S^{f}(p)\right]\right).$
(13)

Similarly, we next evaluate the $\delta$ meson susceptibility, which is
defined as

$\chi_{\delta}=\int_{T}d^{4}x\left[\langle(\bar{u}(0)u(0))(\bar{u}(x)u(x))\rangle_{\rm
conn}+\langle(\bar{d}(0)d(0))(\bar{d}(x)d(x))\rangle_{\rm conn}\right]\,.$
(14)

The explicit formula for $\chi_{\delta}$ reads [50]

$\chi_{\delta}=\frac{\Pi_{\delta}\left(0,0\right)}{1-\left[2G+K\left\langle\bar{s}s\right\rangle\right]\Pi_{\delta}\left(0,0\right))}\,.$
(15)

Focusing on the neutral $\delta$ meson component, we find the corresponding
polarization function in the $\delta^{0}$ meson channel:

$\displaystyle\Pi_{\delta}\left(0,0\right)=-i\cdot\sum_{f=u,d}\left(\int\frac{d^{4}p}{(2\pi)^{4}}\operatorname{tr}\left[S^{f}(p)S^{f}(p)\right]\right).$
(16)

The more detailed expressions for $\Pi_{\delta}$ as well as $\Pi_{\pi}$ are
presented in Appendix A, which are useful for performing numerical analysis.

It is crucial to note that $\chi_{\pi}$ is related to the light quark
condensates, through the chiral Ward-identity [52, 53, 54] as

$\displaystyle\langle\bar{u}u\rangle+\langle\bar{d}d\rangle=-m_{0}\chi_{\pi}\,.$
(17)

This is operative even at finite temperature [54], and can also work even with
a constant magnetic field, though it provides extra explicit-chiral breaking
term 444 The magnetic field effect turns out to come in the Ward identity as a
subleading order correction in terms of perturbation with respect to the
electromagnetic coupling, so that Eq.(17) is still intact to the leading
order. Detailed derivation and analysis wil be presented in another
publucation. . Indeed, we have checked that Eq.(17) is satisfied with
$\chi_{\pi}$ and light quark condensates.

Making difference of $\chi_{\pi}$ and $\chi_{\delta}$, we define the axial
susceptibility as

$\displaystyle\chi_{\pi-\delta}\equiv\chi_{\pi}-\chi_{\delta}.$ (18)

This $\chi_{\pi-\delta}$ becomes zero, when the $U(1)_{A}$ symmetry is exact,
because $\chi_{\pi}\leftrightarrow\chi_{\delta}$ by the $U(1)_{A}$
transformation, as is manifest in the definitions, Eq.(11) and (14). In Fig. 3
we plot the magnitude of $\chi_{\pi-\delta}$ as a function of $eB$ at $T=0$,
with the AMM values fixed inside (for $v=v_{s}=1.4\,{\rm GeV}^{-2}$ ) or
outside (for $v=v_{s}=0$) the allowed regime in the “phase” diagram, Fig. 2.
We observe that $|\chi_{\pi-\delta}|$ gets larger as $eB$ grows, namely, the
MC for the axial symmetry at $T=0$. Realization of the AMC at $T=0$ is
somewhat insensitive to the size of the AMMs, in sharp contrast to the case of
the CMC at $T=0$.

Figure 3:  The magnitude of the axial susceptibility, $|\chi_{\pi-\delta}|$,
versus the strength of $eB$, at zero temperature, with a couple of reference
AMM values inside and outside the “Allowed” regime in Fig. 2. Both cases
realize the AMC.  Figure 4: The AMM dependence on $|\chi_{\pi-\delta}|$ at
finite temperature, with $eB$ fixed.

From Fig. 4, we also notice the trend of monotonic reduction for the magnitude
by finite AMMs, with fixed $eB$, which is observed irrespective to $v$ or
$v_{s}$. This is because all the AMMs play a destructive interference in
$\chi_{\pi-\delta}$, against contributions from the current quark masses and
the KMT-determinant $U(1)_{A}$ anomaly, to drive faster $U(1)_{A}$
restoration.

This trend is observed even at finite temperature. See Fig. 4, which shows
$|\chi_{\pi-\delta}|$ as a function of $T$, with $eB$ fixed and the sizes of
the AMMs being flavor symmetrically ($v=v_{s}$) or asymmetrically ($v\neq
v_{s}$) varied. Larger AMMs tend to reduce the magnitude of
$|\chi_{\pi-\delta}|$ at any temperature, and the AMMs generically play a role
of catalizer toward the $U(1)_{A}$ symmetry restoration.

Figure 5 shows the $T$-dependence of $|\chi_{\pi-\delta}|$ with $eB$ varied,
at a reference allowed point for the AMMs $(v,v_{s})$ in Fig. 2. We see that
$|\chi_{\pi-\delta}|$ starts to drop faster at higher temperatures, as $eB$
gets larger, while it develops with $eB$ at lower temperatures. This implies
the IMC for the $U(1)_{A}$ symmetry, i.e., AIMC, in perfect analogy to the IMC
for the chiral symemtry (CIMC). This AIMC can further be quantified by
observing the $eB$ dependence on the pseudo-critical temperature,
$T_{pc}^{A}$, which is defined as the inflection point of the
$|\chi_{\pi-\delta}|$ curve with respect to $T$ as

$\displaystyle\frac{\partial^{2}|\chi_{\pi-\delta}|}{\partial
T^{2}}\Bigg{|}_{T=T_{pc}^{A}}=0\,.$ (19)

Figure 6 plots this $T_{pc}^{A}$ as a function of $eB$, at the same reference
point for AMMS as in Fig. 5. A monotonic decrease trend for $T_{pc}^{A}$ with
growing $eB$ is indeed observed, so it manifests the AIMC, just like the case
of the CIMC.

Figure 5: Plots of $|\chi_{\pi-\delta}|$ as a function of $T$, with $eB$
varied, at a viable reference point for the AMMS $(v,v_{s})$ in the “phase”
diagram, Fig. 2. The AIMC is observed.  Figure 6: The $eB$ scaling of the
pseudo-critical temperature for $|\chi_{\pi-\delta}|$, $T_{pc}^{A}$ defined in
the text. The AMM values have been set to the same viable reference point as
in Fig. 5. The observed decreasing trend manifests presence of the AIMC.

Finally, in Fig. 7 we show an extended “phase” diagram on the AMM $(v,v_{S})$
plane of Fig. 2, by incorporating the parameter space to realize the AIMC at
high temperatures. The figure tells us that the AIMC at high temperature is
necessarily realized when the desired CMC at lower $T$ and the CIMC at higher
$T$ are present (filled in green), except for domains with a larger AMM for
strange quark (in yellow) including the TCP (red blob), or with a larger AMM
for up and down quarks (in orange).

The former discrepancy is due to the flavor-universal destructive contribution
from the AMMs to $|\chi_{\pi-\delta}|$, hence even so large $v_{s}$ can still
act as a destructive interference in $|\chi_{\pi-\delta}|$ against the other
axial breaking contributions from the current quark mass and $U(1)_{A}$
anomaly. This feature is contrast to the $v_{s}$ sensitivity to the chiral
symmetry, as emphasized above, for which the property of the $v_{s}$
interference changes in low or high temperatures.

The latter case would involve limitation of the present analysis based on the
NJL description. The boundary separating the “CIMC and AIMC” (in green) and
“CIMC” (in orange) domains has been created by the present calculability:
going over the “CIMC and AIMC” domain to the right (i.e. to a larger $v$
regime), $|\chi_{\pi-\delta}|$ actually starts to show non-monotoic dumping
scaling at higher $T$, which we would regard as unphysical or an artifact of
the present NJL model as a low energy description of QCD. This lack of
calculability has also affected the existence of a top endpoint of the AIMC
regime (at $(v,v_{s})\simeq(0.9,22)$), and the right-side boundary curve in
Fig. 7. (The left-side boundary curve is present because the AMC at higher
temperatures is realized due to too small AMMs.)

Lattice simulations in the near future will clarify our prediction of the
AIMC, and give a conclusive answer to whether the AIMC at high temperatures is
necessarily realized when the CIMC at high temperatures is present.

Figure 7:  The “phase” diagram extended from Fig. 2 with the property of the
axial sector incorporated. In the same way as in Fig. 2, the “phase” [1] is
excluded by CIMC at $T$=0; [2] is ruled out by CMC at higher temperatures; [3]
is disfavored because of CIMC at $T$ =0 and CMC at higher temperatures. The
TCP is placed at the same point as in Fig. 2. The domain filled in yellow,
which realizes the AIMC at higher temperatures, almost overlaps with the
“Allowed” regime in Fig. 2, but separates into two, to create the orange
regime out of the original green “Allowed” regime. The reason would be related
to the calculability of the present analysis. For more details, see the text.

##  5 Conclusion and Discussion

In conclusion, we proposed the axial inverse magnetic catalysis at high
temperatures, which can be driven by the 2 + 1 flavor-universal destructive
interference against the current quark mass and the $U(1)_{A}$ anomaly in the
axial susceptibility, when the chiral inverse magnetic catalysis is present.
One possible candidate to realize this kind of destructive effects involves
the anomalous magnetic moments of quarks, which, in the present paper, we have
incorporated into a chiral effective model, an NJL model with 2 + 1 flavors,
to demonstrate emergence of the axial inverse magnetic catalysis at the
physical point.

Measuring the axial inverse magnetic catalysis with a moderate anomalous
magnetic moment for lightest quarks would give a complementary probe of the
chiral inverse magnetic catalysis (See Fig. 6), and imply a definite
interpretation on how the inverse magnetic catalysis is realized: it is due to
the destructive axial- and chiral-breaking driven in thermomagnetic QCD.

The axial inverse magnetic catalysis may provide a hint to reveal whether in a
sense of early thermomagnetic universe, the remnant of the $U(1)_{A}$ breaking
in the origin of mass might be comparable with what the chiral breaking
leaves, in contrast to the pure-thermal QCD in which the former might highly
dominate [5].

Our findings are shortly testable by lattice simulations in the near future,
by which the anomalous-magnetic moment values in the NJL model can be
constrained and a part of the model parameter space will be excluded or
probed.

The presently proposed AIMC would also give impact on modeling chiral
effective models monitoring the thermomagnetic QCD: confirmation by lattice
QCD simulations for the AIMC at high temperatures would give evidence of the
destructive chiral and axial breaking in the chiral and the axial
susceptibilities, and effective chiral models other than the present NJL with
AMMs would be severely constrained, and, without such destructive interference
leading to the CIMC and AIMC, would be excluded. It would be for instance
intriguing to see if a recently proposed NJL model with a lattice data-driven
NJL coupling, in light of the CIMC [51], can survive the AIMC as well.

Possible prospected studies along this axial inverse magnetic catalysis also
include correlation between the dual inverse magnetic catalysis (CIMC and
AIMC) and the topological susceptibility: in the case without magnetic fields,
the susceptibility differences for the chiral and axial partners are firmly
linked to the topological susceptibility, through the chiral Ward identities
[52, 53, 54], and can form what is called the QCD trilemma [55]. It would be
noteworthy to look into the magnetic dependence on the topological
susceptibility, when a strong magnetic field is applied, and its sensitivity
to the emergence of the chiral and axial inverse magnetic catalyses, through
the chiral Ward identities.

Then, some cosmological implications to QCD axion coupled to thermomagnetic
QCD could also be derived: if the topological susceptibility could drop faster
around the chiral crossover regime, due to the strong magnetic field along
with the CIMC and AIMC, the axion mass (with fixed axion decay constant) could
also get smaller, implying a significant modification of estimate on the
thermal relic abundance of axion as a dark matter candidate.

Other possible application of the axial inverse magnetic catalysis may be
related to a stable neutral $a_{0}$ meson around the chiral crossover regime:
note first that the susceptibilities scale with the associated meson masses
like $\sim 1/m_{\rm mesons}^{2}$, hence the degeneracy in the meson masses
actually signals the (effective) restoration of the associated symmetry. Since
we expect $m_{\pi^{0}}\simeq m_{(\delta^{0}=a_{0}^{0})}$ due to the axial
inverse magnetic catalysis, the dominant decay channel
$a_{0}^{0}\to\pi^{0}+\eta^{0}$ will almost be closed (no matter how the
$\eta^{0}$ mass gets changed from the vacuum value around the chiral crossover
regime), so that the neutral $a_{0}$ meson can be somewhat long-lived. This
might give phenomenological impact on meson physics relevant to heavy ion
collision experiments.

Exploring those interesting issues are to be left, and pursed elsewhere.

## Acknowledgements

We are grateful to Mamiya Kawaguchi, Sheng-Tai Li, Akio Tomiya, and Lang Yu
for useful comments. This work was supported in part by the National Science
Foundation of China (NSFC) under Grant No.11747308, 11975108, 12047569, and
the Seeds Funding of Jilin University (S.M.).

##  Appendix A Useful formulas for polarization functions

Substituting the quark propagator under the magnetic field in Eq.(5), which
includes the AMM terms, into the polarization functions, we work on the Dirac
trace quantities. For the pion polarization function in Eq.(13), we have

$\displaystyle\operatorname{Tr}\left[\gamma_{5}D_{n}\left(p_{\|},p_{\perp}\right)F_{n}\left(p_{\|},p_{\perp}\right)\gamma_{5}D_{m}\left(p_{\|},p_{\perp}\right)F_{m}\left(p_{\|},p_{\perp}\right)\right]$
(20) $\displaystyle=-8\left[f_{1}(n)f_{1}(m)-f_{2}^{2}\cdot
p_{\|}^{2}\right]\left[\left(L_{n}L_{m}+L_{n-1}L_{m-1}\right)\cdot\left[p_{\|}^{2}-\left(M_{f}^{2}+\kappa_{f}^{2}q_{f}^{2}B^{2}\right)\right]\right.$
$\displaystyle\left.-2s\left(q_{f}B\right)M_{f}\kappa_{f}q_{f}B\left(L_{n}L_{m}-L_{n-1}L_{m-1}\right)-8p_{\perp}^{2}L_{n-1}^{1}L_{m-1}^{1}\right],$

where $p_{||}=(p_{0},p_{3})$; we have omitted the arguments for the
generalized Laguerre polynomials; and we defined

$\displaystyle f_{1}(n)$ $\displaystyle\equiv
p_{\|}^{2}+\left(\kappa_{f}q_{f}B\right)^{2}-M_{f}^{2}-2n\left|q_{f}B\right|,$
(21) $\displaystyle f_{2}$ $\displaystyle\equiv-2\kappa_{f}q_{f}B.$

In a similar way, for the delta meson polarization function in Eq.(15), we
have

$\displaystyle\operatorname{Tr}\left[D_{n}\left(p_{\|},p_{\perp}\right)F_{n}\left(p_{\|},p_{\perp}\right)D_{m}\left(p_{\|},p_{\perp}\right)F_{m}\left(p_{\|},p_{\perp}\right)\right]$
(22) $\displaystyle=8\left[f_{1}(n)f_{1}(m)+f_{2}^{2}\cdot
p_{\|}^{2}\right]\left[\left(L_{n}L_{m}+L_{n-1}L_{m-1}\right)\cdot\left[p_{\|}^{2}+\left(M_{f}^{2}+\kappa_{f}^{2}q_{f}^{2}B^{2}\right)\right]\right.$
$\displaystyle\left.+2s\left(q_{f}B\right)M_{f}\kappa_{f}q_{f}B\left(L_{n}L_{m}-L_{n-1}L_{m-1}\right)-8p_{\perp}^{2}L_{n-1}^{1}L_{m-1}^{1}\right]$
$\displaystyle+16\left(f_{1}(n)+f_{1}(m)\right)f_{2}p_{\|}^{2}M_{f}s\left(q_{f}B\right)\left(L_{n}L_{m}-L_{n-1}L_{m-1}\right)$
$\displaystyle-8p_{\|}^{2}\left(f_{1}(n)+f_{1}(m)\right)f_{2}^{2}\left(L_{n}L_{m}+L_{n-1}L_{m-1}\right),$

where

$\displaystyle f_{1}(n)$
$\displaystyle=p_{\|}^{2}+\left(\kappa_{f}q_{f}B\right)^{2}-M_{f}^{2}-2n\left|q_{f}B\right|,$
(23) $\displaystyle f_{2}$ $\displaystyle=-2\kappa_{f}q_{f}B.$

Plugging those into Eq.(18) with Eqs.(11) and (14), and working on momentum
integration with respect to $p_{\perp}$, and summing over Matsubara
frequencies, we are then ready to evaluate the axial susceptibility
numerically. Through this procedure, a couple of the results have been
presented in the main text.

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11institutetext:  Konkoly Observatory, HUN-REN Research Centre for Astronomy
and Earth Sciences, Konkoly Thege Miklós út 15-17, H-1121 Budapest, Hungary
11email<EMAIL_ADDRESS>22institutetext: CSFK, MTA Centre of Excellence,
Konkoly Thege Miklós út 15-17, H-1121 Budapest, Hungary 33institutetext:
Institute of Physics and Astronomy, ELTE Eötvös Loránd University, Pázmány
Péter sétány 1/A, H-1117 Budapest, Hungary 44institutetext: Shanghai
Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan Road,
Shanghai 200030, P.R. China 55institutetext: Key Laboratory of Radio Astronomy
and Technology, Chinese Academy of Sciences, A20 Datun Road, Chaoyang
District, Beijing 100101, P.R. China 66institutetext: Department of Astronomy,
Institute of Physics and Astronomy, ELTE Eötvös Loránd University, Pázmány
Péter sétány 1/A, H-1117 Budapest, Hungary 77institutetext: HUN-REN–ELTE
Extragalactic Astrophysics Research Group, Eötvös Loránd University, Pázmány
Péter sétány 1/A, H-1117 Budapest, Hungary 88institutetext: Joint Institute
for VLBI ERIC, Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, The Netherlands
99institutetext: Faculty of Aerospace Engineering, Delft University of
Technology, Kluyverweg 1, 2629 HS Delft, The Netherlands 1010institutetext:
Graduate Institute of Astronomy, National Central University, Taoyuan City,
32001, Taiwan

# Milliarcsecond-scale radio structure of the most distant BL Lac object
candidate at redshift 6.57

S. Frey 112233 Y. Zhang 4455 K. Perger 1122 T. An 4455 K. É. Gabányi 66771122
L. I. Gurvits 8899 C.-Y. Hwang 1010 E. Koptelova 1010 Z. Paragi 88 J. Fogasy
1122

(Received 14 November 2023 / Accepted 25 December 2023)

###### Abstract

Context. The existence of accreting supermassive black holes up to billions of
solar masses at early cosmological epochs (in the context of this work,
redshifts $z\gtrsim 6$) requires very fast growth rates which is challenging
to explain. The presence of a relativistic jet can be a direct indication of
activity and accretion status in active galactic nuclei (AGN), constraining
the radiative properties of these extreme objects. However, known jetted AGN
beyond $z\sim 6$ are still very rare.

Aims. The radio-emitting AGN J2331$+$1129 has recently been claimed as a
candidate BL Lac object at redshift $z=6.57$, based on its synchrotron-
dominated emission spectrum and the lack of ultraviolet/optical emission
lines. It is a promising candidate for the highest-redshift blazar known to
date. The aim of the observations described here was to support or refute the
blazar classification of this peculiar source.

Methods. We performed high-resolution radio interferometric imaging
observations of J2331$+$1129 using the Very Long Baseline Array at $1.6$ and
$4.9$ GHz in 2022 Feb.

Results. The images revealed a compact but slightly resolved, flat-spectrum
core feature at both frequencies, indicating that the total radio emission is
produced by a compact jet and originates from within a central 10-pc scale
region. While these are consistent with the radio properties of a BL Lac
object, the inferred brightness temperatures are at least an order of
magnitude lower than expected from a Doppler-boosted radio jet, leaving the
high-redshift BL Lac identification still an open question.

###### Key Words.:

radio continuum: galaxies – galaxies: high redshift – BL Lacertae objects:
individual: J2331$+$1129 – techniques: interferometric

##  1 Introduction

The most powerful active galactic nuclei (AGN) can be observed from vast
distances, at redshifts beyond $z=10$ (Goulding et al., 2023; Bogdán et al.,
2023), corresponding to about $3.5\%$ of the present age of the Universe.
These objects are powered by material accretion onto the central supermassive
black holes (SMBHs) residing in their host galaxies. The existence of black
holes reaching the mass of billions of solar masses at $z\sim 6-7$ poses
challenges for models describing the formation and early growth of SMBHs (e.g.
Volonteri et al., 2021). Recent discoveries of AGN at extremely high redshifts
($z\approx 10$) favour models with massive ($\sim
10^{4}-10^{5}\,\mathrm{M}_{\odot}$) black hole seeds such as resulting from
direct collapse of gravitationally unstable gas clouds (e.g. Lodato &
Natarajan, 2006), merger of/accretion onto primordial black holes (e.g.
Cappelluti et al., 2022), or spherically symmetric accretion into the combined
potential of a stellar-mass seed black hole at the centre of a dark matter
halo (Sharma & Sharma, 2023).

According to the updated catalogue111https://cdsarc.cds.unistra.fr/viz-
bin/cat/J/other/FrASS/4.9, accessed on 2023 Nov 14 of Perger et al. (2017),
the number of currently known $z>6$ quasars with measured spectroscopic
redshift is nearly 300. From these, only 15 sources have been individually
detected in radio bands, although many of the remaining objects may have some
level of weak AGN-related radio emission (Perger et al., 2019, 2024). It is
expected that radio-emitting AGN will be found up to $z\approx 15$ in the
coming decades, thanks to new sensitive radio and optical instruments (Agudo
et al., 2015; Ighina et al., 2023; Latif et al., 2024).

In general, the relative minority, about one-tenth of quasars (e.g. Ivezić et
al., 2002) produce relativistic plasma jets that originate from the close
vicinity of the central SMBH. These are strong radio emitters via the
synchrotron process of charged particles accelerated to relativistic speeds
and moving in magnetic fields. Such sources are important targets for very-
long-baseline interferometric (VLBI) observations that are capable of imaging
the radio emission from compact jets down to pc scales – better than any other
technique in astronomy at present. Radio measurements are not affected by dust
obscuration in the young host galaxies and provide a unique insight into jet
launching and emission. These distant AGN are signposts of the earliest
occurrence of SMBH activity in the Universe. Studies of these objects
contribute to understanding the evolution of jetted AGN, the role jets play in
the co-evolution SMBHs and their host galaxies, and may even help evaluate
cosmological models.

Blazars form a special class of AGN with their jets believed to be closely
aligned with the line of sight and thus their emission is enhanced by Doppler
boosting. About half of the currently known $21$ distant radio quasars at
$z\geq 5.8$, $10$ objects have been observed, and $9$ detected with VLBI (Frey
et al., 2003, 2005, 2008, 2011; Momjian et al., 2008, 2018, 2021; Cao et al.,
2014; Wang et al., 2017; Spingola et al., 2020; Zhang et al., 2022; Liu et
al., 2022). Among the latter, the only blazar candidate is PSO
J030947.49$+$271757.31 at $z=6.10$ (Spingola et al., 2020), based on its
prominent one-sided core–jet structure. The other typical $z\ga 6$ VLBI radio
sources known to date have compact but somewhat resolved structure and steep
radio spectrum, indicating unbeamed jet emission and suggesting young age. The
double structure of the quasar NDWFS J14276$+$3312 $(z=6.12)$ with components
separated by $\sim 160$ pc (Frey et al., 2008; Momjian et al., 2008) is
reminiscent of compact symmetric objects known to be young ($\la
10^{3}-10^{4}$ yr) radio sources.

FIRST J233153.20$+$112952.11 (J2331$+$1129 hereafter) is a source from the
Faint Images of the Radio Sky at Twenty centimeters (FIRST) survey (Becker et
al., 1995) catalogue, with flux density $S_{\mathrm{1.4\,GHz}}=1.85$ mJy. This
object was recently identified as a high-redshift AGN candidate by cross-
matching $z_{\mathrm{PS1}}$-band dropouts from the Panoramic Survey Telescope
and Rapid Response System 1 (Pan-STARRS1, Chambers et al., 2016) with FIRST
radio sources (Koptelova & Hwang, 2022). The source has counterparts in the
near- and mid-infrared, as well as in other radio surveys. Its radio spectrum
is flat within a relatively narrow range of frequencies, between the observed
$888$ MHz and $3$ GHz (spectral index $\alpha=-0.01\pm 0.06$, defined here as
$S\propto\nu^{\alpha}$, where $\nu$ is the frequency), with flux densities
around $2$ mJy. The optical-to-radio spectral energy distribution of
J2331$+$1129 is dominated by the synchrotron emission of the jet and closely
resembles those of BL Lac objects. The non-thermal ultraviolet/optical
continuum has a spectral index $\alpha_{\rm opt}=1.43\pm 0.23$. The source
appears variable in near-infrared and possibly in radio as well, on timescales
of months to years in the observer’s frame. The observed near-infrared
spectrum shows no detected emission lines (Koptelova & Hwang, 2022). Its
redshift was estimated based on the Gunn–Peterson trough (Gunn & Peterson,
1965) found at the wavelength $\lambda=0.921\leavevmode\nobreak\
\mu\mathrm{m}$, giving a lower redshift limit of $z=6.57$. The observed
properties together suggest that this AGN belongs to a subclass of blazars, BL
Lac objects, and thus might be the highest-redshift BL Lac discovered to date.
Notably, no BL Lac objects have been found so far beyond $z=4$ (Koptelova &
Hwang, 2022).

VLBI imaging observations with milliarcsec (mas) resolution are essential for
verifying the truly compact radio structure and the high brightness
temperature expected from a relativistically beamed jet of a BL Lac object.
Here we report on our dual-frequency ($1.6$ and $4.9$ GHz) observations of
J2331$+$1129 with the U.S. National Radio Astronomy Observatory (NRAO) Very
Long Baseline Array (VLBA). Throughout this Letter, we assume a flat $\Lambda$
Cold Dark Matter cosmological model with Hubble constant $H_{0}=70$ km s-1
Mpc-1, matter density parameter $\Omega_{\mathrm{m}}=0.3$, and vacuum energy
density parameter $\Omega_{\Lambda}=0.7$. In this model, the luminosity
distance at $z=6.57$ is $D_{\mathrm{L}}=64.119$ Gpc, the angular scale $5.425$
pc mas-1, and the age of the Universe $813$ Myr (Wright, 2006).

##  2 Observations and data reduction

Figure 1: Naturally weighted VLBA images of J2331$+$1129 at 1.6 GHz (left) and
4.9 GHz (right). The intensity colour scales are displayed on the right-hand
side of the panels. The lowest contours start at $\pm 3\sigma$ rms noise, and
the positive levels increase by a factor of 2. Image rms and peak intensity
values are shown in the bottom of the panels, and the elliptical Gaussian
restoring beams in the lower left corners. The beam sizes are
$11.7\leavevmode\nobreak\ \mathrm{mas}\times 5.7\leavevmode\nobreak\
\mathrm{mas}$ and $4.0\leavevmode\nobreak\ \mathrm{mas}\times
1.8\leavevmode\nobreak\ \mathrm{mas}$ (FWHM), with major axis position angles
$3\aas@@fstack{\circ}2$ and $-1\aas@@fstack{\circ}0$ (measured from north
through east) at 1.6 and 4.9 GHz, respectively.

We conducted observations of J2331$+$1129 with the VLBA at the central
frequencies of $1.6$ and $4.9$ GHz on 2022 Feb 1 and 4, respectively (project
code: BF132, P.I.: S. Frey). The experiment was performed in phase-referencing
mode (Beasley & Conway, 1995) using J2330$+$1100 as phase calibrator. This
source is listed in the 3rd edition of the International Celestial Reference
Frame (ICRF3, Charlot et al., 2020) and its angular separation from the target
is $0\aas@@fstack{\circ}59$. Both experiment segments lasted for $4$ h and
recorded data in left and right circular polarizations. The observing time
spent on the target, J2331$+$1129, was $165$ min in each segment. For the
lower frequency band, the observations were conducted in $2$ intermediate
frequency channels (IFs) around $1.4$ and $1.7$ GHz. The data rate was $2$
Gbps. At $4.9$ GHz, the data rate was $4$ Gbps, and the number of IFs $4$. In
both experiment segments, the bandwidth was $128$ MHz per IF and polarization.
The raw data recorded at each VLBA telescope were correlated at the DiFX
correlator (Deller et al., 2011) in Socorro (New Mexico, USA) with $2$ s
integration time.

The VLBA data were calibrated in the standard way in the NRAO Astronomical
Image Processing System package (aips, Greisen, 2003). The appropriate tasks
and procedures were used to correct for the dispersive ionospheric delays
based on models from Global Navigation Satellite Systems data, and the
accurate Earth orientation parameters. After digital sampling corrections, we
performed manual phase calibration to remove instrumental delays using a short
1-min scan spent on the secondary calibrator source J2327$+$0940 also
scheduled in the experiment. Bandpass correction was performed using the same
section of data. Then amplitude calibration followed, using antenna-based
system temperatures and gain curves supplied along with the correlated
visibility data. Finally, global fringe-fitting was attempted for the bright
sources observed, J2330$+$1100, J2327$+$0940, and the fringe-finder 3C 454.3.
The phase, delay and delay-rate solutions were applied to the respective
source data.

The calibrated single-source visibility files were then exported to the
Caltech Difmap program (Shepherd, 1997) for hybrid mapping. The procedure
started with several iterations of clean image decomposition and phase self-
calibration. To begin the amplitude and phase self-calibration, the overall
antenna gain correction factors were determined. In a few cases when gain
corrections exceeded $\pm 5\%$, those values were fed back into aips to refine
the amplitude calibration.

Fringe-fitting was then repeated for the phase-reference calibrator
J2330$+$1100 in aips. Now the clean image of the source obtained in Difmap was
also used as an input, to account for a potential small phase contribution of
the source structure. The solutions obtained for the calibrator were then
interpolated to the target source.

During imaging of J2331$+$1129 in Difmap, we did not attempt any self-
calibration because of the weakness of the target source. Radio emission was
clearly detected both at $1.6$ and $4.9$ GHz. However, the source position was
offset by $\sim 0.1\arcsec$ in both right ascension (RA) and declination (Dec)
with respect to the a-priori pointing position. To minimize smearing effects,
we repeated the calibration steps in aips but with the target source
coordinates shifted to the newly determined accurate position.

##  3 Results and discussion

Results of the clean imaging of J2331$+$1129 are displayed in Fig. 1. We found
that the source is compact but slightly resolved at both observing
frequencies. To characterise its brightness distribution quantitatively, we
fitted circular Gaussian model components directly to the interferometric
visibility data in Difmap. At $1.6$ GHz, the flux density is
$S_{\mathrm{1.6\,GHz}}=(1.8\pm 0.2)$ mJy and the component diameter is
$\theta_{\mathrm{1.6\,GHz}}=(3.67\pm 0.05)$ mas (full width at half maximum,
FWHM). At $4.9$ GHz, the parameters of the circular Gaussian component that
adequately describe the source are $S_{\mathrm{4.9\,GHz}}=(1.6\pm 0.2)$ mJy
and $\theta_{\mathrm{4.9\,GHz}}=(0.68\pm 0.01)$ mas. The fitted component
sizes exceed the minimum resolvable angular size of the interferometer
(Kovalev et al., 2005). These angular sizes correspond to about $(4-20)$ pc
linear extent if we assume that the source is at $z=6.57$.

The accurate coordinates of the target source calculated by finding the
4.9-GHz image brightness peak position with the aips verb maxfit are
$\mathrm{RA}=23^{\mathrm{h}}31^{\mathrm{m}}53\aas@@fstack{s}21007$ and
$\mathrm{Dec}=11\degr 29\arcmin 52\aas@@fstack{\prime\prime}0089$. The
estimated uncertainty of $0.2$ mas in both coordinates is dominated by the
effect arising from the angular distance of the phase-reference calibrator
source (cf. Chatterjee et al., 2004; Rioja & Dodson, 2020). The position
determined at $1.6$ GHz is slightly less accurate but the coordinates are
consistent with the above values within the errors.

Based on the fitted flux densities, the radio spectrum of the $\sim 10$-pc
scale structure is flat ($\alpha_{\mathrm{pc}}=-0.11$), similar to what is
known from the total flux density measurements (Koptelova & Hwang, 2022).
Irrespective of whether we consider a $\sim 10-20\%$ coherence loss (Martí-
Vidal et al., 2010) in phase-referenced observations or not, the measured flux
densities match well those obtained with lower-resolution connected-element
interferometers, indicating that the total radio emission of J2331$+$1129 is
essentially confined to the region imaged with VLBI.

For comparison with other high-redshift radio AGN, we calculated the $1.4$-GHz
rest-frame radio power as $P=4\pi
S_{\mathrm{1.4\,GHz}}D_{\mathrm{L}}^{2}(1+z)^{-\alpha-1}$. The high value of
$P=1.4\times 10^{26}\leavevmode\nobreak\ $W Hz-1 suggests that the emission
originates from a powerful active galactic nucleus (see Perger et al., 2019,
and references therein), provided that the source is indeed at such a high
redshift.

The brightness temperatures were calculated as

$T_{\mathrm{b}}=1.22\cdot
10^{12}(1+z)\frac{S_{\nu}}{\nu^{2}\theta^{2}}\,\text{K,}$ (1)

where the flux density $S_{\nu}$ is measured in Jy, the observing frequency
$\nu$ in GHz, and the circular Gaussian component diameter $\theta$ (FWHM) in
mas. The values ($T_{\mathrm{b}}\approx 5\times 10^{8}$ K at $1.6$ GHz and
$T_{\mathrm{b}}\approx 1.5\times 10^{9}$ K at $4.9$ GHz) also confirm the non-
thermal nature of the radio emission that originates from AGN activity. This
conclusion is so robust that it would hold true even if $z=0$ because
brightness temperatures exceeding $T_{\mathrm{b}}\sim 10^{5}$ K cannot be
attributed to star-forming activity in the host galaxy (Condon, 1992).

Even the higher measured brightness temperature for J2331$+$1129 is an order
of magnitude below the equipartition limit, $T_{\mathrm{b,eq}}\approx 5\times
10^{10}$ K (Readhead, 1994), when the energy density in the radiating
particles matches that of the magnetic field. This is conventionally assumed
as the intrinsic value in the core region of the jet. In our case, the Doppler
boosting factor, i.e. the ratio of measured and intrinsic brightness
temperatures,
$\delta=T_{\mathrm{b}}/T_{\mathrm{b,int}}=T_{\mathrm{b}}/T_{\mathrm{b,eq}}$,
is smaller than $1$. This would indicate that the radio emission is not
relativistically boosted, contrary to one would expect from a powerful blazar
jet pointing nearly to the line of sight. Even considering somewhat lower
intrinsic brightness temperature, $T_{\mathrm{b,int}}\approx 3\times 10^{10}$
K (Homan et al., 2006), found in samples of powerful jetted AGN in their
median-low state (i.e. not in outburst when brightness temperatures can reach
an order of magnitude higher values), $\delta$ remains well below unity.

Based on a large sample of jetted AGN (mostly blazars) studied with VLBI,
Cheng et al. (2020) found that brightness temperatures are generally
frequency-dependent, reaching their maximum at the emitted (rest-frame)
frequency $\nu_{\mathrm{em}}\approx 6.8$ GHz. If the redshift of J2331$+$1129
is indeed extremely high, the observed $\nu=4.9\leavevmode\nobreak\
\mathrm{GHz}$ frequency corresponds to
$\nu_{\mathrm{em}}=\nu\,(1+z)=37\leavevmode\nobreak\ \mathrm{GHz}$ in the
expanding Universe. According to the phenomenological dependence derived by
Cheng et al. (2020), the maximum brightness temperature of the source could
then be a factor of $\sim 4-5$ higher than what we measured at $\nu=4.9$ GHz,
i.e. $\sim 5\times 10^{9}$ K. In any case, this is still not sufficient for
having observed Doppler-boosted emission in the J2331$+$1129 jet. We note,
however, that the core brightness temperatures (or lower limits) of the only
other claimed blazar candidate known to date at $z>6$, PSO
J030947.49$+$271757.31, are also in the order of $T_{\mathrm{b}}\sim 10^{8}$ K
(Spingola et al., 2020). On the other hand, for the previously known second
most distant blazar, J0906$+$6930 $(z=5.47)$, the brightness temperature
measured with VLBI ($3\times 10^{11}$ K) clearly exceeds the equipartition
limit (An et al., 2020).

The flat section of the radio spectrum of J2331$+$1129 around $\nu\ga 1$ GHz
(see also Koptelova & Hwang, 2022), although not yet sampled in a sufficiently
broad range of frequencies, could possibly be interpreted as similar to those
of Gigahertz-Peaked Spectrum (GPS) radio sources (e.g. O’Dea & Saikia, 2021,
and references therein). However, GPS sources typically have more extended
mas-scale radio structure, within about $500$ pc in size (O’Dea & Saikia,
2021). Indeed, a high-redshift GPS source recently studied with VLBI,
J1606$+$3124 $(z=4.56)$, shows a relatively compact ($\sim 70$ pc) triple
structure (An et al., 2022), much different from the single component ($<20$
pc) we see in J2331$+$1129.

Note that, assuming $z=6.57$ for our source, the observed $\sim 1$ GHz
frequency range translates to above $5$ GHz, where turnover frequencies are
characteristic to High-Frequency Peaker (HFP) sources (O’Dea & Saikia, 2021).
Typically both HFPs and GPS sources have less compact structures that blazar
core–jets. However, HFP and blazar properties are not necessarily mutually
exclusive. A good example is J0906$+$6930 $(z=5.47)$ which has a peaked broad-
band radio spectrum with a turnover frequency $\nu_{\mathrm{em}}\approx 40$
GHz (Coppejans et al., 2017; Zhang et al., 2017) but also very compact
Doppler-boosted jet emission (An et al., 2020). Future flux density
measurements of J2331$+$1129 in a frequency range wider than available at
present (like in the case of a sample of $z>5$ radio AGN by Shao et al., 2022)
would provide decisive information about the possible HFP nature of this
source. At present, the previous low-resolution and our recent VLBI data
indicate that the radio spectrum of J2331$+$1129 is likely flat between the
observed $888$ MHz and $4.9$ GHz, as expected for blazars. The currently
available data do not show evidence for a spectral turnover at gigahertz
frequencies.

##  4 Summary and conclusions

The redshift of $z=6.57$ for the radio-emitting AGN J2331$+$1129 was inferred
from the shape of the near-infrared continuum spectrum (Koptelova & Hwang,
2022). The lack of identifiable emission lines, as well as the flat radio
spectrum in the observed GHz range, the spectral energy distribution, and the
indication of variability led Koptelova & Hwang (2022) to classify the source
as a candidate BL Lac object, by far the most distant known to date. However,
since the redshift determination is based on locating the break in the
continuum spectrum and not emission lines, the value should be considered
approximate.

We performed high-resolution radio interferometric observations with the VLBA
to reveal whether there is a compact core–jet structure in J2331$+$1129\.
Nearly simultaneous dual-frequency ($1.6$\- and $4.9$-GHz) imaging showed a
compact, flat-spectrum radio core whose flux density matches the total flux
density, indicating that the entire radio emission is confined within $\sim
20$ pc. Via traditional nodding-style phase-referencing to a nearby ICRF3
radio quasar, we precisely determined the astrometric position of the source
with sub-mas accuracy.

The inferred maximum brightness temperature is about $5\times 10^{9}$ K,
confirming that this is an AGN source but providing no clear evidence for
Doppler-boosted radio jet emission. This leaves the BL Lac identification of
the source an open question. Comparing the brightness temperature of
J2331$+$1129 with those of other high-redshift ($z\ga 6$) blazars would be
critical for understanding this. However, currently the sample of known
sources with similar properties is very small. It is possible that the
observed jet properties in extremely distant radio sources differ from those
at lower redshifts, due to yet poorly understood evolutionary and
environmental effects. The rarity of $z\ga 6$ blazars makes each new discovery
and study valuable for our understanding of the behaviour of jetted SMBHs in
the early Universe. Observational limitations could also play a role, as
phase-referenced VLBI observations of weak radio sources may only provide an
upper limit of the resolved source size, and thus a lower limit to the
brightness temperature.

To verify the BL Lac nature of J2331$+$1129, sensitive VLBI observations at
higher frequencies and thus higher resolution might prove useful. Also,
densely time-sampled total flux density monitoring could possibly reveal rapid
variations indicative of high brightness temperature. Finally, it cannot be
excluded that the source is a misidentified BL Lac object and/or it is not at
extremely high redshift. This should be tested in the future with sensitive
spectroscopic observations, to look for emission lines and possibly re-
evaluate the redshift.

###### Acknowledgements.

The National Radio Astronomy Observatory is a facility of the National Science
Foundation operated under cooperative agreement by Associated Universities,
Inc. This work made use of the Swinburne University of Technology software
correlator (Deller et al., 2011), developed as part of the Australian Major
National Research Facilities Programme and operated under licence. This work
was supported by the Hungarian National Research, Development and Innovation
Office (OTKA K134213 & PD146947). This project has received funding from the
HUN-REN Hungarian Research Network. T.A. thanks for the financial support from
the Pinghu Laboratory. E.K. thanks for support from Ministry of Science and
Technology of Taiwan grants MOST 109-2112-M-008-021-MY3 and
112-2112-M-008-017-MY3. Y.Z. is supported by the National SKA Programme of
China (grant no. 2022SKA0120102), Shanghai Sailing Program (grant no.
22YF1456100).

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