11institutetext: IRAP, Université de Toulouse, CNRS, CNES, UPS, 31400
Toulouse, France
11email<EMAIL_ADDRESS><EMAIL_ADDRESS>22institutetext:
Institute of Astronomy, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium
33institutetext: Institute for Frontiers in Astronomy and Astrophysics,
Beijing Normal University, Beijing 102206, China 44institutetext: Department
of Astronomy, Beijing Normal University, Beijing 100875, China
55institutetext: School of Physics and Astronomy, The University of
Birmingham, UK, B15 2TT

# Internal magnetic fields in 13 red giants detected by asteroseismology

Gang Li (李刚) 1122 Sébastien Deheuvels 11 Tanda Li (李坦达) 334455 Jérôme Ballot
11 François Lignières 11

###### Abstract

Context. Magnetic fields affect stars at all evolutionary stages. While
surface fields have been measured for stars across the HR diagram, internal
magnetic fields remain largely unknown. The recent seismic detection of
magnetic fields in the cores of several Kepler red giants has opened a new
avenue to better understand the origin of magnetic fields and their impact on
stellar structure and evolution.

Aims. The goal of our study is to use asteroseismology to systematically
search for internal magnetic fields in red giant stars observed with the
Kepler satellite, and to determine the strengths and geometries of these
fields.

Methods. Magnetic fields are known to break the symmetry of rotational
multiplets. In red giants, oscillation modes are mixed, behaving as pressure
modes in the envelope and as gravity modes in the core. Magnetism-induced
asymmetries are expected to be stronger for gravity-dominated modes than for
pressure-dominated modes, and to decrease with frequency. Among Kepler red
giants, we searched for stars that exhibit asymmetries satisfying these
properties.

Results. After collecting a sample of $\sim$2500 Kepler red giant stars with
clear mixed-mode patterns, we specifically searched for targets among
$\sim$1200 stars with dipole triplets. We identified 13 stars exhibiting clear
asymmetric multiplets and measured their parameters, especially the asymmetry
parameter $a$ and the magnetic frequency shift $\delta\nu_{\mathrm{g}}$. By
combining these estimates with best-fitting stellar models, we measured
average core magnetic fields ranging from $\sim$20 to $\sim$150 kG,
corresponding to $\sim 5\%$ to $\sim 30\%$ of the critical field strengths. We
showed that the detected core fields have various horizontal geometries, some
of which significantly differ from a dipolar configuration. We found that the
field strengths decrease with stellar evolution, despite the fact that the
cores of these stars are contracting. Additionally, even though these stars
have strong internal magnetic fields, they display normal core rotation rates,
suggesting no significantly different histories of angular momentum transport
compared to other red giant stars. We also discuss the possible origin of the
detected fields.

###### Key Words.:

asteroseismology – stars: magnetic field - stars: rotation

##  1 Introduction

Understanding the creation and evolution of magnetic fields is one of the main
challenges faced by modern stellar physics. An important effect of magnetic
fields on stellar evolution is that they are efficient at transporting angular
momentum (Cantiello et al., 2014; Rüdiger et al., 2015; Fuller et al., 2019;
Gouhier et al., 2022). Thus, they influence the internal rotation of stars,
and in turn the transport of chemical elements. Surface magnetic fields have
been observed across the Hertzsprung-Russell diagram (Landstreet, 1992; Donati
& Landstreet, 2009). These measurements, together with numerical simulations,
establish the presence of dynamo-generated magnetic fields in convective
regions (Donati & Landstreet, 2009; Aurière et al., 2015). A small fraction
($5-10\%$) of intermediate- and high-mass stars with radiative envelopes
harbours strong kilogauss surface fields that remain stable over decades (Wade
et al., 2012; Braithwaite & Spruit, 2017). These fields, which are thought to
result from the star formation process, can subsist thanks to negligible ohmic
diffusion and are potential progenitors for the magnetic white dwarfs and
neutron stars (Ferrario et al., 2020). Another potentially widespread class of
magnetic intermediate-mass stars has been identified with the detection of
$\sim\\!\\!1$ Gauss fields in a few stars (Lignières et al., 2009; Blazère et
al., 2016). Although it seems plausible that magnetic fields pervade much of
stellar interior plasmas, the absence of direct measurements of internal
fields has posed a significant obstacle to the study of their properties and
their impact on stellar evolution. Fortunately, we have asteroseismology to
help us explore the interior of stars (e.g. Aerts et al., 2010).

Asteroseismology has yielded measurements of various physical processes within
stars, one of which is stellar rotation at various stages of their evolution:
on the main sequence (e.g., Kurtz et al. 2014, Benomar et al. 2015, Van Reeth
et al. 2016), in the subgiant and red giant branch (RGB) phases (Beck et al.
2012, Mosser et al. 2012b, Deheuvels et al. 2014, Triana et al. 2017, Gehan et
al. 2018, Deheuvels et al. 2020, Kuszlewicz et al. 2023), in the core-He
burning phase (Mosser et al. 2012b, Deheuvels et al. 2015) and in white dwarfs
(Hermes et al. 2017). In all of these phases, it was concluded that stars
rotate more slowly than predicted by theoretical models (Zahn, 1992;
Eggenberger et al., 2012; Ceillier et al., 2013; Marques et al., 2013;
Ouazzani et al., 2019; Li et al., 2020). This shows that there must be
additional not-yet-identified processes that efficiently carry angular
momentum inside stars, beyond the classical hydrodynamic processes (Cantiello
et al., 2014; Fuller et al., 2014; Belkacem et al., 2015; Spada et al., 2016;
Pinçon et al., 2016; Eggenberger et al., 2017, 2019). One of the main
solutions proposed is the presence of magnetic fields in radiative cores.

The advent of space missions partly dedicated to asteroseismology has yielded
the high frequency resolution and photometric accuracy required to
characterise stellar oscillation modes. Thanks to the Kepler mission (Borucki
et al., 2010), solar-like oscillators have been discovered in tens of
thousands of stars (Bedding et al., 2010; Yu et al., 2018). Their oscillations
are excited stochastically by the outer envelope convection similar to the
Sun. Most of these solar-like oscillators are post-main-sequence stars that
exhibit mixed dipole ($l=1$) modes, which arise from the coupling between the
outer pressure modes and the interior gravity modes (Bedding, 2014). Mixed
modes enable us to probe the physics from the stellar core to surface, for
example, to distinguish evolutionary stages (Bedding et al., 2011; Mosser et
al., 2011), infer previous processes such as mergers or mass transfers
(Deheuvels et al., 2022; Rui & Fuller, 2021; Li et al., 2022c), or measure the
internal rotation, as mentioned above. Around 20% of red giants exhibit
suppressed dipole mixed modes (Mosser et al. 2012a, García et al. 2014, Stello
et al. 2016). It was suggested that this phenomenon could be caused by central
magnetic fields exceeding the critical field intensity $B_{\rm c}$ above which
magneto-gravity waves no longer propagate in the core (Fuller et al., 2015;
Stello et al., 2016; Rui & Fuller, 2023). However, this interpretation is
still a topic of debate (Mosser et al., 2017a; Loi, 2020).

From a theoretical perspective, it has been known for decades that magnetic
fields impact stellar oscillations (Gough & Thompson, 1990; Hasan et al.,
2005). Similarly to rotation, they break the degeneracy of oscillation modes
with same degree $l$ but different azimuthal order $m$. Rotational effects
alone produce multiplets (triplets for $l=1$ modes) that are generally
symmetric with respect to the central $m=0$ component when the rotation rate
is not too fast (that is, when second-order effects are negligible). The
effects of magnetic fields on mixed modes in red giants has been addressed,
considering the simple case of dipolar fields with different radial profiles,
either aligned with the rotation axis (Gomes & Lopes 2020, Mathis et al. 2021,
Bugnet et al. 2021) or inclined (Loi, 2021; Mathis & Bugnet, 2023). These
studies showed that magnetic fields are expected to break the symmetry of
rotational multiplets, and they produced estimates of the minimal field
intensities required to detect magnetic asymmetries, defined as
$\delta_{\mathrm{asym}}=\nu_{m=-1}+\nu_{m=1}-2\nu_{m=0}$ (Deheuvels et al.
2017).

Observation breakthrough has only been recently achieved. Li et al. (2022a)
reported the detection of clear asymmetries in the $l=1$ multiplets of three
Kepler red giants, which they found could only be accounted for by the
presence of strong magnetic fields in their cores. For this purpose, they
extended previous theoretical works to magnetic fields with arbitrary
configurations and found that the magnetic asymmetries of multiplets can be
either positive or negative depending on the field topology (all the
configurations studied before them yielded positive asymmetries). They
measured radial field intensities ranging from 30 to 130 kG in these stars,
and placed constraints on their topology. If the magnetic field is strong
enough, it can also significantly modify the regular spacing of g-mode period,
which can be used to detect them (Li et al. 2022a; Bugnet 2022). Deheuvels et
al. (2023) thus detected even stronger core fields in 11 Kepler red giants,
with intensities that are comparable to the critical field strength $B_{\rm
c}$.

These studies naturally raise the question of the prevalence of magnetic red
giants, which can shed light on the origin of these fields. In this study, we
systematically searched for asymmetries in the rotational multiplets of Kepler
red giants with detected oscillations. The paper is organised as follows. In
Sect. 2, we present the method we used to search for multiplet asymmetries
among Kepler red giants and we list the underlying assumptions. This leads us
to identify 13 targets with multiplet asymmetries that exhibit all the
features expected in the presence of an internal magnetic field. In Sect. 3,
we fit asymptotic expressions of mixed modes including rotational and magnetic
perturbations to the observations for these stars. We thus obtain estimates of
the average field strength in the core, as well as constraints on its
horizontal topology. In Sect. 4, we discuss the implications of these results
for the origin and evolution of internal magnetic fields, and their impact on
angular momentum transport. Sect. 5 is dedicated to conclusions.

##  2 Systematic search for multiplet asymmetries in Kepler data

###  2.1 Assumptions

The power spectra of mixed modes in red giants are complex. Specific methods
were derived based on asymptotic expressions of mixed modes (Shibahashi 1979,
Unno et al. 1989) to identify the modes and recover general properties of p
and g modes. In this study, we used methods that are derived from those
prescribed by Mosser et al. (2015a), Vrard et al. (2016) and Gehan et al.
(2018) in order to search for asymmetric multiplets within Kepler data. These
methods assume that rotational multiplets are symmetric, so they needed to be
adapted. They also make use of the regularity in the period spacings of
asymptotic g modes. For field intensities comparable to those found by Li et
al. (2022a), this assumption remains approximately correct. However, stronger
fields can significantly modify this regularity, as already mentioned in
Section 1, and it is likely that the methods that we use in this study are
ill-suited to detect such strong fields. This introduces an observational
bias, which is further discussed in Section 4.6.

We also assume that the effects of non-axisymmetry of the magnetic field on
oscillations are small. If it is not the case, multiplets can be split into
$(2l+1)^{2}$ components, instead of $(2l+1)$ in the axysymmetric case (Gough &
Thompson 1990,Loi 2021,Li et al. 2022a). Dipole multiplets can thus have up to
nine components instead of three. Li et al. (2022a) have shown that this
effect arises only if the ratio between the magnetic frequency shift and the
rotational frequency shift exceeds unity. We note that this was not the case
for the three red giants studied in Li et al. (2022a) (we found ratios that do
not exceed $\sim 0.6$). For these stars, the magnetic perturbations of the
oscillation frequencies are expected to be indistinguishable, whether the
field is axisymmetric or not. In this study, we restrict our search to stars
in the same regime, postponing the search for stars showing the seismic
signature of non-axisymmetric magnetic fields to a future work. This also
introduces an observational bias (see Section 4.6).

The following subsections describe the different steps of the method that we
applied to search for multiplet asymmetries, in the framework exposed above.
The results obtained in this section serve as the basis for the measurements
of magnetic field strengths and topology reported in Section 3.

###  2.2 Data reduction and sample selection

We used the Kepler 4-yr long-cadence data to calculate the power spectra,
which were downloaded from the Mikulski Archive for Space Telescopes (MAST,
https://archive.stsci.edu). We calculated the power spectra density (PSD)
(Lomb, 1976; Scargle, 1982; Kjeldsen & Bedding, 1995), the global
asteroseismic parameters ($\nu_{\mathrm{max}}$, $\Delta\nu$), and the
background properties following the processes used in the SYD pipeline (Huber
et al., 2009; Chontos et al., 2021).

We visually inspected about 8000 red giant branch (RGB) stars reported by Yu
et al. (2018) and Gehan et al. (2018) to select the stars with good patterns
of $l=1$ mixed modes. During the visual inspection, our primary focus is on
identifying stars that exhibit clear and distinct peaks between each $l=0$ and
$l=2$ p mode. Some stars do not display any discernible peaks in their mixed-
mode regions, a phenomenon referred to as ‘suppressed $l=1$ mixed modes’ as
mentioned in the Introduction. Additionally, the pattern of mixed modes
becomes unclear for more evolved red giants because of their smaller
asymptotic period spacing and the effects of radiative damping, which become
large for g-dominated modes (e.g. Grosjean et al., 2014). To ensure the
selection of stars with well-defined mixed-mode features, we excluded red
giant stars in such unfavourable cases, resulting in a cutoff for $\Delta\nu$
at approximately $7\,\mathrm{\mu Hz}$. Consequently, we have identified and
retained around 2500 RGB stars for further investigation.

###  2.3 Identification of azimuthal order $m$

####  2.3.1 Streched periods

The first step of the method consists of identifying the azimuthal order $m$
of the detected modes. For this purpose, it is convenient to use the so-called
“stretched” periods introduced by Mosser et al. (2015a). These authors have
shown that the period spacing between consecutive dipole mixed modes can be
expressed as $\Delta P=\zeta\Delta\Pi_{1}$, where $\zeta$ represents the
fraction of the g-mode inertia over the whole inertia (Goupil et al., 2013)
($\zeta$ tends to unity for pure g modes and to zero for pure p modes), and
$\Delta\Pi_{1}$ is the asymptotic period spacing of pure g modes. The
frequencies $\nu$ are transformed into the so-called “stretched periods”
$\tau$ using the differential equation

$\mathrm{d}\tau=\frac{1}{\zeta}\frac{\mathrm{d}\nu}{\nu^{2}},$ (1)

so that the mixed modes are equally spaced by $\Delta\Pi_{1}$. When
representing the stretched periods in an échelle diagram folded with
$\Delta\Pi_{1}$, the modes of the same azimuthal order $m$ are expected to
align nearly vertically in this diagram.

####  2.3.2 Asymptotic expression of $\zeta$

To estimate the value of $\zeta$ for the detected modes, we used an asymptotic
expression of this quantity, as is now commonly done (e.g., Mosser et al.
2015a, Li et al. 2022a). We briefly recall the steps of the procedure.
Following the work of Shibahashi (1979) and Unno et al. (1989), the implicit
asymptotic relation of mixed modes is expressed as

$\tan\theta_{\mathrm{p}}=q\tan\theta_{\mathrm{g}},$ (2)

where $q$ is the coupling factor between p and g components of the modes (e.g.
Mosser et al., 2017b), and $\theta_{\mathrm{p}}$ and $\theta_{\mathrm{g}}$ are
the phases for the pure p and g modes.

The phase of the pure p modes is written as

$\theta_{\mathrm{p}}=\pi\frac{\nu-\nu_{\mathrm{p}}}{\Delta\nu(n_{\mathrm{p}})},$
(3)

where $\nu_{\mathrm{p}}$ is the pure p-mode frequency and
$\Delta\nu(n_{\mathrm{p}})$ is the local frequency separation at the radial
order $n_{\mathrm{p}}$ (Mosser et al., 2015b). We used the detected $l=0$ and
$l=2$ mode frequencies 111We fit a Lorentzian profile to each detected $l=0$
and $l=2$ modes to obtain their frequencies (Anderson et al., 1990). to derive
an expression of $\nu_{\mathrm{p}}$ for dipole mode frequencies. From
asymptotic expressions,

$\nu_{\mathrm{p}}=\left[n_{\mathrm{p}}+\frac{l}{2}+\varepsilon+\frac{\alpha}{2}(n_{\mathrm{p}}-n_{\mathrm{max}})^{2}\right]\Delta\nu-l(l+1)D,$
(4)

where $\varepsilon$ is the phase term and
$n_{\mathrm{max}}=\nu_{\mathrm{max}}/\Delta\nu$ is the radial order at the max
power frequency. The term $D$ describes the small separation $\delta\nu_{02}$
between $l=0$ and $2$ modes. We fit the expression given by Eq. 4 to the
frequencies of the detected $l=0$ and $l=2$ modes. The fit results are listed
in Table 1 for the stars that are discussed in the following sections of the
paper. Since the pure $l=1$ p modes are not observable, and the small
separation ratio $\delta\nu_{02}/\delta\nu_{01}$ deviate from the solar value
(three) with stellar evolution (Lund et al., 2017), we add an extra free
parameter $f_{\mathrm{shift}}$ to the expression of $\nu_{\mathrm{p}}$ given
in Eq. 4 only for dipole mode frequencies, that is:

$\nu_{\mathrm{p},l=1}=\nu_{\mathrm{p}}+f_{\mathrm{shift}}.$ (5)

We use $\nu_{\mathrm{p},l=1}$ to compute the dipole p-mode phase
$\theta_{\mathrm{p}}$ in eq. 3. The initial guess of $f_{\mathrm{shift}}$ is
0.6 $\mu\mathrm{Hz}$ and it will be set to be a free parameter in further
fitting steps.

The phase of the pure g modes is

$\theta_{\mathrm{g}}=\frac{\pi}{\Delta
P_{\mathrm{g}}}\left(\frac{1}{\nu}-P_{\mathrm{g}}\right),$ (6)

in which $\Delta P_{\mathrm{g}}$ is the local period spacing and
$P_{\mathrm{g}}$ is the pure g-mode period (Mosser et al., 2015b). Without
considering any perturbations by rotation and magnetism, the pure g-mode
period is equally spaced in period, shown as

$P_{\mathrm{g}}=\Delta\Pi_{1}\left(n_{\mathrm{g}}+\varepsilon_{\mathrm{g}}\right)$
(7)

for $l=1$ g modes, where $n_{\mathrm{g}}$ is the g-mode radial order and
$\varepsilon_{\mathrm{g}}$ is the g-mode phase.

As shown by Mosser et al. (2015b) and Hekker & Christensen-Dalsgaard (2017),
the $\zeta$ function can then be expressed as

$\zeta=\left[1+\frac{\nu^{2}}{q}\frac{\Delta\Pi_{1}}{\Delta\nu}\frac{1}{\frac{1}{q}\sin^{2}\theta_{\mathrm{p}}+\cos^{2}\theta_{\mathrm{p}}}\right]^{-1}.$
(8)

####  2.3.3 Identification of $m$ with stretched échelle diagrams

For all the stars of our sample, we selected the peaks with signal-to-noise
ratio larger than ten to plot the initial stretched échelle diagram (eqs. 1
and 8), which is used to measure the asymptotic period spacing $\Delta\Pi_{1}$
and identify azimuthal orders. In this step, we set $q=0.15$, as a typical
value for hydrogen-shell-burning (HSB) stars (Mosser et al., 2017b) and varied
$\Delta\Pi_{1}$ to produce vertical ridges in the stretched échelle diagram.
The modes with different $m$ are not exactly equally spaced by the period
spacing $\Delta\Pi_{1}$. As shown by Mosser et al. (2015a), they have slightly
different spacings $\Delta\tau_{m}$ in the stretched period diagram

$\Delta\tau_{m}=\Delta\Pi_{1}\left(1+2m\frac{\mathcal{N}}{\mathcal{N}+1}\frac{\delta\nu_{\mathrm{rot,core}}}{\nu_{\mathrm{max}}}\right),$
(9)

with $\mathcal{N}=\Delta\nu/(\Delta\Pi_{1}\nu_{\rm max}^{2})$, where
$\delta\nu_{\mathrm{rot,core}}$ is the rotational splitting of pure g modes,
that is, half the core rotation frequency. Multiple ridges may thus appear. We
measured their period spacing $\Delta\tau_{m}$ by slightly changing
$\Delta\Pi_{1}$ and identified the $m$ value as described by Eq. 21 in Mosser
et al. (2015a) ($m=1$ or $-1$ for doublets, and $m=1$, $0$, $-1$ for
triplets). About 1200 stars with clear triplets were used for the subsequent
analysis, and we also obtained about 800 stars with doublets (their
asymmetries cannot be measured due to the lack of $m=0$ modes). The rest
$\sim$500 stars do not show splittings.

Figure 1: Stretched échelle diagram of KIC 5792889 that does not show any
magnetism-induced perturbation. The x-axis is the stretched periods $\tau$
modulo $\Delta\Pi_{1}\approx 81.6\,\mathrm{s}$. The peaks with S/N¿10 are
shown by the grey points ($l=0$ and $l=2$ modes have been removed). The green
‘+’ stands for the $m=1$ modes. The red ‘-’ stands for the $m=-1$ modes. The
blue ‘$\bullet$’ shows the $m=0$ modes. The best-fitting results are plotted
by the cross.

Figure 1 shows the stretched échelle diagram of KIC 5792889, which is selected
randomly from our sample. Three distinct ridges are visible, each marked by a
different symbol denoting their azimuthal order (refer to the caption of Fig.
1). However, this star does not show magnetism-induced asymmetries, as a
result, the ridge corresponding to $m=0$ appears halfway between the $m=1$ and
$m=-1$ ridges.

In the presence of a core magnetic field, the oscillation modes undergo a
frequency shift that depends on $|m|$ (see Sect. 3). If the core field is
strong enough, we anticipate that it can modify the ordering of the
$m$-components in a multiplet. This would invalidate our identification of $m$
base on Eq. 9. For all the stars where multiplet asymmetries were detected, we
thus investigated alternate identifications of $m$, assigning $m=0$ to the
external components of the multiplet. In all these cases, these alternate
identifications led to poor fits to the observations, so we ruled out the
possibility of having a different ordering of the components in the
multipltets.

###  2.4 Identification of the rotational multiplets

In order to measure multiplet asymmetry, we then needed to identify modes that
belong to a same multiplet, that is, modes that share common values of $l$ and
$n$ but have different values of $m$. This step can be complicated when the
rotational splitting is comparable to or larger than the frequency spacing
between modes of consecutive radial order $n$. In such cases, the nearest
three modes no longer form a single multiplet, making the identification more
difficult, as illustrated in Figure 2 for KIC 7749842.

Figure 2: Stretched échelle diagram of KIC 9467102, which does not show any
magnetism-induced perturbation. The symbols are the same as fig. 1. We show
this star as an example because its splittings overlap seriously.

To identify rotational multiplets, we fit an asymptotic expression of mixed
modes including rotational effects to the detected modes. For given values of
$\Delta\Pi_{1}$, $\varepsilon_{\rm g}$, $q$, and $f_{\rm shift}$, the
frequencies of the unperturbed modes are given by solving Eq. 2. We then add
rotational splittings to these modes. The rotational splittings
$\delta\nu_{\mathrm{R}}$ can be expressed as

$2\pi\delta\nu_{\mathrm{R}}=0.5\Omega_{\mathrm{core}}\zeta+\Omega_{\mathrm{env}}\left(1-\zeta\right),$
(10)

where $\Omega_{\mathrm{core}}$ and $\Omega_{\mathrm{env}}$ are the mean
rotation rates (in unit of angular frequency) in the core and the outer
envelope (Goupil et al., 2013; Deheuvels et al., 2014). This expression leads
to symmetric rotational multiplets, contrary to what we generally expect for
red giants harbouring core magnetic fields. However, if the effects of non-
axisymmetry of the magnetic field on the frequency shifts are negligible (as
we have assumed in Sect. 2.1), the mode frequencies of azimuthal order $m=\pm
1$ are affected in the same way. In this case, the frequency spacing between
these two components remains equal to $2\delta\nu_{\mathrm{R}}$, as in the
non-magnetic case. We thus used only the $m=\pm 1$ modes to perform the fit.

We ran a Markov chain Monte Carlo (MCMC) method to optimise the parameters
using the python package emcee (Foreman-Mackey et al., 2013). There are in
total six parameters that will be optimised. We applied uniform priors to the
parameters and defined their ranges for optimisation in the MCMC as follows:

  1. 1.

Asymptotic period spacing $\Delta\Pi_{1}$.
$[\Delta\Pi_{\mathrm{1,init}}-0.5\,\mathrm{s},\Delta\Pi_{\mathrm{1,init}}+0.5\,\mathrm{s}]$
where $\Delta\Pi_{\mathrm{1,init}}$ is the initial guess of the period spacing
from the stretched échelle diagram.

  2. 2.

Coupling factor $q$. [0.08, 0.25].

  3. 3.

Phase of g mode $\varepsilon_{g}$. [-0.1, 1.1].

  4. 4.

Dipole ($l=1$) frequency shift $f_{\mathrm{shift}}$. [0.3$\,\mathrm{\mu Hz}$,
1.0$\,\mathrm{\mu Hz}$].

  5. 5.

Core angular frequency $\Omega_{\mathrm{core}}$. [0$\,\mathrm{\mu Hz}$,
20$\,\mathrm{\mu Hz}$].

  6. 6.

Envelope angular frequency $\Omega_{\mathrm{env}}$. [-1$\,\mathrm{\mu Hz}$,
1$\,\mathrm{\mu Hz}$].

The prior ranges of these parameters were determined based on several previous
analyses of large samples of red giant stars (Mosser et al., 2017b, 2018;
Gehan et al., 2018; Triana et al., 2017). The MCMC algorithm maximises the
likelihood function defined as follows:

$\ln
L=-\frac{1}{2}\sum_{m=1,0,-1}\sum_{i}\left[\frac{(\nu_{m,i}^{\mathrm{obs}}-\nu_{m,i}^{\mathrm{cal}})^{2}}{\sigma_{m,i}^{2}}+\ln\left(2\pi\sigma_{m,i}^{2}\right)\right],$
(11)

where $\nu_{m,i}^{\mathrm{obs}}$ is the $i^{\mathrm{th}}$ observed frequency
with azimuthal order of $m$ and $\nu_{m,i}^{\mathrm{cal}}$ is the calculated
frequency. At this stage, the observed frequencies are estimated as the mean
of the nearby points in the PSD whose signal-to-noise ratio (S/N) is greater
than 10, which is more conservative than the criterion applied in previous
studies, such as Mosser et al. (2015b).

Currently, we assume that the uncertainty in the observed frequency
$\sigma_{m,i}$ is 0.02 $\mu$Hz. More proper estimates of the mode frequencies
and their uncertainties are obtained in section 2.5 by fitting Lorentzian
profiles to the PSD.

We ran 14 parallel chains with length of 5000 steps. The first 50% samples are
discarded. Finally, we obtained the best-fitting results that give the
identification of the multiplets and allow us to run an automated algorithm
(in section 2.5) to measure the asymmetric splittings We show the best-fitting
results of KIC 9467102 in Fig. 15. The identified triplets are marked by the
horizontal red lines in each panel. Even though there is significant overlap
between multiplets, we still can distinguish them. We also find that the
splitting identification works well even for the stars with asymmetric
splittings.

###  2.5 Asymmetry measurement

We measured the asymmetries of the identified multiplets by fitting three
Lorentzian profiles, whose initial locations are given by the MCMC algorithm
in section 2.4. In this fitting for the Lorentzian profile, we can determine
the following parameters: frequencies of components, linewidths assuming
identical for three components, amplitudes, and inclinations. The amplitudes
of the three components were determined by a relation that was modified by
inclination (Gizon & Solanki, 2003), and the best-fitting results were
obtained by maximising the likelihood function defined by Anderson et al.
(1990). We allowed the $m=0$ component to shift freely to reproduce the
asymmetry.

Among the stars showing significant multiplet asymmetries, we selected those
that share the features expected for magnetic asymmetries, namely:

  * •

They should have the same sign (either positive or negative) for all
multiplets in a given star.

  * •

They should decrease with frequency (in absolute value).

  * •

They should be larger for g-dominated modes than for p-dominated modes.

Finally, 13 stars were found to show magnetism-induced asymmetries, including
the three stars reported by Li et al. (2022a). Here we show KIC 5696081 as an
example. The three panels in Fig. 3 show three continuous asymmetric
splittings, whose x-axes are aligned by the $m=1$ and $m=-1$ mode frequencies.
The modes in the top and the bottom panels are g-dominated, so they show
narrower linewidths and larger asymmetries. While the mode in the middle panel
is p-dominated, hence it has wider linewidth and smaller asymmetry. Although
the asymmetries vary with different modes, all the asymmetries keep positive.
Figure 4 shows the variations in the asymmetry as a function of the frequency
in KIC 5696081. We find that they follow all the characteristics expected from
a magnetic perturbation (as listed in Sect. 1): the asymmetries are all
positive in this star, decrease with frequency, and are smaller for
p-dominated modes. We show all the asymmetry measurements in Appendix B.

Figure 3: Three continuous asymmetric splittings in KIC 5696081. The top and
bottom panels show g-dominated modes while the middle panel displays a
p-dominated one. Note that the x-axes of three panels are aligned by the $m=1$
and $-1$ mode frequencies. Figure 4: The asymmetries as a function of
frequency in KIC 5696081. The dark fringes show where the p-dominated modes
are, while the white background shows the locations of g-dominated modes.

##  3 Magnetic perturbation and field strength measurement

###  3.1 Magnetic perturbation

For the stars identified as showing multiplet asymmetries of magnetic origin
in Sect. 2.5, we estimated the properties of the field that could reproduce
the observations. For this purpose, we followed the approach that we proposed
in Li et al. (2022a). We again solved Eq. 2 to obtain the asymptotic
expression of mixed mode frequencies, but here the frequencies of pure p and g
modes include perturbations arising from rotation and magnetic field.

In the case of axisymmetric fields, or the non-axisymmetric effects are
negligible, the frequency perturbation of pure g modes caused by both
magnetism and rotation is given by

$\delta\nu_{\mathrm{g\leavevmode\nobreak\
mode},m=0}=\left(1-a\right)\delta\nu_{\mathrm{g}}\left(\frac{\nu_{\mathrm{max}}}{\nu}\right)^{3}$
(12)

for $m=0$ modes, and

$\delta\nu_{\mathrm{g\leavevmode\nobreak\ mode},m=\pm
1}=\left(1+\frac{a}{2}\right)\delta\nu_{\mathrm{g}}\left(\frac{\nu_{\mathrm{max}}}{\nu}\right)^{3}\mp\frac{\Omega_{\mathrm{core}}}{4\pi}$
(13)

for $m=\pm 1$ modes (Li et al., 2022a). In Eqs. 12 and 13,
$\delta\nu_{\mathrm{g}}$ is the magnetic shift for pure g modes at the
frequency at maximum power of the oscillations. The asymmetry parameter $a$ is
a dimensionless average of $B_{r}^{2}$ in the oscillation cavity weighted by
the second order Legendre polynomial $P_{2}(\cos\theta)$:

$a=\frac{\displaystyle\int_{r_{\mathrm{i}}}^{r_{\mathrm{o}}}K(r)\iint
B_{r}^{2}P_{2}(\cos\theta)\sin\theta\,\hbox{d}\theta\hbox{d}\phi\,\hbox{d}r}{\displaystyle\int_{r_{\mathrm{i}}}^{r_{\mathrm{o}}}K(r)\iint
B_{r}^{2}\sin\theta\,\hbox{d}\theta\hbox{d}\phi\,\hbox{d}r}.$ (14)

It verifies $-0.5<a<1$, its exact value depending on the latitudinal
distribution of $B_{r}^{2}$ in the oscillation cavity (Li et al., 2022a).

The pure g-mode periods in Eq. 7 can then be rewritten as

$P^{\prime}_{\mathrm{g},m}=\left(\frac{1}{P_{\mathrm{g}}}+\delta\nu_{\mathrm{g\leavevmode\nobreak\
mode},m}\right)^{-1}.$ (15)

For pure p modes, the perturbation only arises from rotation, since the
magnetic field being buried inside the star acts mainly on the g-mode parts of
mixed modes. The effect on p modes is still negligible even if the magnetic
field extends to the p-mode cavity (Mathis et al., 2021).Therefore, the pure
p-mode frequency in Eq. 4 is rewritten as

$\nu^{\prime}_{\mathrm{p},m}=\nu_{\mathrm{p}}-m\frac{\Omega_{\mathrm{env}}}{2\pi}.$
(16)

We then solved the asymptotic expressions for $m=1$, $0$, and $-1$
respectively,

$\tan\theta^{\prime}_{\mathrm{p},m}=q\tan\theta^{\prime}_{\mathrm{g},m}\leavevmode\nobreak\
\mathrm{for}\leavevmode\nobreak\ m=1,\leavevmode\nobreak\
0,\leavevmode\nobreak\ -1,$ (17)

where $\theta^{\prime}_{\mathrm{p},m}$ and $\theta^{\prime}_{\mathrm{g},m}$
are the perturbed phases for pure p and g modes. Their expressions are similar
to those of the unperturbed phases $\theta_{\mathrm{p}}$ and
$\theta_{\mathrm{g}}$ given by Eqs. 3 and 6, but the frequencies of pure p and
g modes are now replaced with the perturbed expressions given by Eqs. 12, 13,
15, and 16. Eq. 17 can then be solved to obtain the perturbed mixed mode
frequencies for any set of parameters characterising
$(\Delta\Pi_{1},q,\varepsilon_{\rm
g},f_{\mathrm{shift}},\Omega_{\mathrm{core}},\Omega_{\mathrm{env}},a,\delta\nu_{\mathrm{g}})$.

###  3.2 Fit to the observations

In order to fit our asymptotic expression of mixed modes including rotational
and magnetic perturbations to the observations, we ran an MCMC algorithm
similar to the one used in Sect. 2.4 The priors for the first six parameters
($\Delta\Pi_{1},q,\varepsilon_{\rm
g},f_{\mathrm{shift}},\Omega_{\mathrm{core}},\Omega_{\mathrm{env}}$) are the
same as in Sect. 2.4, and for the two additional parameters $a$ and
$\delta\nu_{\mathrm{g}}$, we considered

  * •

Asymmetry parameter $a$: uniform prior [-0.5, 1.0].

  * •

Magnetic shift $\delta\nu_{\mathrm{g}}$: uniform prior [0$\,\mathrm{\mu Hz}$,
1$\,\mathrm{\mu Hz}$].

The same likelihood function as eq. 11 was used with the uncertainty defined
as the quadratic summation of model and observation errors:

$\sigma_{m,i}^{2}=\left(\sigma_{m,i}^{\mathrm{mod}}\right)^{2}+\left(\sigma_{m,i}^{\mathrm{obs}}\right)^{2}.$
(18)

When fitting asymptotic expressions of mixed modes to the observations, we
found that the optimal solutions have a typical residual spread of 0.02
$\mathrm{\mu Hz}$, which we consider as some residual spread caused by the WKB
approximation since the asymptotic expression is not expected to be exact.
Hence we included $\sigma_{m,i}^{\mathrm{mod}}=0.02\,\mathrm{\mu Hz}$ as
additional white noise into the frequency uncertainties to account for the
model uncertainty, assuming they are completely uncorrelated. The
uncertainties of the mode frequencies ($\sigma_{m,i}^{\mathrm{obs}}$) were
obtained in Section 2.5.

In the MCMC algorithm, 18 parallel chains were used with length of 4000 steps,
and we discarded the first 50% sample results. The posterior distributions of
the eight parameters are shown in Fig. 5. We obtained good agreement between
the formula including both rotational and magnetic perturbation and the
observed asymmetric splittings. To illustrate this, Figure 6 displays the
stretched échelle diagram of KIC 5696081. The dipole triplets form three
nearly vertical ridges, and the $m=0$ ridge does not fall exactly halfway
between $m=1$ and $m=-1$ ridges due to the magnetic perturbations. Our best-
fitting frequencies (shown by the crosses) follow the observations well,
showing a good agreement between the theory and observations. We display the
stretched échelle diagrams of the 13 stars in Appendix B. They show that our
fits are in good agreement with the observations and reproduce the asymmetries
well.

The best-fitting parameters for all the 13 stars are listed in Table 2. Here
we show KIC 5696081 as an example. The corner diagram of the MCMC result is
shown in Fig. 5, where the best-fitting result is found with
$a=0.74_{-0.18}^{+0.17}$ and
$\delta\nu_{\mathrm{g}}=0.104_{-0.019}^{+0.03}\,\mu\mathrm{Hz}$. In Fig. 5, as
well as the corner diagrams for the other stars in Appendix B, we identify
correlations between several parameters. As in the case of non-magnetic red
giants, the measurement of $\Delta\Pi_{1}$ is strongly anti-correlated with
the measurement of $\varepsilon_{\mathrm{g}}$. This is a direct result from
the linear relation given by Eq. 7. We also find a clear correlation between
$\Delta\Pi_{1}$ and $\delta\nu_{\mathrm{g}}$. This can be understood as
follows: starting from a best-fit solution, if we increase the value of
$\delta\nu_{\mathrm{g}}$ (that is, if we increase the intensity of the field),
the magnetic frequency perturbations increase, which tends to decrease the
period spacing between consecutive g modes. Therefore, to correctly reproduce
the observations, the asymptotic period spacing of unperturbed g modes
$\Delta\Pi_{1}$ needs to be increased. Also, the parameters
$\delta\nu_{\mathrm{g}}$ and $a$ are found to be strongly anti-correlated.
Again, this was expected. Indeed, using Eqs. 12 and 13, we find that the
asymmetry of g-mode multiplets near $\nu_{\rm max}$ corresponds to
$3a\delta\nu_{\mathrm{g}}$. Therefore, if $\delta\nu_{\mathrm{g}}$ increases,
one needs to decrease $a$ to reproduce the observed asymmetries. Finally, we
observe a slight anti-correlation between the measurements of
$\Omega_{\mathrm{core}}$ and $\Omega_{\mathrm{env}}$. This can be understood
from Eq. 10 (even though this relation has not been used in our fits here): in
this linear relation, $\Omega_{\mathrm{core}}$ and $\Omega_{\mathrm{env}}$ are
related to the slope and the intercept, respectively, and the measurements of
these quantities are not independent.

The asymmetry parameter $a$ generally has larger uncertainties and broader
distributions within its prior range (from -0.5 to 1) compared to the other
parameters, suggesting that the constraints on $a$ are weaker. The parameter
$\delta\nu_{\mathrm{g}}$ often had an asymmetric distribution, while the other
parameters have symmetric distributions and generally tighter constraints.

These stars display normal values for the other parameters of the fits
($\Delta\Pi_{1}$, $q$, $\varepsilon_{\mathrm{g}}$, and $f_{\mathrm{shift}}$),
as shown in Appendix C. Despite the fact that we used non-informative priors
for $\varepsilon_{\mathrm{g}}$, we obtained results that are in line with
typical values for other red giants ($0.28\pm 0.08$ Takata, 2016; Mosser et
al., 2018).

Figure 5: The corner diagram of the magnetic fitting of KIC 5696081. The
vertical dashed lines mark the median values and $\pm 1\sigma$ ranges.  Figure
6: Same as fig. 1 but for KIC 5696081. Note that this star shows clear
asymmetries by magnetic field.

###  3.3 Stellar model and field strength

After obtaining the magnetic shift $\delta\nu_{\mathrm{g}}$, we can calculate
the magnetic field strength in the stellar interior. What we can measure about
the magnetic field strength is a weighted integral of the horizontal average
of the squared radial magnetic field, $\overline{B_{r}^{2}}$, given by (Li et
al., 2022a):

$\langle
B_{r}^{2}\rangle=\int_{r_{\mathrm{i}}}^{r_{\mathrm{o}}}K\left(r\right)\overline{B_{r}^{2}}\mathrm{d}r=\frac{16\pi^{4}\mu_{0}\delta\nu_{\mathrm{g}}\nu_{\rm
max}^{3}}{\mathcal{I}},$ (19)

with $K(r)$ the weight function and where $r_{\mathrm{i}}$ and
$r_{\mathrm{o}}$ are the inner and outer turning points of the g-mode cavity,
respectively, and $\mu_{0}$ is the vacuum permeability. The core factor
$\mathcal{I}$ is determined by the internal structure of the star:

$\mathcal{I}=\frac{\int_{r_{\mathrm{i}}}^{r_{\mathrm{o}}}\left(\frac{N}{r}\right)^{3}\frac{\mathrm{d}r}{\rho}}{\int_{r_{\mathrm{i}}}^{r_{\mathrm{o}}}\left(\frac{N}{r}\right)\,\mathrm{d}r},$
(20)

where $N$ is the buoyancy frequency and $\rho$ is the local density. The
weighted function,

$K\left(r\right)=\frac{\frac{1}{\rho}\left(\frac{N}{r}\right)^{3}}{\int_{r_{\mathrm{i}}}^{r_{\mathrm{o}}}\left(\frac{N}{r}\right)^{3}\frac{\mathrm{d}r}{\rho}},$
(21)

sharply peaks at the hydrogen-burning shell, with a much lower sensitivity in
the layers below.

To estimate the intensity of the detected magnetic fields, we needed to
characterise the internal structures of these stars (such as the buoyancy
frequency $N$ and the density profile $\rho$). For this purpose, we adopted
the seismology-modelling pipeline and the model grid introduced by Li et al.
(2022b) to search for the best-fitting models. The input constraints were
based on global parameters, such as effective temperature, luminosity, and
metallicity, which were reported by Berger et al. (2020) and listed in table
3. We also used the observed radial-mode frequencies given by Li et al.
(2022b), and the asymptotic period spacing of g modes ($\Delta\Pi_{1}$)
measured by this work as additional constraints. By applying the pipeline, we
determined the best-fitting structural model, whose parameters (masses, ages,
and radii) are listed in Table 3.

Using our best-fit stellar models, we could derive estimates of the core
factor $\mathcal{I}$, and we were thus able to obtain measurements of the
average radial magnetic fields $\langle B_{r}^{2}\rangle^{0.5}$ using Eq. 19
(see Table 3). We found field strengths ranging from about 20 kG to 150 kG.

Since this work and Li et al. (2022a) used different approaches in the stellar
modelling and the fitting of the asymptotic expression of mixed mode
frequencies to the observations, there are slight differences in the inferred
field strengths for the three stars that they have in common. We compared the
two sets of results and found that the field strengths derived by both works
are generally consistent, with slightly larger strengths obtained in this
work. For KIC 7518143 and KIC 8684542, they have consistent field strengths
within 1-$\sigma$ ranges. While for KIC 11515377, the field strength obtained
by this work is approximately 30% times larger than the strength by Li et al.
(2022a), though still within the 2-$\sigma$ range. The ratios between the
field strength and the critical field strength show good agreement between the
two sets of results, likely due to this ratio being more sensitive to the
stellar structure rather than the observed frequencies or the fitting
strategies.

Figure 7: Top panel: measured field strengths by this work and by Li et al.
(2022a). Bottom: the ratio between the measured field strengths and the
critical strengths. The dotted lines show 1:1 relations. Note that the value
of the field strength of KIC 7518143 reported by Li et al. (2022a) is smaller
than 41 kG, while we use the median here (20.5 kG).

In Appendix D, we also offer a linear relation between $\mathcal{I}$ and
$\Delta\Pi_{1}$. This relation can be used for model-independent calculations
of field strength, particularly when a best-fitting stellar model is
unavailable.

Table 1: The fit results of the p-mode asymptotic relations in eq. 4.  KIC |  $\nu_{\mathrm{max}}$, $\mu$Hz |  $\Delta\nu$, $\mu$Hz | $\varepsilon$ | $\alpha$ |  $D$, $\mu\mathrm{Hz}$
---|---|---|---|---|---
4458118 | $207.6\pm 1.6$ |  $16.442\pm 0.004$ |  $0.326\pm 0.003$ |  $0.0057\pm 0.0005$ | $0.3620\pm 0.0015$
5196300 | $207.6\pm 1.1$ |  $15.138\pm 0.003$ |  $0.340\pm 0.003$ | $0.00406\pm 0.00018$ | $0.3215\pm 0.0011$
5696081 | $248.0\pm 1.3$ |  $17.901\pm 0.004$ |  $0.362\pm 0.003$ | $0.00115\pm 0.00024$ | $0.3544\pm 0.0012$
6936091 | $94.3\pm 0.4$ |  $8.731\pm 0.004$ |  $0.174\pm 0.006$ |  $0.0057\pm 0.0005$ | $0.2090\pm 0.0011$
7009365 | $210.5\pm 1.2$ | $15.6082\pm 0.0029$ | $0.3332\pm 0.0027$ | $0.00679\pm 0.00019$ | $0.3147\pm 0.0011$
7518143 | $159.2\pm 0.5$ |  $12.394\pm 0.003$ |  $0.216\pm 0.003$ | $0.00388\pm 0.00021$ | $0.2555\pm 0.0012$
8540034 | $195.4\pm 0.7$ |  $15.108\pm 0.003$ | $0.2517\pm 0.0029$ | $0.00563\pm 0.00025$ | $0.3285\pm 0.0012$
8619145 | $130.0\pm 0.4$ |  $10.896\pm 0.004$ |  $0.163\pm 0.004$ |  $0.0097\pm 0.0004$ | $0.2428\pm 0.0012$
8684542 | $179.4\pm 0.7$ | $13.4921\pm 0.0027$ | $0.2741\pm 0.0028$ | $0.00345\pm 0.00020$ | $0.282\pm 0.0010$
9202471 | $218.5\pm 1.3$ |  $15.946\pm 0.004$ |  $0.358\pm 0.003$ |  $-0.0005\pm 0.0003$ | $0.3138\pm 0.0012$
9589420 | $109.9\pm 0.5$ |  $9.473\pm 0.004$ |  $0.187\pm 0.005$ |  $-0.0015\pm 0.0005$ | $0.1993\pm 0.0012$
10801792 | $199.7\pm 1.0$ | $15.0092\pm 0.0028$ | $0.3358\pm 0.0026$ | $0.00244\pm 0.00018$ | $0.326\pm 0.0010$
11515377 | $194.6\pm 0.5$ |  $14.739\pm 0.005$ |  $0.340\pm 0.005$ |  $0.0011\pm 0.0003$ | $0.3204\pm 0.0012$
Table 2: The best-fitting results of the $l=1$ mixed mode frequencies considering the magnetism-induced perturbations in eqs. 12, 13, and 16. The parameters are: period spacing $\Delta\Pi_{1}$, the coupling factor $q$, g-mode phase $\varepsilon_{\mathrm{g}}$, the frequency correction of $l=1$ pure p mode $f_{\mathrm{shift}}$, core and envelope rotation rates $\Omega_{\mathrm{core}}/2\pi$ and $\Omega_{\mathrm{env}}/2\pi$, the asymmetry parameter $a$, and the magnetic shift $\delta\nu_{\mathrm{g}}$.  KIC |  $\Delta\Pi_{1}$ (s) | $q$ | $\varepsilon_{\mathrm{g}}$ |  $f_{\mathrm{shift}}$ ($\mu$Hz) |  $\Omega_{\mathrm{core}}/2\pi$ ($\mu$Hz) |  $\Omega_{\mathrm{env}}/2\pi$ ($\mu$Hz) | $a$ |  $\delta\nu_{\mathrm{g}}$ ($\mu$Hz)
---|---|---|---|---|---|---|---|---
4458118 | $88.71_{-0.06}^{+0.06}$ | $0.1617_{-0.0025}^{+0.0025}$ | $0.27_{-0.03}^{+0.03}$ | $0.734_{-0.015}^{+0.016}$ | $0.867_{-0.020}^{+0.019}$ |  $0.019_{-0.022}^{+0.020}$ |  $0.42_{-0.26}^{+0.3}$ |  $0.023_{-0.012}^{+0.022}$
5196300 | $82.99_{-0.05}^{+0.07}$ | $0.1304_{-0.0020}^{+0.0019}$ | $0.306_{-0.04}^{+0.028}$ | $0.620_{-0.013}^{+0.014}$ | $1.251_{-0.012}^{+0.012}$ |  $0.031_{-0.014}^{+0.014}$ |  $0.71_{-0.22}^{+0.20}$ |  $0.080_{-0.018}^{+0.029}$
5696081 | $88.26_{-0.05}^{+0.07}$ | $0.1617_{-0.0019}^{+0.0017}$ | $0.375_{-0.029}^{+0.022}$ | $0.653_{-0.013}^{+0.014}$ | $0.826_{-0.014}^{+0.013}$ |  $0.035_{-0.014}^{+0.015}$ |  $0.74_{-0.18}^{+0.17}$ |  $0.104_{-0.019}^{+0.03}$
6936091 | $75.29_{-0.13}^{+0.16}$ | $0.132_{-0.008}^{+0.007}$ | $0.39_{-0.24}^{+0.21}$ | $0.550_{-0.024}^{+0.023}$ | $0.362_{-0.012}^{+0.013}$ | $-0.01_{-0.03}^{+0.03}$ |  $0.30_{-0.13}^{+0.23}$ |  $0.064_{-0.029}^{+0.04}$
7009365 | $83.298_{-0.027}^{+0.029}$ | $0.1377_{-0.0016}^{+0.0017}$ | $0.270_{-0.018}^{+0.017}$ | $0.600_{-0.012}^{+0.012}$ | $0.546_{-0.012}^{+0.013}$ |  $0.023_{-0.012}^{+0.012}$ |  $0.83_{-0.19}^{+0.12}$ |  $0.036_{-0.006}^{+0.010}$
7518143 | $78.50_{-0.03}^{+0.04}$ | $0.1209_{-0.0022}^{+0.0025}$ | $0.29_{-0.04}^{+0.03}$ | $0.611_{-0.013}^{+0.016}$ | $0.315_{-0.011}^{+0.013}$ |  $0.068_{-0.018}^{+0.018}$ |  $0.49_{-0.23}^{+0.3}$ |  $0.020_{-0.008}^{+0.015}$
8540034 | $84.25_{-0.03}^{+0.03}$ | $0.1524_{-0.0022}^{+0.0024}$ | $0.308_{-0.022}^{+0.020}$ | $0.612_{-0.011}^{+0.014}$ | $1.138_{-0.010}^{+0.011}$ |  $0.009_{-0.014}^{+0.012}$ |  $0.57_{-0.27}^{+0.29}$ |  $0.020_{-0.009}^{+0.012}$
8619145 | $76.49_{-0.14}^{+0.19}$ | $0.126_{-0.005}^{+0.005}$ | $0.54_{-0.20}^{+0.15}$ | $0.645_{-0.019}^{+0.020}$ | $0.920_{-0.013}^{+0.013}$ |  $0.020_{-0.018}^{+0.023}$ |  $0.58_{-0.22}^{+0.29}$ |  $0.11_{-0.04}^{+0.06}$
8684542 | $80.55_{-0.09}^{+0.08}$ | $0.1213_{-0.0021}^{+0.0023}$ | $0.22_{-0.06}^{+0.06}$ | $0.680_{-0.013}^{+0.014}$ | $0.663_{-0.016}^{+0.016}$ |  $0.016_{-0.018}^{+0.017}$ |  $0.38_{-0.05}^{+0.05}$ |  $0.21_{-0.03}^{+0.03}$
9202471 | $84.66_{-0.09}^{+0.14}$ | $0.1351_{-0.0019}^{+0.0016}$ | $0.26_{-0.07}^{+0.05}$ | $0.684_{-0.013}^{+0.015}$ | $0.950_{-0.018}^{+0.019}$ |  $0.014_{-0.016}^{+0.015}$ |  $0.21_{-0.09}^{+0.19}$ |  $0.10_{-0.04}^{+0.07}$
9589420 | $74.44_{-0.08}^{+0.08}$ | $0.130_{-0.004}^{+0.006}$ | $0.28_{-0.12}^{+0.12}$ | $0.612_{-0.018}^{+0.018}$ | $0.645_{-0.008}^{+0.010}$ | $-0.008_{-0.018}^{+0.017}$ |  $0.37_{-0.12}^{+0.17}$ |  $0.070_{-0.018}^{+0.020}$
10801792 | $83.17_{-0.10}^{+0.13}$ | $0.1416_{-0.0019}^{+0.0020}$ | $0.25_{-0.08}^{+0.06}$ | $0.640_{-0.015}^{+0.016}$ | $0.918_{-0.016}^{+0.015}$ |  $0.028_{-0.016}^{+0.016}$ |  $0.37_{-0.12}^{+0.16}$ |  $0.12_{-0.04}^{+0.06}$
11515377 | $83.7_{-0.10}^{+0.10}$ | $0.1542_{-0.0021}^{+0.0020}$ | $0.21_{-0.06}^{+0.06}$ | $0.615_{-0.014}^{+0.013}$ | $0.649_{-0.014}^{+0.014}$ |  $0.050_{-0.016}^{+0.016}$ | $-0.16_{-0.05}^{+0.03}$ |  $0.18_{-0.04}^{+0.04}$
Table 3: In this table, we list the values and their uncertaintiers of $T_{\mathrm{eff}}$, $\log L/L_{\odot}$, and [Fe/H] given by Berger et al. (2020) that were used to constrain the theoretical models. Then we list the model-inferred masses, radii, and ages. We also provide the core structure factor $\mathcal{I}$, the measured field strengths $\left<B_{\mathrm{r}}^{2}\right>^{0.5}$, and the model-inferred critical strengths at the hydrogen-burning shells $B_{\mathrm{c}}$ of the 13 stars in this work. KIC | $T_{\mathrm{eff}}$ | $\log L/L_{\odot}$ | [Fe/H] | Mass | radius | Age | $\mathcal{I}$ | $\langle B_{\mathrm{r}}^{2}\rangle^{0.5}$ | $B_{\mathrm{c}}$
---|---|---|---|---|---|---|---|---|---
| K |  | dex | $\mathrm{M_{\odot}}$ | $\mathrm{R_{\odot}}$ | Gyr | $10^{-23}\frac{\mathrm{m}}{\mathrm{s^{2}kg}}$ | kG | kG
4458118 | $4900_{-80}^{+90}$ | $0.97_{-0.15}^{+0.11}$ | $-0.18_{-0.15}^{+0.13}$ | $1.077_{-0.04}^{+0.021}$ | $4.10_{-0.05}^{+0.03}$ | $8.4_{-0.4}^{+1.5}$ | 1.1732 | $\phantom{0}58_{-19}^{+24}$ | 704.7
5196300 | $4840_{-80}^{+90}$ | $1.08_{-0.03}^{+0.03}$ | $\phantom{-}0.20_{-0.12}^{+0.11}$ | $1.44_{-0.03}^{+0.08}$ | $4.78_{-0.05}^{+0.09}$ | $3.6_{-0.6}^{+0.7}$ | 1.7617 | $\phantom{0}89_{-12}^{+16}$ | 564.0
5696081 | $4910_{-80}^{+80}$ | $0.93_{-0.28}^{+0.26}$ | $-0.02_{-0.11}^{+0.15}$ | $1.34_{-0.04}^{+0.04}$ | $4.19_{-0.05}^{+0.04}$ | $4.5_{-0.6}^{+0.6}$ | 1.3404 | $152_{-16}^{+23}$ | 957.4
6936091 | $4840_{-90}^{+90}$ | $1.24_{-0.18}^{+0.22}$ | $-0.17_{-0.15}^{+0.12}$ | $1.04_{-0.03}^{+0.04}$ | $6.13_{-0.09}^{+0.10}$ | $8.1_{-1.2}^{+1.7}$ | 2.0698 | $\phantom{0}23_{-6}^{+7}$ | 107.9
7009365 | $4880_{-80}^{+80}$ | $1.04_{-0.17}^{+0.14}$ | $\phantom{-}0.10_{-0.12}^{+0.14}$ | $1.396_{-0.021}^{+0.06}$ | $4.64_{-0.03}^{+0.07}$ | $3.8_{-0.6}^{+0.8}$ | 1.7098 | $\phantom{0}62_{-6}^{+9}$ | 585.2
7518143 | $4780_{-80}^{+80}$ | $1.148_{-0.006}^{+0.028}$ | $\phantom{-}0.15_{-0.14}^{+0.10}$ | $1.40_{-0.05}^{+0.06}$ | $5.41_{-0.08}^{+0.08}$ | $3.8_{-0.8}^{+1.0}$ | 2.1362 | $\phantom{0}27_{-6}^{+9}$ | 294.1
8540034 | $4940_{-80}^{+80}$ | $1.06_{-0.11}^{+0.08}$ | $-0.12_{-0.15}^{+0.11}$ | $1.20_{-0.03}^{+0.08}$ | $4.52_{-0.06}^{+0.09}$ | $5.0_{-0.3}^{+1.3}$ | 1.5042 | $\phantom{0}44_{-11}^{+12}$ | 545.0
8619145 | $4850_{-80}^{+80}$ | $1.19_{-0.04}^{+0.07}$ | $-0.03_{-0.14}^{+0.12}$ | $1.26_{-0.04}^{+0.04}$ | $5.68_{-0.06}^{+0.06}$ | $5.6_{-1.1}^{+0.6}$ | 2.1872 | $\phantom{0}47_{-9}^{+12}$ | 197.3
8684542 | $4850_{-80}^{+100}$ | $1.112_{-0.06}^{+0.014}$ | $\phantom{-}0.19_{-0.15}^{+0.12}$ | $1.40_{-0.06}^{+0.07}$ | $5.11_{-0.08}^{+0.09}$ | $3.6_{-0.6}^{+0.7}$ | 1.9372 | $111_{-9}^{+9}$ | 397.6
9202471 | $4720_{-80}^{+80}$ | $0.97_{-0.17}^{+0.14}$ | $\phantom{-}0.36_{-0.11}^{+0.12}$ | $1.40_{-0.04}^{+0.03}$ | $4.57_{-0.05}^{+0.05}$ | $4.0_{-0.4}^{+1.3}$ | 1.5490 | $110_{-30}^{+40}$ | 663.3
9589420 | $4780_{-80}^{+80}$ | $1.31_{-0.18}^{+0.21}$ | $-0.01_{-0.15}^{+0.11}$ | $1.30_{-0.03}^{+0.06}$ | $6.27_{-0.08}^{+0.08}$ | $5.6_{-1.2}^{+0.6}$ | 2.6778 | $\phantom{0}26_{-4}^{+4}$ | 124.7
10801792 | $4890_{-80}^{+80}$ | $1.08_{-0.05}^{+0.010}$ | $-0.06_{-0.16}^{+0.15}$ | $1.34_{-0.05}^{+0.04}$ | $4.68_{-0.06}^{+0.05}$ | $4.21_{-0.28}^{+1.2}$ | 1.7296 | $102_{-19}^{+24}$ | 535.6
11515377 | $4960_{-80}^{+80}$ | $1.11_{-0.08}^{+0.07}$ | $-0.21_{-0.12}^{+0.15}$ | $1.30_{-0.07}^{+0.06}$ | $4.70_{-0.10}^{+0.06}$ | $4.2_{-0.9}^{+0.4}$ | 1.4839 | $133_{-17}^{+14}$ | 544.3

##  4 Discussion

###  4.1 Asymmetry parameter and magnetic shift

Figure 8 displays the relation between the asymmetry parameter $a$ and the
magnetic shift $\delta\nu_{\mathrm{g}}$. We find that there is no obvious
correlation between these two parameters and there is also no correlation with
the period spacing $\Delta\Pi_{1}$. Figure 9 shows the relation between the
asymmetry parameter $a$ and the field strength of our sample, and we still do
not find any correlation between them, meaning that the field strength does
not correlate with the field topology.

The asymmetry parameter $a$ reaches 1 when the field is entirely concentrated
to the poles and it reaches $-0.5$ when the field is concentrated to the
equator. Dipolar fields have values of $a$ ranging from $-0.2$ (corresponding
to a dipolar field aligned with the equator) to $0.4$ (corresponding to a
dipolar field aligned with the rotation axis). The asymmetry parameter can
also vanish, even in the presence of a strong field, for example if a dipolar
field is inclined by about 55∘ with respect to the rotation axis or if the
latitudinal variations of $B_{r}^{2}$ only occur at length scales much smaller
than the star radius (Li et al., 2022a). Most of the stars exhibit $a$ values
between 0.2 and 0.8. KIC 11515377 is the only star that shows negative
asymmetries, which lead to $a$ close to $-0.2$. This suggests that this
configuration might be rarer among red giants. We note the diversity of the
values obtained for the asymmetry parameter $a$, which clearly shows that the
core fields of red giants have various horizontal geometries. In some stars,
the posterior probabilities for the asymmetry parameter $a$ nearly vanish
below $0.4$, which means that for these stars the magnetic fields are more
sharply concentrated on the poles than a purely dipolar field aligned with the
rotation axis. Our measurements of $a$ will be useful to constrain future
models of red giant core magnetic fields.

Figure 8: Relation between the asymmetry parameter $a$ and the magnetic shift
$\delta\nu_{\mathrm{g}}$. KIC 11515377 is the only star that shows negative
asymmetries.  Figure 9: Relation between the asymmetry parameter $a$ and the
field strength.

###  4.2 Core and envelope rotation rates

We display the relation between $\Omega_{\mathrm{core}}$ and $\Delta\Pi_{1}$
in Fig. 10. The figure shows that the rotation rates of our stars are typical
for red giants and are consistent with those of other studies (shown as the
grey circles by Gehan et al., 2018). Our observations thus suggest that these
stars have not experienced significantly different histories of angular
momentum transport compared to other red giant stars. We note that this does
not contradict the hypothesis that magnetic fields could be responsible for
angular momentum transport in red giants because we cannot exclude that
magnetic fields currently escaping detection might exist in other red giants.

We compared our measurements of the core rotation rates with those of Gehan et
al. (2018) for the five stars in our sample that were also in their study. We
find that our results are consistent in four stars (KIC 7518143, 6936091,
11515377, and 8540034), while a large discrepancy appears in KIC 8684542. The
reason for the consistency is that the magnetism-induced perturbation does not
change the frequency separation between $m=1$ and $m=-1$ modes, hence it does
not affect the measurements of splittings if the mode identification is
correct. However, the discrepancy for KIC 8684542 arises because only
symmetric rotational splittings were considered in the previous work, which
resulted in an incorrect identification of the splitting of $m=1$ and $-1$
modes (see the online peer review file of Li et al., 2022a).

Figure 10: The core rotation rates $\Omega_{\mathrm{g}}$ as a function of
$\Delta\Pi_{1}$. The black dots are the stars in this work, and the grey
circles are reported by Gehan et al. (2018).

The top panel of fig. 11 displays the measurements of the envelope rotation
rates as a function of their model-inferred radii. The method used in section
3 does not require the use of the $\zeta$ function, which is different from
the method used by Li et al. (2022a) to measure the internal rotation.
Consequently, the envelope rotation rates obtained using the current method
exhibit differences compared to the results reported by Li et al. (2022a). Due
to the expansion of the envelopes, red giant stars show very slow envelope
rotations. Most of the stars have surface rotation rates around
$0.02\,\mathrm{\mu Hz}$, equivalent to about $\sim 600\,\mathrm{d}$. KIC
7518143 has the fastest surface rotation rate with a rotation period of about
170 days. For the two stars KIC 6936091 and KIC 9589420, with relatively large
radii inferred from the modelling (larger than six solar radii), the rotation
rates are slower than the detection limit of the seismic signal, resulting in
surface rotation rates that are compatible with zero within $1\sigma$ ranges.

The bottom panel of fig. 11 shows the ratio between the core and the envelope
rotations ($\Omega_{\mathrm{core}}/\Omega_{\mathrm{env}}$), which represents
the level of differential rotation. For the two stars with near-zero envelope
rotations (with radii larger than 6 solar radii), we can only obtain a lower
limit to this ratio. Among the remaining stars, we find that the rotation
ratio lies between 10 and 100. Interestingly, KIC 7518143, the star with the
fastest envelope rotation, also shows a slow core-rotation rate, which leads
to the rotation ratio $\Omega_{\mathrm{core}}/\Omega_{\mathrm{env}}$ is only
around 4. This finding suggests that a strong transfer of angular momentum
occurs inside the star, but the relation with magnetic field is unclear, as
the measured field strength of this star is not the strongest.

Figure 11: Top panel: surface rotation rates as a function of model-inferred
radii. Bottom panel: variations in the ratio
$\Omega_{\mathrm{core}}/\Omega_{\mathrm{env}}$ with stellar radii.

###  4.3 Comparison with critical field strength

Fuller et al. (2015) showed that magnetic fields that exceed a critical value
$B_{\rm c}$ can prevent the propagation of gravity waves. This phenomenon was
invoked as a possible explanation for the suppression of $l=1$ mixed modes in
a fraction of red giant stars (see also in Stello et al., 2016). Fuller et al.
(2015) suggested that when the g-mode cavity harbours a magnetic field
stronger than $B_{\rm c}$, all the mode energy that reaches the core is
dissipated, leading to modes that have a pure p-like behaviour. This
interpretation was challenged by Mosser et al. (2017a), who found that for red
giants that show only partially-suppressed dipole modes, the modes still have
a g-like character. This question is currently under debate. Even though we do
not yet have a clear picture of how global oscillation modes are affected by a
magnetic field exceeding $B_{\rm c}$, it is clear that it will have an impact.
We thus compared the measured field strength to the critical field strength,
which we computed using our best-fit stellar models. In practice, this
critical field varies as a function of radius. The minimum appears at the
hydrogen-burning shell (HBS), as given by

$B_{\mathrm{c,min}}=\left(\frac{16\pi^{4}\mu_{0}\rho_{\mathrm{hbs}}r_{\mathrm{hbs}}^{2}\nu_{\mathrm{max}}^{4}}{8N_{\mathrm{hbs}}^{2}}\right)^{0.5},$
(22)

where $\rho_{\mathrm{hbs}}$, $r_{\mathrm{hbs}}$, and $N_{\mathrm{hbs}}$ are
the density, radius, and the Brunt-Väisälä frequency at the hydrogen-burning
shell. Using our optimal stellar models, we computed $B_{\mathrm{c,min}}$ for
the 13 stars of our sample. The obtained values ore listed in Table 3.

Using the values listed in Table 3, we can calculate the ratio between the
measured and critical field strengths as a function of $\Delta\Pi_{1}$. The
ratios range from approximately 0.1 to 0.3. It is important to note that the
critical field strength varies with radius, and we use the minimum value,
which occurs at the hydrogen-burning shell and is consistent with the layer
where we measure the field strength. Additionally, the measured field strength
is the average over the weight function $K(r)$, which means that some
contribution from the field strength inside the hydrogen-burning shell is also
included (see extended data Figure 1 in Li et al., 2022a). Therefore, the
reported ratios may not be representative of the fields ratio at the location
of hydrogen-burning shell.

Figure 12: Field strength as a function of mixed mode density. The red
hexagons with errorbars are the field strengths of the 13 stars reported by
this work, and the black dots are their critical field strengths. The grey
vertical arrows show the lower limits of the field strengths reported by
Deheuvels et al. (2023), where the stars do not show any asymmetric
splittings, but only $m=0$ curved ridges in their stretched échelle diagrams.

Fig. 12 shows the observed and critical field strength with evolution,
indicated by the mixed mode density
$\mathcal{N}=\Delta\nu/(\nu_{\mathrm{max}}^{2}\Delta\Pi_{1})$ (Gehan et al.,
2018). It is evident from Fig. 12 that the critical field strengths tend to
decrease as stars evolve along the red giant branch (this was already shown by
Fuller et al., 2015). We also observe an overall decrease in the observed
field strengths with the evolution (albeit with relatively large scatter).
This is in contrast to the theoretical prediction that the field strength
should increase with evolution, assuming the magnetic flux is conserved while
the core contracts. However, if the field strength exceeds the critical field
strength, the energy of gravity waves is thought to be completely transferred
to Alfvén waves, and we cannot observe any dipole mixed modes (Fuller et al.,
2015) Moreover, a strong field leads to a curvature in the stretched échelle
diagram (Li et al., 2022a; Bugnet, 2022; Deheuvels et al., 2023), which could
obscure the observations. Therefore, only a field that is $\sim 10\%$ to $\sim
30\%$ of the critical field strength can generate observable asymmetric
splittings. This could explain why the decrease in the detected field strength
with evolution seems to follow the decrease in the critical field $B_{\rm c}$
(Fig. 12).

Deheuvels et al. (2023) reported 11 stars showing curved stretched échelle
diagrams caused by their central magnetic fields. In this case, only the lower
limit of the field strengths are derived. We plot the results by the grey
vertical arrows in Fig. 12, and find that they also follow a decreasing trend
with evolution but with much stronger field strengths. Figure 12 reveals a
large gap of field strength between the stars exhibiting asymmetric
splittings, as seen in this work, and those with curved stretched échelle
diagrams, as observed by Deheuvels et al. (2023). The search for such power
spectra may be hindered by an observational bias, as explained in Sect. 4.6.

###  4.4 Origin of the fields

Figure 13 depicts the measured field strengths plotted against model-inferred
age and mass. In the top panel, most stars have ages smaller than 6 Gyr and
the field strengths of these stars show large scatter. However, for the two
stars with longer ages (between 8 and 9 Gyr), the field strengths are small.
As stellar age is strongly related to mass, we examined the correlation
between field strength and mass in the bottom panel of Fig. 13. We find that
most stars have masses larger than 1.3 $\mathrm{M_{\odot}}$, meaning that they
had convective cores in their main-sequence stages and can generate strong
central fields by the dynamo processes. Since the Ohmic timescale, over which
magnetic field dissipate, is longer than the evolution timescale (Cantiello et
al. 2016), such fields could survive until the red-giant phase, where they may
relax into stable mixed poloidal-toroidal configurations (Braithwaite & Spruit
2004).

However, we observed two stars with lower masses (1.08 $\mathrm{M_{\odot}}$
for KIC 4458118 and 1.04 $\mathrm{M_{\odot}}$ for KIC 6936091). Our best-fit
models for these stars had a radiative core during the bulk of their main
sequence, which challenges the interpretation that their central fields may
have been generated by the core dynamo. To further study the origin of the
field, we calculated the mass and the radius of the convective core at the
beginning of stellar life. For the two low-mass stars (KIC 4458118 and
6936091), we found that they show convective cores with masses of $\sim$12% of
the total stellar masses at the very beginning of its evolution ($\sim 30$
Myr), owing to the burning of ${}^{3}\mathrm{He}$ and ${}^{12}\mathrm{C}$
outside of equilibrium (Deheuvels et al., 2010). However, the convective core
was too small to reach the shell that is currently burning hydrogen in red-
giant phase (which is located at about $\sim$20% of the total stellar masses).
Even though the hydrogen-burning shell in these two stars was never
convective, we cannot rule out the dynamo origin because the weight function
$K(r)$ involved in the expression of the measured field has a contribution
from the deeper layers of the star. If we assume that the magnetic field is
confined to the layers that were convective at the beginning of the main
sequence (i.e. the layers whose mass is smaller than 12% of the total mass),
we find that fields of $\sim$300 kG for KIC 4458118 and $\sim$200 kG for KIC
6936091 are needed to reproduce the observations. These values are several
times larger than the values given in Table 3. However, we cannot exclude that
such fields might result from a dynamo process in main-sequence cores because
these field strengths remain compatible with the amplitudes predicted by
numerical simulations of core convection (Brun et al. 2005) and dynamo scaling
laws (Bugnet et al. 2021). In addition, the determination of convective core
size still shows an uncertainty due to the poorly-understanding internal
chemical mixing process (e.g. Johnston, 2021). Another possibility for the
origin of the magnetic field is that it is inherited from a fossil field.

Figure 13: Top panel: field strengths with model-inferred ages. Bottom panel:
field strengths but with model-inferred masses.

###  4.5 Ongoing dynamo or stable fields?

Figure 14: Ratio of the Alfvèn frequency $\Omega_{\rm
A}=\sqrt{\frac{<B_{r}^{2}>}{\mu_{0}\rho r^{2}}}$ to the core rotation rate
$\Omega_{\rm core}$ as a function of mixed mode density. The density $\rho$
and the radius $r$ have been computed at the HBS of the star models.

The ratio between the Alfvèn frequency $\Omega_{\rm
A}=\sqrt{\frac{<B_{r}^{2}>}{\mu_{0}\rho r^{2}}}$ and the rotation rate
$\Omega$ is an important parameter of the dynamical interactions between
magnetic field and rotation. Its value can provide some clues about the nature
of the detected magnetism. We computed this ratio at the HBS of the 13 red
giants and, as shown in Fig. 14, found that it is comprised between $0.05$ to
$0.25$.

A first consequence is that a Tayler-Spruit dynamo is probably not presently
at work in these layers. Indeed, the Fuller et al. (2019) model of the Tayler-
Spruit dynamo finds that $\Omega_{\rm A}/\Omega$ should scale with
$(\Omega/N)^{5/3}$ where $N$, the Brunt-Vaisala frequency, measures the
strength of stable stratification in radiative zones. As $(\Omega/N)^{5/3}\sim
10^{-7}$ at the HBS of a typical 1.2 M☉ and 4 R☉ star at the base of the red
giant branch Fuller et al. (2019), the predicted $\Omega_{\rm A}/\Omega$ are
extremely small and thus incompatible with our measurements.

The observed ratios are rather compatible with a stable magnetic field
configuration. Indeed, as anticipated in Spruit (1999), numerical studies of a
poloidal magnetic field embedded in a differentially rotating radiative zone
have shown that, if the ratio $\Omega_{\rm A}/\Omega_{\rm core}$ exceeds a
certain value, the field is not affected by instabilities and evolves towards
stable configurations while being only subjected to ohmic dissipation (Jouve
et al., 2015, 2020; Gouhier et al., 2022). The values obtained in numerical
simulations vary between $10^{-3}$ for a free differential rotation (Jouve et
al., 2020) and $10^{-2}$ for a differential rotation forced by a radial flow
simulating a star contraction (Gouhier et al., 2022). Another argument in
favour of a stable magnetic field configuration comes from the magnetic fields
observed at the surface of intermediate-mass and massive main-sequence stars.
In this mass range, the so-called fossil magnetic fields are stable at least
over decades and their $\Omega_{A}/\Omega$ ratio, computed at the star’s
surface, is always greater than $\sim 1$ (Aurière et al., 2007).

###  4.6 Observational biases

Our method of analysis of Kepler data presented in Sect. 2 leads to several
observational biases that need to be acknowledged.

####  4.6.1 Limitation on the measurable field strength

As mentioned in Sect. 2.1, we have assumed that the magnetic field is not
strong enough to significantly alter the regularity of g-mode period spacings.
Deheuvels et al. (2023) estimated the magnetic intensity threshold $B_{\rm
th}$ above which deviations from the regular period spacing of pure g modes
become detectable (see their Appendix B). They found that $B_{\rm th}$ follows
a similar decreasing trend with evolution as that of the critical field
$B_{\rm c}$. Overall, a magnetic field that exceeds about 40% of the critical
field should produce significant deviations. We can thus expect that our
method may start failing for field strengths above $0.4\,B_{\rm th}$. This
could at least partly explain the gap that we found between the field
measurements in this work and those obtained by Deheuvels et al. (2023).

Additionally, we assumed in Sect. 2.3 that the magnetic asymmetries are not
large enough to push the $m=0$ component of the multiplets outside the
interval formed by $m=\pm 1$ components. This phenomenon occurs if
$\nu_{\mathrm{g\leavevmode\nobreak\
mode},m=0}<\nu_{\mathrm{g\leavevmode\nobreak\ mode},m=+1}$ when $a>0$, and if
$\nu_{\mathrm{g\leavevmode\nobreak\
mode},m=0}>\nu_{\mathrm{g\leavevmode\nobreak\ mode},m=-1}$ when $a<0$. Using
Eq. 12 and 13, one finds that near $\nu_{\rm max}$, this condition is
equivalent to

$\delta\nu_{\rm g}\frac{3|a|}{2}\frac{4\pi}{\Omega_{\rm core}}>1.$ (23)

For the 13 stars of our sample, the quantity on the left-hand-side of Eq. 23
takes values between 0.03 and 0.36. It is not straightforward to translate the
condition given by Eq. 23 in terms of a constraint on the field intensity
because it depends on properties that vary from star to star (namely $a$,
$\mathcal{I}$, $\nu_{\rm max}$, and $\Omega_{\rm core}$). If we consider the
values of these parameters that were obtained for the stars of our sample, we
find that a change in the ordering of components in a multiplet would occur
for these stars if the field strengths were multiplied by a factor ranging
from 1.6 to 5.8. This corresponds to field intensities that are intermediate
between the measured values and the critical field strengths. We thus conclude
that our assumption that the $m=0$ component lies between the $m=\pm 1$
components could partly explain the lack of red giants with measured fields
closer to the critical field strength.

####  4.6.2 Non-axisymmetric effects

Equations 12 and 13 hold when the non-axisymmetric effects of magnetic fields
are negligible. This is true (i) if $B_{r}^{2}$ is axisymmetric, but also (ii)
if the ratio $b$ between the magnetic frequency and the rotational splitting
($b=4\pi\delta\nu_{\mathrm{g}}/\Omega_{\mathrm{core}}$) is smaller than $\sim
1$ (see the supplementary material S2.7 in Li et al., 2022a). Our results in
Table 2 show that the ratio $b$ is much smaller than one. The stars with the
largest values of $b$ are KIC 8684542 ($b=0.64\pm 0.09$) and KIC 11515377
($b=0.56\pm 0.12$), the other stars having $b$ values below $\sim 0.35$.
Hence, we state that case (ii) is valid for our stars, so that even if
$B_{r}^{2}$ is non-axisymmetric, we still cannot see any significant non-
axisymmetric effect in the oscillation spectra.

As mentioned in Sect. 2.1, when $b\gtrsim 1$, non-axisymmetric magnetic fields
can produce up to nine components for dipole modes (Li et al. 2022a).
Dedicated methods need to be devised and applied to search for such features
in the oscillation spectra of red giants. This work is currently undertaken by
our team and will be the subject of a next publication. Meanwhile, our
assumption that non-axisymmetric effects are weak limits the strength of
detectable fields. Similarly to Sect. 4.6.1, the condition $b\gtrsim 1$ cannot
directly be translated in terms of field strength. We thus estimated the
minimal field strengths that the stars of our sample should have in order to
produce significant non-axisymmetric effects. We found that the measured
fields would need to be multiplied by a factor ranging from 1.2 to 5.4. Again,
this corresponds to field intensities that lie between the measurements
obtained in this study and those found by Deheuvels et al. (2023). Taking non-
axisymmetric effects into account could thus populate the gap observed in Fig.
12.

##  5 Conclusions

In this study, we conducted a systematic search for magnetism-induced
asymmetric splittings in Kepler data. We successfully identified 13 stars
(including the three stars previously reported by Li et al. 2022a) exhibiting
clear multiplet asymmetries with properties matching those expected in the
presence of a core magnetic field. Notably, we found that only one star (KIC
11515377) displayed negative asymmetries, while the remaining stars exhibited
positive asymmetries. By fitting an asymptotic expression of mixed mode
frequencies including rotational and magnetic effects to the observations, we
were able to measure the magnetic frequency shift $\delta\nu_{\mathrm{g}}$
(which is related to the field strength), and the asymmetry parameter $a$
(which yields constraints on the field topology).

Using the best-fitting stellar structure model, we were able to measure the
average radial field strength in the core ($\langle B_{r}^{2}\rangle^{0.5}$)
for the 13 stars. These field strengths were found to lie between
approximately 20 and 150 kG, with maximal sensitivity in the vicinity of the
hydrogen-burning shells. These values represent about 10% to 30% of the
critical field strength above which gravity modes are no longer expected to
propagate in the core (Fuller et al. 2015).

We also obtained estimates of the asymmetry parameter $a$, which provides a
horizontal average of $B_{r}^{2}$ weighted by the second-order Legendre
polynomial (see Eq. 14). For the 13 stars, we found values of $a$ between
about $-0.2$ and $0.95$, nearly spanning the entire possible range for this
parameter ($-0.5\leqslant a\leqslant 1$). We recall that large negative values
of $a$ are reached for fields that are concentrated near the equator, while
large positive values of $a$ correspond to fields concentrated on the poles.
Our results thus show that the core fields of red giants have various
horizontal geometries. For some stars, we find values of $a$ that
significantly exceed 0.4, which is the highest value that can be obtained with
a dipolar magnetic field. This means that for these stars, the fields are more
sharply concentrated on the poles than a pure dipolar field aligned with the
rotation axis. The fact that a negative value of $a$ was obtained for only one
star (KIC 11515377) suggests that this configuration might be rare among red
giants.

In agreement with the results of Deheuvels et al. (2023), we found that the
magnetic field strength in the core of red giants decreases with evolution
(see Fig. 12). This is in contradiction with the general expectation that the
contraction of the core should increase the magnetic field, if we assume a
conservation of the magnetic flux. As was already pointed out in Deheuvels et
al. (2023), the observed decrease seems to follow the overall decrease of the
critical field strength $B_{\rm c}$ with evolution. One possible
interpretation is thus that for a given star, the core magnetic field
increases with evolution until it reaches $B_{\rm c}$. At this point, gravity
waves no longer propagate in the core so that dipole modes do not have a mixed
behaviour, and therefore core magnetic fields can no longer be detected. This
could account for the observed decrease in the measured field strength,
although further work is clearly needed to test this interpretation.

This work presents a larger sample of red giant stars with central magnetic
fields, comprising 13 stars out of approximately 1200 red giant stars with
triplets. The prevalence of such magnetic fields is thus currently found to be
only 1%. Stello et al. (2016) claimed that about 5 to 15% of red giants in the
mass range of our sample are magnetised based on the assumption that
suppressed dipole mixed modes are caused by magnetic fields exceeding the
critical field strength. So far, this assumption remains challenged (Mosser et
al. 2017a), but if it were correct, the stars of Stello et al. (2016) (which
lie in a different range of field intensities compared to this work) could be
added to the list of magnetic red giants. The low prevalence of magnetic
giants in our sample may be partly due to observational biases related to the
analysis method that we adopted in this study. We assumed that the field
strength was not strong enough to significantly alter the regularity in the
g-mode period spacings, modify the ordering of the components within dipole
multiplets, or produce detectable effects related to the non-axisymmetric
component of the magnetic field. We estimated that field intensities exceeding
the measured field strengths by a factor of a few would be enough to make at
least one of these assumptions invalid. Therefore, the present study might
miss stars with stronger core fields. We also stress that in this study,
magnetic giants have been identified by searching for multiplet asymmetries.
This method can thus not detect magnetic fields with horizontal geometries
that correspond to vanishing values of $a$.

Regarding the origin of the detected fields, one of the main scenarios is that
they were produced by a dynamo in the main-sequence convective core. After the
end of the main sequence, these fields would have relaxed into stable
configurations (Braithwaite & Spruit 2004), undergoing only weak Ohmic
diffusion (Cantiello et al. 2016). In this study, we found magnetic fields in
two low-mass stars ($1.04_{-0.03}^{+0.04}\,M_{\odot}$ for KIC 6936091, and
$1.08_{-0.04}^{+0.02}\,M_{\odot}$ for KIC 4458118), who had a small convective
core only at the very beginning of the main sequence, owing to nuclear
reactions outside of equilibrium. These convective cores never reached the
layers where our magnetic field measurements have maximal sensitivity, namely
the hydrogen-burning shell. Assuming that the core magnetic field is confined
to the layers that were once convective, we found that field strengths of 200
to 300 kG need to be invoked. These intensities remain compatible with order-
of-magnitude predictions of field strengths produced by dynamo in main-
sequence convective cores (Brun et al. 2005, Bugnet et al. 2021). Another
possible interpretation is that the detected fields might be inherited from
fossil magnetic fields.

Internal magnetic fields have been proposed as a candidate to provide
additional transport of angular momentum inside stars. In this study, we were
also able to measure average core rotation rates for the 13 stars. We found
values that are in line with the typical rotation rates of red giant cores, as
obtained by Gehan et al. (2018). This suggests that the stars of our sample do
not undergo enhanced angular momentum redistribution compared to other red
giants. This does not rule out magnetic fields as the origin of the angular
momentum transport in red giants, as other red giants may harbour core
magnetic fields that were not detected, either because of our observational
biases or because they are below the detection threshold. The relatively high
ratios between the Alfvèn frequency and the rotation rate rule out that the
observed fields are fed by an ongoing Tayler-Spruit dynamo. They rather point
towards fields that have settled into stable configurations.

This work constrains the strength and topology of the magnetic fields, hence
it can be used in future numerical simulations and provides the possibility to
further investigate the evolution of magnetic fields and their interaction
with stellar rotation.

###### Acknowledgements.

The authors acknowledge support from the project BEAMING ANR-18-CE31-0001 of
the French National Research Agency (ANR) and from the Centre National
d’Etudes Spatiales (CNES). Gang Li received funding from the KU Leuven
Research Council under grant C16/18/005: PARADISE. TL acknowledges support
from the Joint Research Fund in Astronomy (U2031203) under cooperative
agreement between the National Natural Science Foundation of China (NSFC) and
Chinese Academy of Sciences (CAS), NSFC grants (12090040, 12090042), and the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (CartographY GA. 804752). This paper
includes data collected by the Kepler mission and obtained from the MAST data
archive at the Space Telescope Science Institute (STScI). Funding for the
Kepler mission is provided by the NASA Science Mission Directorate. STScI is
operated by the Association of Universities for Research in Astronomy, Inc.,
under NASA contract NAS 5–26555.

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##  Appendix A Échelle diagram of KIC 9467102

We show the normal échelle diagram of KIC 9467102 as an example of serious
splitting overlap discussed in section 2.4.

Figure 15: Normal échelle diagram for KIC 9467102. The frequencies are folded
by the large separation $\Delta\nu=10.4278\,\mu\mathrm{Hz}$. The modes that
are used in the fitting procedure are marked by the colour symbols: green ‘+’
for $m=1$, purple ‘$\bullet$’ for m=0, and red ‘-’ for m=-1 modes. The best-
fitting results are shown by the horizontal red lines, which connect the
components in each triplet. Note that the overlap is serious.

##  Appendix B All the diagrams

We show the related diagrams of all the 13 stars. For each star, there are
four diagrams shown by the following order:

  * •

Normal échelle diagram.

  * •

Stretched échelle diagram.

  * •

Splitting asymmetry as a function of frequency.

  * •

Corner diagram of the MCMC fit result.

Figure 16: Normal échelle diagram of KIC 4458118. The frequencies are folded
by the large separation $\Delta\nu=16.4420\,\mu\mathrm{Hz}$. The modes which
are used in the fitting procedure are marked by the colour symbols: green ‘+’
for $m=1$, purple ‘$\bullet$’ for m=0, and red ‘-’ for m=-1 modes. The best-
fitting frequencies are marked by the black ‘+’, ‘$\bullet$’, ‘-’ symbols at
the top of each panel. Figure 17: Stretched échelle diagram of KIC 4458118.
Symbols are explained in Fig. 1.  Figure 18: Splitting asymmetries as a
function of frequency of KIC 4458118. Symbols are explained in Fig. 4. Figure
19: Corner diagram of the MCMC fitting result of KIC 4458118. Figure 20:
Normal échelle diagram of KIC 5196300. See Fig. 16 for the explanations of the
symbols. Figure 21: Stretched échelle diagram of KIC 5196300. Symbols are
explained in Fig. 1.  Figure 22: Splitting asymmetries as a function of
frequency of KIC 5196300. Symbols are explained in Fig. 4. Figure 23: Corner
diagram of the MCMC fitting result of KIC 5196300. Figure 24: Normal échelle
diagram of KIC 5696081. See Fig. 16 for the explanations of the symbols.
Figure 25: Stretched échelle diagram of KIC 5696081. Symbols are explained in
Fig. 1.  Figure 26: Splitting asymmetries as a function of frequency of KIC
5696081. Symbols are explained in Fig. 4. Figure 27: Corner diagram of the
MCMC fitting result of KIC 5696081. Figure 28: Normal échelle diagram of KIC
6936091. See Fig. 16 for the explanations of the symbols. Figure 29: Stretched
échelle diagram of KIC 6936091. Symbols are explained in Fig. 1.  Figure 30:
Splitting asymmetries as a function of frequency of KIC 6936091. Symbols are
explained in Fig. 4. Figure 31: Corner diagram of the MCMC fitting result of
KIC 6936091. Figure 32: Normal échelle diagram of KIC 7009365. See Fig. 16 for
the explanations of the symbols. Figure 33: Stretched échelle diagram of KIC
7009365. Symbols are explained in Fig. 1.  Figure 34: Splitting asymmetries as
a function of frequency of KIC 7009365. Symbols are explained in Fig. 4.
Figure 35: Corner diagram of the MCMC fitting result of KIC 7009365. Figure
36: Normal échelle diagram of KIC 7518143. See Fig. 16 for the explanations of
the symbols. Figure 37: Stretched échelle diagram of KIC 7518143. Symbols are
explained in Fig. 1.  Figure 38: Splitting asymmetries as a function of
frequency of KIC 7518143. Symbols are explained in Fig. 4. Figure 39: Corner
diagram of the MCMC fitting result of KIC 7518143. Figure 40: Normal échelle
diagram of KIC 8540034. See Fig. 16 for the explanations of the symbols.
Figure 41: Stretched échelle diagram of KIC 8540034. Symbols are explained in
Fig. 1.  Figure 42: Splitting asymmetries as a function of frequency of KIC
8540034. Symbols are explained in Fig. 4. Figure 43: Corner diagram of the
MCMC fitting result of KIC 8540034. Figure 44: Normal échelle diagram of KIC
8619145. See Fig. 16 for the explanations of the symbols. Figure 45: Stretched
échelle diagram of KIC 8619145. Symbols are explained in Fig. 1.  Figure 46:
Splitting asymmetries as a function of frequency of KIC 8619145. Symbols are
explained in Fig. 4. Figure 47: Corner diagram of the MCMC fitting result of
KIC 8619145. Figure 48: Normal échelle diagram of KIC 8684542. See Fig. 16 for
the explanations of the symbols. Figure 49: Stretched échelle diagram of KIC
8684542. Symbols are explained in Fig. 1.  Figure 50: Splitting asymmetries as
a function of frequency of KIC 8684542. Symbols are explained in Fig. 4.
Figure 51: Corner diagram of the MCMC fitting result of KIC 8684542. Figure
52: Normal échelle diagram of KIC 9202471. See Fig. 16 for the explanations of
the symbols. Figure 53: Stretched échelle diagram of KIC 9202471. Symbols are
explained in Fig. 1.  Figure 54: Splitting asymmetries as a function of
frequency of KIC 9202471. Symbols are explained in Fig. 4. Figure 55: Corner
diagram of the MCMC fitting result of KIC 9202471. Figure 56: Normal échelle
diagram of KIC 9589420. See Fig. 16 for the explanations of the symbols.
Figure 57: Stretched échelle diagram of KIC 9589420. Symbols are explained in
Fig. 1.  Figure 58: Splitting asymmetries as a function of frequency of KIC
9589420. Symbols are explained in Fig. 4. Figure 59: Corner diagram of the
MCMC fitting result of KIC 9589420. Figure 60: Normal échelle diagram of KIC
10801792. See Fig. 16 for the explanations of the symbols. Figure 61:
Stretched échelle diagram of KIC 10801792. Symbols are explained in Fig. 1.
Figure 62: Splitting asymmetries as a function of frequency of KIC 10801792.
Symbols are explained in Fig. 4. Figure 63: Corner diagram of the MCMC fitting
result of KIC 10801792. Figure 64: Normal échelle diagram of KIC 11515377. See
Fig. 16 for the explanations of the symbols. Figure 65: Stretched échelle
diagram of KIC 11515377. Symbols are explained in Fig. 1.  Figure 66:
Splitting asymmetries as a function of frequency of KIC 11515377. Symbols are
explained in Fig. 4. Figure 67: Corner diagram of the MCMC fitting result of
KIC 11515377.

##  Appendix C Other parameters

###  C.1 $\Delta\nu$ vs $\Delta\Pi_{1}$

Figure 68 displays the correlation between the measured period spacing
$\Delta\Pi_{1}$ and the frequency separation $\Delta\nu$. The frequency
separations are distributed between $\sim\\!\\!8$ to
$\sim\\!\\!18\,\mathrm{\mu Hz}$ and the period spacings range from
$\sim\\!\\!75$ to $\sim\\!\\!90\,\mathrm{s}$. All the stars are located at the
well-defined $\Delta\Pi_{1}$–$\Delta\nu$ degenerate ridge, as shown by the
grey dots by Mosser et al. (2018), which implies that these stars did not
undergo any mass transfer or binary merger (e.g. Deheuvels et al. 2022; Rui &
Fuller 2021). We find that our stars tend to locate close to the lower
boundary of the $\Delta\Pi_{1}$–$\Delta\nu$ ridge, implying that they have
higher masses (Deheuvels et al. 2022).

Figure 68: The correlation between $\Delta\nu$ and $\Delta\Pi_{1}$. The grey
circles are given by Mosser et al. (2018).

###  C.2 $q$, $\varepsilon_{\mathrm{g}}$, and $f_{\mathrm{shift}}$

We show the values of the coupling factor $q$ in Fig. 69. The $q$ values of
our stars are in the typical range of red giant stars, and we also reproduce
the increasing trend between $q$ and $\Delta\Pi_{1}$ (e.g. Mosser et al.
2018).

Figure 69: The coupling factor $q$ vs $\Delta\Pi_{1}$. The grey circles are
reported by Mosser et al. (2018).

The g-mode phases $\varepsilon_{\mathrm{g}}$ are displayed in Fig. 70. We only
applied a uniform prior for $\varepsilon_{\mathrm{g}}$, but we still find that
most of the stars have the typical $\varepsilon_{\mathrm{g}}$ values of red
giant stars, which is $0.28\pm 0.08$ reported by both observations (Mosser et
al. 2018) and theoretical predictions that do not include magnetic
perturbations (Takata 2016). Therefore, the magnetism-induced perturbation
does not affect the distribution of g-mode phase. KIC 8619145 has a large
$\varepsilon_{\mathrm{g}}$ with large uncertainty, and we cannot provide
strong constraint on $\varepsilon_{\mathrm{g}}$ of KIC 6936091 either.

Figure 70: The g-mode phase $\varepsilon_{\mathrm{g}}$ as a function of
$\Delta\Pi_{1}$.

The frequency correction of $l=1$ pure p modes $f_{\mathrm{shift}}$ in Eq. 4
is a new parameter that has not been studied in previous work. We plot
$f_{\mathrm{shift}}$ as function of $\Delta\nu$ in Fig. 71. The best values of
$f_{\mathrm{shift}}$ lie between $\sim 0.5$ to $\sim 0.75\,\mathrm{\mu Hz}$.
It seems that there is a rapid drop when $\Delta\nu\lesssim 10\,\mathrm{\mu
Hz}$, but this feature needs more stars to confirm.

Figure 71: $f_{\mathrm{shift}}$ as a function of $\Delta\nu$.

##  Appendix D Linear relation between $\mathcal{I}$ and $\Delta\Pi_{1}$

Using the best-fitting models, we calculated the core factors $\mathcal{I}$
and derived the field strengths, listed in table 3. The core factor
$\mathcal{I}$ is vital to calculate the field strength in Eq. 19, but it is
time-consuming to run a stellar evolution code to derive it. Hence, we report
an empirical relation between $\mathcal{I}$ and the period spacing
$\Delta\Pi_{1}$. As shown in Fig. 72, there is a linear relation between
$\mathcal{I}$ and $\Delta\Pi_{1}$, which is characterised as

$\frac{\mathcal{I}\left(\mathrm{\frac{m}{s^{2}kg}}\right)}{10^{-23}}=\left(-0.0864\Delta\Pi_{1}+8.914\right)\pm
0.11.$ (24)

Figure 72: The core factor $\mathcal{I}$ as a function of $\Delta\Pi_{1}$,
colour coded by the model-inferred masses. The grey dashed line shows the
linear fit and the shaded area gives the $1\sigma$ range.