1]ASI Agenzia Spaziale Italiana, Via del Politecnico snc, 00133 Roma, Italy
2]MIT Kavli Institute for Astrophysics and Space Research, Massachusetts
Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
3]NASA Marshall Space Flight Center, Huntsville, AL 35812, USA 4]INAF Istituto
di Astrofisica e Planetologia Spaziali, Via del Fosso del Cavaliere 100, 00133
Roma, Italy 5]Dipartimento di Fisica, Università degli Studi di Roma “La
Sapienza", Piazzale Aldo Moro 5, I00185 Roma, Italy 6]Dipartimento di Fisica,
Università degli Studi di Roma “Tor Vergata", Via della Ricerca Scientifica 1,
00133 Roma, Italy 7]INAF Osservatorio Astronomico di Brera, Via E. Bianchi 46,
23807 Merate (LC), Italy 8]Finnish Centre for Astronomy with ESO, 20014
University of Turku, Finland 9]Institute of Astrophysics, Foundation for
Research and Technology-Hellas, GR-71110 Heraklion, Greece 10]Department of
Physics, University of Crete, 70013, Heraklion, Greece 11]Instituto de
Astrofísica de Andalucía-CSIC, Glorieta de la Astronomía s/n, 18008, Granada,
Spain 12]Institute for Astrophysical Research, Boston University, 725
Commonwealth Avenue, Boston, MA 02215, USA 13]Department of Astrophysics, St.
Petersburg State University, Universitetsky pr. 28, Petrodvoretz, 198504 St.
Petersburg, Russia 14]Space Science Data Center, Agenzia Spaziale Italiana,
Via del Politecnico snc, 00133 Roma, Italy 15]INAF Osservatorio Astronomico di
Roma, Via Frascati 33, 00078 Monte Porzio Catone (RM), Italy 16]University of
Maryland, Baltimore County, Baltimore, MD 21250, USA 17]NASA Goddard Space
Flight Center, Greenbelt, MD 20771, USA 18]Center for Research and Exploration
in Space Science and Technology, NASA/GSFC, Greenbelt, MD 20771, USA
19]Department of Physics and Kavli Institute for Particle Astrophysics and
Cosmology, Stanford University, Stanford, California 94305, USA 20]Owens
Valley Radio Observatory, California Institute of Technology, MC 249-17,
Pasadena, CA 91125, USA 21]Institut de Radioastronomie Millimétrique, Avenida
Divina Pastora, 7, Local 20, E18012 Granada, Spain 22]Center for Astrophysics
$|$ Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA
23]University of Siena, Astronomical Observatory, Via Roma 56, 53100 Siena,
Italy 24]Department of Physics, Graduate School of Advanced Science and
Engineering, Hiroshima University Kagamiyama, 1-3-1 Higashi-Hiroshima,
Hiroshima 739-8526, Japan 25]Department of Physics, Tokyo Institute of
Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan 26]Hiroshima
Astrophysical Science Center, Hiroshima University, 1-3-1 Kagamiyama, Higashi-
Hiroshima, Hiroshima 739-8526, Japan 27]Core Research for Energetic Universe
(Core-U), Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima
739-8526, Japan 28]Planetary Exploration Research Center, Chiba Institute of
Technology 2-17-1 Tsudanuma, Narashino, Chiba 275-0016, Japan 29]Astronomical
Institute, St. Petersburg State University, 28 Universitetskiy prospekt,
Peterhof, St. Petersburg 198504, Russia 30]Special Astrophysical Observatory,
Russian Academy of Sciences, 369167, Nizhnii Arkhyz, Russia 31]Pulkovo
Observatory, St.Petersburg, 196140, Russia 32]Université de Strasbourg, CNRS,
Observatoire Astronomique de Strasbourg, UMR 7550, 67000 Strasbourg, France
33]Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Via Pietro Giuria
1, 10125 Torino, Italy 34]Dipartimento di Fisica, Università degli Studi di
Torino, Via Pietro Giuria 1, 10125 Torino, Italy 35]Science and Technology
Institute, Universities Space Research Association, Huntsville, AL 35805, USA
36]Physics Department and McDonnell Center for the Space Sciences, Washington
University in St, Louis, MO, 63130, USA 37]Astronomical Institute of the Czech
Academy of Sciences, Boční II 1401/1, 14100 Praha 4, Czech Republic
38]Physics Department and McDonnell Center for the Space Sciences, Washington
University in St. Louis, St. Louis, MO 63130, USA 39]International Space
Science Institute (ISSI),Hallerstrasse 6, 3012 Bern, Switzerland
40]Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, Via
della Vasca Navale 84, 00146 Roma, Italy 41]Graduate School of Science,
Division of Particle and Astrophysical Science, Nagoya University, Furo-cho,
Chikusa-ku, Nagoya, Aichi 464-8602, Japan 42]Université Grenoble Alpes, CNRS,
IPAG, 38000 Grenoble, France 43]Mullard Space Science Laboratory, University
College London, Holmbury St Mary, Dorking, Surrey RH5 6NT, UK 44]INAF
Osservatorio Astronomico di Cagliari, Via della Scienza 5, 09047 Selargius
(CA), Italy 45]Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Largo
B. Pontecorvo 3, 56127 Pisa, Italy 46]Dipartimento di Fisica, Università di
Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy 47]INAF Osservatorio
Astrofisico di Arcetri, Largo Enrico Fermi 5, 50125 Firenze, Italy
48]Dipartimento di Fisica e Astronomia, Università degli Studi di Firenze, Via
Sansone 1, 50019 Sesto Fiorentino (FI), Italy 49]Istituto Nazionale di Fisica
Nucleare, Sezione di Firenze, Via Sansone 1, 50019 Sesto Fiorentino (FI),
Italy 50]Istituto Nazionale di Fisica Nucleare, Sezione di Roma “Tor Vergata",
Via della Ricerca Scientifica 1, 00133 Roma, Italy 51]Institut für Astronomie
und Astrophysik, Universität Tübingen, Sand 1, 72076 Tübingen, Germany
52]RIKEN Cluster for Pioneering Research, 2-1 Hirosawa, Wako, Saitama
351-0198, Japan 53]California Institute of Technology, Pasadena, CA 91125, USA
54]Yamagata University,1-4-12 Kojirakawa-machi, Yamagata-shi 990-8560, Japan
55]Osaka University, 1-1 Yamadaoka, Suita, Osaka 565-0871, Japan 56]University
of British Columbia, Vancouver, BC V6T 1Z4, Canada 57]International Center for
Hadron Astrophysics, Chiba University, Chiba 263-8522, Japan 58]Department of
Physics and Astronomy and Space Science Center, University of New Hampshire,
Durham, NH 03824, USA 59]Istituto Nazionale di Fisica Nucleare, Sezione di
Napoli, Strada Comunale Cinthia, 80126 Napoli, Italy 60]Department of Physics,
The University of Hong Kong, Pokfulam, Hong Kong 61]Department of Astronomy
and Astrophysics, Pennsylvania State University, University Park, PA 16802,
USA 62]Department of Physics and Astronomy, 20014 University of Turku, Finland
63]Dipartimento di Fisica e Astronomia, degli Studi di Padova, Via Marzolo 8,
35131 Padova, Italy 64]Department of Astronomy, University of Maryland,
College Park, Maryland 20742, USA 65]Anton Pannekoek Institute for Astronomy &
GRAPPA, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The
Netherlands 66]Guangxi Key Laboratory for Relativistic Astrophysics, School of
Physical Science and Technology, Guangxi University, Nanning 530004, China

# Discovery of X-ray polarization angle rotation in active galaxy Mrk 421

Laura Di Gesu  Herman L. Marshall  Steven R. Ehlert  Dawoon E. Kim  Immacolata
Donnarumma  Fabrizio Tavecchio  Ioannis Liodakis  Sebastian Kiehlmann  Iván
Agudo 11 Svetlana G. Jorstad  Fabio Muleri  Alan P. Marscher  Simonetta
Puccetti  Riccardo Middei  Matteo Perri  Luigi Pacciani  Michela Negro  Roger
W. Romani  Alessandro Di Marco  Dmitry Blinov  Ioakeim G. Bourbah  Evangelos
Kontopodis  Nikos Mandarakas  Stylianos Romanopoulos  Raphael Skalidis  Anna
Vervelaki  Carolina Casadio  Juan Escudero 11 Ioannis Myserlis  Mark A.
Gurwell 22 Ramprasad Rao 22 Garrett K. Keating 22 Pouya M. Kouch  Elina
Lindfors  Francisco José Aceituno 11 Maria I. Bernardos 11 Giacomo Bonnoli 7,
11 Víctor Casanova 11 Maya García-Comas 11 Beatriz Agís-González 11 César
Husillos 11 Alessandro Marchini  Alfredo Sota 11 Ryo Imazawa  Mahito Sasada
Yasushi Fukazawa  Koji S. Kawabata  Makoto Uemura  Tsunefumi Mizuno  Tatsuya
Nakaoka  Hiroshi Akitaya  Sergey S. Savchenko  Andrey A. Vasilyev  José L.
Gómez 11 Lucio A. Antonelli  Thibault Barnouin  Raffaella Bonino 33,34
Elisabetta Cavazzuti  Luigi Costamante  Chien-Ting Chen  Nicolò Cibrario 34,33
Alessandra De Rosa  Federico Di Pierro 33 Manel Errando  Philip Kaaret
Vladimir Karas  Henric Krawczynski  Lindsey Lisalda  Grzegorz Madejski
Christian Malacaria  Frédéric Marin  Andrea Marinucci  Francesco Massaro 33,34
Giorgio Matt  Ikuyuki Mitsuishi  Stephen L. O’Dell  Alessandro Paggi  Abel L.
Peirson  Pierre-Olivier Petrucci  Brian D. Ramsey  Allyn F. Tennant  Kinwah Wu
Matteo Bachetti 44 Luca Baldini  Wayne H. Baumgartner  Ronaldo Bellazzini
Stefano Bianchi  Stephen D. Bongiorno  Alessandro Brez  c Bucciantini  Fiamma
Capitanio  Simone Castellano  Stefano Ciprini  Enrico Costa  Ettore Del Monte
Niccolò Bucciantini Di Lalla  Victor Doroshenko  Michal Dovčiak  Teruaki Enoto
Yuri Evangelista  Sergio Fabiani  Riccardo Ferrazzoli  Javier A. Garcia
Shuichi Gunji  Kiyoshi Hayashida 55 Jeremy Heyl  Wataru Iwakiri  Fabian Kislat
Takao Kitaguchi  Jeffery J. Kolodziejczak  Fabio La Monaca  Luca Latronico 33
Simone Maldera 33 Alberto Manfreda  C.-Y. Ng  Nicola Omodei  Chiara Oppedisano
33 Alessandro Papitto  George G. Pavlov  Melissa Pesce-Rollins  Maura Pilia 44
Andrea Possenti44 Juri Poutanen  John Rankin  Ajay Ratheesh  Oliver J. Roberts
Carmelo Sgrò  Patrick Slane  Paolo Soffitta  Gloria Spandre  Douglas A. Swartz
Toru Tamagawa  Roberto Taverna  Yuzuru Tawara  Nicholas E. Thomas  Francesco
Tombesi  Alessio Trois 44 Sergey S. Tsygankov  Roberto Turolla  Jacco Vink
Martin C. Weisskopf  Fei Xie 66,4 Silvia Zane  [  [  [  [  [  [  [  [  [  [  [
[  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [
[  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [  [
[  [  [

###### Abstract

The magnetic field conditions in astrophysical relativistic jets can be probed
by multiwavelength polarimetry, which has been recently extended to X-rays.
For example, one can track how the magnetic field changes in the flow of the
radiating particles by observing rotations of the electric vector position
angle $\Psi$. Here we report the discovery of a $\Psi_{\mathrm{x}}$ rotation
in the X-ray band in the blazar Mrk 421 at an average flux state. Across the 5
days of Imaging X-ray Polarimetry Explorer (IXPE) observations of 4-6 and 7-9
June 2022, $\Psi_{\mathrm{x}}$ rotated in total by $\geq 360^{\circ}$. Over
the two respective date ranges, we find constant, within uncertainties,
rotation rates ($80\pm 9$ and $91\pm 8^{\circ}/\rm day$) and polarization
degrees ($\Pi_{\mathrm{x}}=10\%\pm 1\%$). Simulations of a random walk of the
polarization vector indicate that it is unlikely that such rotation(s) are
produced by a stochastic process. The X-ray emitting site does not completely
overlap the radio/infrared/optical emission sites, as no similar rotation of
$\Psi$ was observed in quasi-simultaneous data at longer wavelengths. We
propose that the observed rotation was caused by a helical magnetic structure
in the jet, illuminated in the X-rays by a localized shock propagating along
this helix. The optically emitting region likely lies in a sheath surrounding
an inner spine where the X-ray radiation is released.

##  1 Introduction

Despite decades of effort, the physical processes shaping the dynamics and
emission of relativistic jets are in large part still unclear (e.g. Blandford
et al. (2019)). However, in the last decade, several important clues have been
obtained thanks to new facilities that allow the possibility of following in
detail the time-variable emission properties of jets, especially at high
photon energies. A similar leap is now expected from the newly-opened window
of X-ray polarimetry, which provides us with an unprecedented view of the
physical sites of particle acceleration and emission.

Blazars, whose relativistic jets point close to the Earth, provide excellent
laboratories to investigate the physics of jets and test current theoretical
ideas (e.g. Romero et al. (2017)). Among blazars, the so-called high-peaked BL
Lac objects (HBLs), whose synchrotron and Compton components peak in the X-ray
and very-high-energy $\gamma$-ray band, respectively, are the ideal sources to
investigate acceleration and cooling processes acting on ultra-relativistic
electrons (with Lorentz factors $\gamma\gtrsim 10^{6}$). From the theoretical
point of view, the main mechanisms advocated for the energization of particles
in relativistic jets are diffusive shock acceleration (DSA; e.g., Blandford
and Eichler (1987)), magnetic reconnection (e.g., Sironi and Spitkovsky
(2014)), possibly triggered by instabilities (e.g., Bodo et al. (2021)) or
specific anti-parallel magnetic field structures Zhang et al. (2020), or
reconnection plus stochastic acceleration in relativistic turbulence in a
highly magnetized plasma (e.g., Comisso and Sironi (2018)).

Figure 1: From top to bottom: time evolution of the X-ray polarization angle,
degree, IXPE photon count rate in time bins of $\sim$3 hours within the IXPE
pointings of 4 June (red diamonds) and 7 June (blue diamonds) 2022, and
hardness ratio in the X-ray band (black squares) measured with Swift-XRT. As a
comparison, we display the polarization properties from simultaneous radio
(IRAM: black stars), infrared (Perkins: magenta diamonds), and optical
observations (NOT/RoboPol: green crosses).

Polarimetry is a powerful tool that can break the degeneracies involved in the
modeling of the spectral energy distribution (SED) and gain insight into the
magnetic field and emission region geometry Angel and Stockman (1980). Recent
theoretical efforts have allowed us to identify polarimetric signatures
expected under different scenarios of blazar emission regions that can be
contrasted with observational evidence. Polarimetric measurements are very
sensitive to the geometrical structure of the magnetic field permeating the
flow and to its global properties (e.g., chaotic vs. globally ordered). In
particular, the evidence for systematic and large variations of the
polarization angle, potentially associated with powerful $\gamma$-ray flares
(e.g., Blinov et al. (2015, 2018)), has been interpreted in terms of a
helically twisted jet (e.g., Larionov et al. (2013)), an emission region
moving along helical field lines that propagate down the jet (e.g., Vlahakis
(2006); Marscher et al. (2008, 2010)), or light-travel delays when a shock
forms in a jet with a predominantly helical/toroidal field Zhang et al. (2015,
2016). Such a structure for the global field is supported by (spatially
resolved) polarimetric studies of parsec-scale jets in the radio band Hovatta
et al. (2012); Gabuzda (2021). Interaction between a moving and a stationary
shock can also produce large rotations (e.g., Liodakis et al., 2020). An
alternative explanation is that the polarization behavior is stochastic,
related to turbulence in the flow D’Arcangelo et al. (2007); Peirson and
Romani (2018), possibly combined with a standing shock Marscher (2014, 2015).
In this framework, the observed polarimetric parameters do not carry any
direct information on the structure of the magnetic field in the jet, since
they are mainly related to the turbulent nature of the flow. Turbulence can
play an important role even if an average ordered magnetic field is present.
In this case, while the polarization angle can display rotations related to
the effect of the ordered field, the overall level of polarization can be
decreased by the depolarizing effect of turbulence (e.g., Marscher and Jorstad
(2021); Di Gesu et al. (2022)).

Mrk 421 is a nearby (redshift $z=0.0308$) HBL that has been intensively
studied at many wavelengths (e.g., Abdo et al. (2011)). It is among the first
blazars detected at both GeV (by EGRET onboard the Compton Gamma Ray
Observatory; Lin et al. (1992)) and TeV energies (by the Whipple Observatory;
Punch et al. (1992)). It is bright and well-monitored in the X-ray band (e.g.,
Giommi et al. (2021); Middei et al. (2022)), where the synchrotron SED peaks
at a high flux level, making it a prime target for linear polarization
observations by IXPE.

##  2 Results

In the first year of IXPE operation, we observed the blazar Mrk 421 three
times (on 4 May, 4 June, and 7 June 2022) to search for multi-epoch
variability of the X-ray polarization properties. In this case, a single IXPE
pointing extends over $\sim$2 days, which permits a search for intra-day
variations of the polarization properties as well Di Gesu et al. (2022). The
IXPE observations were complemented with radio-millimetre [Very Long Baseline
Array (VLBA), Institut de Radioastronomie Millimetrique (IRAM) and the
SubMillimeter Array (SMA)], infrared (KANATA, Perkins Telescope), and optical
polarization measurements [KANATA, Nordic Optical Telescope, Observatorio de
Sierra Nevada (OSN), and Skinakas Observatory], as detailed in Table 1 and
displayed in Figures 1 and 5. We captured the source in an average activity
state relative to its flux history (see Methods), discounting occasional
major, multi-band outbursts Donnarumma et al. (2009).

In May 2022, Mrk 421 was significantly X-ray polarized (with a polarization
degree $\Pi_{\rm x}\sim 15\%$ at a position angle $\Psi_{\rm x}\sim
35\hbox{${}^{\circ}$}$) and did not exhibit any substantial variability of the
X-ray polarization properties within the IXPE pointing Di Gesu et al. (2022).
This is consistent with the X-ray polarization of Mrk 501, the other
prototypical HBL observed by IXPE Liodakis et al. (2022). In both cases, the
X-ray polarization degree was larger than at radio, infrared, and optical
wavelengths. This suggested that the X-rays are produced by rapidly cooling,
high-energy electrons accelerated at a shock, while the emission at longer
wavelengths is emitted in a larger, downstream region containing a more
disordered magnetic field and lower-energy electrons at increased distances
from the shock.

In contrast, during two subsequent IXPE observations in June 2022, the X-ray
polarization was undetected in the time-averaged IXPE data (see Methods
section). In these observations, the X-ray flux was twice as high as in May
2022 ($\sim 1.5\times 10^{-10}$ and $\sim 3.02\times 10^{-10}$ erg s-1 cm-2 in
the 2-8 keV IXPE band on 4-6 June and 7-9 June, respectively). A similar
increase in flux (by a factor of $\sim$1.2; see Table 1 and Fig. 5) was also
measured in the radio (IRAM), infrared (Perkins), and optical (NOT) data
between 4-5 May and 6-7 June 2022.

When measured over shorter time intervals of the IXPE observations of June
2022 (Fig. 1), the X-ray polarization angle varied significantly in a manner
consistent with a smooth rotation, as often observed for blazars in the
optical band (e.g., Marscher et al., 2008, 2010; Blinov et al., 2015, 2018,
2021). For example, on 4-6 June the X-ray polarization angle
$\Psi_{\mathrm{x}}$ varied from $9\hbox{${}^{\circ}$}\pm 1\hbox{${}^{\circ}$}$
at the beginning of the IXPE pointing to $130\hbox{${}^{\circ}$}\pm
11\hbox{${}^{\circ}$}$ at the end of the observation, while on June 7-9
$\Psi_{\mathrm{x}}$ changed from $222\hbox{${}^{\circ}$}\pm
7\hbox{${}^{\circ}$}$ to $360\hbox{${}^{\circ}$}\pm 20\hbox{${}^{\circ}$}$
(considering time bins of $\sim$3 hours, as in Fig. 1). We adopt a simple
model where the polarization angle rotates at a constant rate, while the
polarization degree remains constant, to test the hypothesis that a
polarization angle rotation caused canceling of the polarization degree in the
time-averaged data. Using a maximum-likelihood method, a fit of binned Stokes
parameter time-series, and a fit in the Q-U plane (see Methods section), three
independent analyses by subsets of the authors support this hypothesis. The
data of 4-6 June are consistent with a rotation rate $\dot{\Psi}_{X}$ of
$80\pm 9\hbox{${}^{\circ}$}/\rm day$ at constant $\Pi=10\%\pm 1\%$, while for
the data of 7-9 June the parameters are $\dot{\Psi}_{X}=91\pm
8\hbox{${}^{\circ}$}/\rm day$ and $\Pi_{\rm X}=10\%\pm 1\%$. Our analysis
suggests that both $\Pi_{\rm X}$ and $\dot{\Psi}_{X}$ probably varied somewhat
with time about the above average values.

While the X-ray polarization angle was rotating, the millimetre-wave,
infrared, and optical polarization angles did not vary substantially (Fig. 1).
In the VLBA images of the radio core of 5 June (see Methods), some fanning out
of the polarization vectors on the southeast side of the the core is apparent,
which indicates an azimuthal component to the magnetic field, as one would
expect if that component were helical. In the optical, the scatter of
$\Psi_{\mathrm{O}}$ during the June IXPE observations is $\sim
20\hbox{${}^{\circ}$}$, with no significant trend. Given the gaps in the
optical observations, and considering the intrinsic 180∘ ambiguity, it is
possible that a poorly sampled rapid rotation occurred. However, no
straightforward scenario makes all the radio/infrared/optical data points
consistent with the time series of the X-ray polarization angle. Therefore, we
conclude that polarization angle rotation at the longer wavelengths was either
absent or proceeded at a different rate compared to the X-ray band.
Simultaneously with the polarization angle rotation, the X-ray spectrum
changed significantly (see Fig. 1, bottom panel, and Methods section). Swift
X-ray Telescope (XRT) data taken over the same time frame as the IXPE pointing
show that the flux in the 2.0-10.0 keV band increased by a factor of $\sim$3,
while in the 0.3-2.0 band the increase in flux was by a factor of $\sim$1.2.
The maximum flux was recorded by the Swift-XRT in correspondence with the
beginning of the second IXPE pointing. Then, by the end of the IXPE
observation, the source brightness relaxed back to the same level as in the
beginning. The spectral shape also changed in a harder-when-brighter fashion,
with the brightest point corresponding to the flattest (hardest) spectrum. In
contrast, in May 2022 the X-ray flux during the IXPE pointing changed in a
similar fashion, but with no change in X-ray spectral shape or polarization
angle Di Gesu et al. (2022). At $\sim$GeV $\gamma$-ray energies, Mrk 421 was
in a quiescent state during the IXPE pointing: the Fermi Large Area Telescope
(LAT) (Atwood et al., 2009) observed minor activity in the 6-month period
around the IXPE observations, never reaching more than twice the average
catalog flux (see fig. 3).

Figure 2: Sketches of the scenario proposed to explain the X-ray polarization
angle rotation in Mrk 421. Left: an off axis emission feature, e.g., a
magnetosonic shock, propagates along helical magnetic field lines down the
jet. Right: the appearance of the emission feature, magnetic field, and
polarization vector at 4 azimuthal positions along its spiral path as viewed
by a distant observer aligned with the jet. The red circle represents the
emission feature, while the blue-shaded region is the ambient jet.

##  3 Discussion

Virtually all of the processes shaping the dynamics of particles in blazar
jets predict specific characteristics of variability of the optical-to-X-ray
polarization properties Tavecchio (2021), which we can compare against our
observations of Mrk 421. Our basic finding conforms to the scenario of an
energy-stratified jet, as discussed above for Mrk 421 in May 2022 and for Mrk
501. We find that the roughly constant value of $\Pi_{\rm X}$, while the
polarization angle was rotating, is higher than $\Pi_{\rm O}$, $\Pi_{\rm IR}$,
and $\Pi_{\rm R}$, and similar to the level observed at non-rotating epochs.
In addition, simultaneous radio, infrared, and optical polarization
measurements do not show evidence of a similar polarization angle rotation.
Due to the limited sampling and intrinsic 180∘ ambiguity, we cannot exclude
the possibility that the optical polarization angle was rotating at some
level. However, we consider a rotation of the optical polarization angle at
the same rate as the X-ray polarization angle unlikely. Indeed, typical
optical polarization rotation rates of blazars vary between a few and a few
tens of degrees per day Kiehlmann et al. (2021), in contrast to the $\sim
85\hbox{${}^{\circ}$}/\rm day$ that we estimated in the X-ray. The optical
rotation rates may be higher during bright outbursts associated with the
passage of knots through the radio core MAGIC Collaboration et al. (2018);
Liodakis et al. (2020). No such outburst was recorded for Mrk 421 at the time
of the IXPE pointings. Moreover, no solution of the 180∘ ambiguity can
completely reconcile the radio/optical/infrared polarization angle time series
with the X-ray curve. Therefore, as the polarization angle was likely not
rotating (or rotating in a different way) at longer wavelengths, we suggest
that the radio/infrared/optical emission sites are, at most, only partially
co-spatial with the X-ray emission region.

In an energy-stratified jet, the X-rays are emitted closer to the site of
acceleration of the particles, while the lower-enegy particles emitting at
radio to optical wavelengths span a larger region downstream owing to their
longer radiative cooling times (e.g., Liodakis et al. (2022)). Acceleration of
electrons by a shock can explain the high X-ray polarization, since the shock
front partially aligns a previously disordered magnetic field (e.g., Hughes et
al. (1985); Tavecchio et al. (2020)). The harder-when-brighter spectral
behavior that Mrk 421 displayed while the X-ray polarization angle was
rotating is also expected within the shock scenario Kirk et al. (1998).

The energy stratification of the particles in the jet can be either linear, in
the downstream direction, or radial, in the case of a structured jet
Georganopoulos and Kazanas (2003); Ghisellini et al. (2005); Chhotray et al.
(2017), including an inner fast spine surrounded by a slower sheath. Such a
spine-sheath jet explains, for example, the limb-brightening morphology that
has been reported in very long-baseline interferometry (VLBI) radio maps of
HBLs, including Mrk 421 Giroletti et al. (2006); Piner et al. (2010). In the
first case, the turbulent component of the magnetic field could become
increasingly dominant over the ordered component with distance from the shock
Marscher and Jorstad (2022). In the second case, turbulence may prevail in the
sheath. In both these frameworks, the particle-energy stratification would
lead to lower polarization of the optical, infrared, and radio emission.
Detailed modeling of the variability of flux and polarization induced by
turbulence predicts erratic variability, as seen in the past at optical
wavelengths in Mrk 421 Marscher and Jorstad (2022), which may by chance result
in an apparent rotation of the polarization angle Marscher (2014). However,
such apparent rotations are highly unlikely to explain, for example, the
systematic rotations of the optical polarization angle in BL Lac and PKS
1510$-$089 Marscher et al. (2008, 2010). This has led to the proposal that
these events occurred upstream of the most turbulent region of the jet. In a
larger study of a number of optical-band rotation events in a sample of
blazars, it was found that a random walk of the polarization vector cannot
reproduce the majority of the observed properties of the population Blinov et
al. (2015); Kiehlmann et al. (2017).

We have investigated whether the rotation of the polarization vector in Mrk
421 could be produced through random walks of the polarization angle by
comparing the observations against simulated polarization light curves,
following Kiehlmann et al. (2016, 2017) (see Methods). We consider two
scenarios: either the rotations in the observations of 4-6 June and 7-9 June
are independent events, or they are part of a single long rotation that
spanned seven days, of which we observed two segments separated by a one-day
gap. In either scenario, we find, under the most favorable conditions, that
only about 2% of the simulated curves of $\dot{\Psi}_{X}$ and $\Pi_{X}$ vs.
time are consistent with the observations. For the second scenario, if the
rotation were actually longer than 7 days, the fraction of simulation trials
that reproduce the observed properties would decrease. This suggests that it
is unlikely that the observed behavior can be attributed to random variations
of the polarization angle, and instead points to a coherent, deterministic
process responsible for the observed rotation(s).

Magnetic reconnection episodes, which can result from the development of a
large-scale kink instability Bodo et al. (2021) or current sheet Zhang et al.
(2021), may also cause rapid polarization angle swings. These could be
associated with $\gamma$-ray flares Zhang et al. (2022) or quasi-periodic
oscillations of the source flux Dong et al. (2020); Jorstad et al. (2022). In
our case, we did not witness any such oscillations or $\gamma$-ray flare while
the polarization angle was rotating. Moreover, in contrast to our observations
of Mrk 421, simulations of magnetic reconnection often display, on average, a
similar level of polarization degree in the X-ray and in the optical Di Gesu
et al. (2022). In some cases, the X-ray polarization degree can also be much
lower than the optical because of time averaging of the polarization vector,
which is the opposite of what is observed Di Gesu et al. (2022).

A specific model for deterministic polarization angle rotation involves an
off-axis emission feature, such as a magnetosonic shock, propagating toward
the observer and down a helical magnetic field Marscher et al. (2008, 2010);
Larionov et al. (2013); Zhang et al. (2016). The observed rotation rate is
determined by the time required for the feature to execute an orbit around the
jet axis, modulated by light-travel delays and possibly other relativistic
effects Peirson and Romani (2018). We display sketches of this scenario in
Fig. 2, both in the reference frame of the host galaxy (left panel) and as
seen by an observer whose line of sight is along the jet axis (right panel).

As a quantitative example of such a model, we find that we can reproduce the
observed rotation rate $\dot{\Psi}_{X}\sim 85$ $\hbox{${}^{\circ}$}/\rm day$
over 5 days if (1) the center of the emission feature is displaced by
$2-4\times 10^{16}$ cm from the jet axis (for an angle to the line of sight
ranging from $4^{\circ}$ to $0^{\circ}$), and (2) moves at a velocity of
$0.99875c$ (Lorentz factor of 20), with a component parallel to the jet axis
of $0.9975c$ and a transverse, rotational component of $0.05c$. During the
rotation, the feature moves through a distance between 0.8 and 1.7 pc
(depending on the line-of-sight angle) parallel to the line of sight in the
observer’s frame.

If the jet is structured, the helical field component may be limited to the
spine. Moreover, the spine can also contribute somewhat to the emission at
longer wavelengths. This would lead to different polarization behaviour in
X-ray than at longer wavelengths, and, at the same time, can explain a
possible rotation of the optical polarization angle and the similarities in
the variability of the X-ray to millimetre fluxes on longer time-scales (i.e.,
between May and June 2022).

The discovery of X-ray polarization angle rotation in HBLs, made possible by
the new capability of IXPE, opens up the possibility of fully investigating
analogies and differences between polarization angle rotations at different
frequencies and for different classes of blazars. For example, Angelakis et
al. (2016) discuss a version of the model in which the polarization angle
rotations occur immediately downstream of a shock, where the compressed
magnetic field is dominated by an ordered (helical) magnetic field superposed
on a turbulent component. As the shock is the site of acceleration of the
particles that radiate at the frequency of the peak of the synchrotron SED,
the highest polarization and main rotation events should be observed near that
frequency. This argument has been invoked to explain why optical polarization
angle rotations preferably occur in low-synchrotron-peak (LSP) blazars Blinov
et al. (2016), and naturally predicts a more frequent occurrence of X-ray
polarization angle rotations in the case of HBLs. In the same speculative
framework, as optical polarization angle rotations are often correlated with
$\sim$GeV $\gamma$-ray flares Blinov et al. (2018), X-ray polarization angle
rotations should be accompanied by $\sim$TeV activity, which is, in the case
of HBLs, the analog of $\sim$GeV activity for LSPs.

In conclusion, this first study of X-ray polarization angle rotation in HBLs
in a multi-wavelength framework shows how the availability of the X-ray
polarization diagnostic enriches our ability to probe the magnetic field
geometry and particle acceleration in different regions of relativistic jets,
and thereby represents a new, fundamental step towards a comprehensive view of
relativistic astrophysical jets.

Acknowledgments

The Imaging X-ray Polarimetry Explorer (IXPE) is a joint US and Italian
mission. The US contribution is supported by the National Aeronautics and
Space Administration (NASA) and led and managed by its Marshall Space Flight
Center (MSFC), with industry partner Ball Aerospace (contract NNM15AA18C). The
Italian contribution is supported by the Italian Space Agency (Agenzia
Spaziale Italiana, ASI) through contract ASI-OHBI-2017-12-I.0, agreements ASI-
INAF-2017-12-H0 and ASI-INFN-2017.13-H0, and its Space Science Data Center
(SSDC), and by the Istituto Nazionale di Astrofisica (INAF) and the Istituto
Nazionale di Fisica Nucleare (INFN) in Italy. This research used data products
provided by the IXPE Team (MSFC, SSDC, INAF, and INFN) and distributed with
additional software tools by the High-Energy Astrophysics Science Archive
Research Center (HEASARC), at NASA Goddard Space Flight Center (GSFC). The
IAA-CSIC group acknowledges financial support from the grant CEX2021-001131-S
funded by MCIN/AEI/10.13039/501100011033 to the Instituto de Astrofísica de
Andalucía-CSIC and through grant PID2019-107847RB-C44. The POLAMI observations
were carried out at the IRAM 30m Telescope. IRAM is supported by INSU/CNRS
(France), MPG (Germany), and IGN (Spain). The Submillimetre Array is a joint
project between the Smithsonian Astrophysical Observatory and the Academia
Sinica Institute of Astronomy and Astrophysics and is funded by the
Smithsonian Institution and the Academia Sinica. Mauna Kea, the location of
the SMA, is a culturally important site for the indigenous Hawaiian people; we
are privileged to study the cosmos from its summit. Some of the data reported
here are based on observations made with the Nordic Optical Telescope, owned
in collaboration with the University of Turku and Aarhus University, and
operated jointly by Aarhus University, the University of Turku, and the
University of Oslo, representing Denmark, Finland, and Norway, the University
of Iceland and Stockholm University at the Observatorio del Roque de los
Muchachos, La Palma, Spain, of the Instituto de Astrofisica de Canarias. E. L.
was supported by Academy of Finland projects 317636 and 320045. The data
presented here were obtained [in part] with ALFOSC, which is provided by the
Instituto de Astrofisica de Andalucia (IAA) under a joint agreement with the
University of Copenhagen and NOT. We are grateful to Vittorio Braga, Matteo
Monelli, and Manuel Sänchez Benavente for performing the observations at the
Nordic Optical Telescope. Part of the French contributions is supported by the
Scientific Research National Center (CNRS) and the French spatial agency
(CNES). The research at Boston University was supported in part by National
Science Foundation grant AST-2108622, NASA Fermi Guest Investigator grants
80NSSC21K1917 and 80NSSC22K1571, and NASA Swift Guest Investigator grant
80NSSC22K0537. This research was conducted in part using the Mimir instrument,
jointly developed at Boston University and Lowell Observatory and supported by
NASA, NSF, and the W.M. Keck Foundation. We thank D. Clemens for guidance in
the analysis of the Mimir data. This work was supported by JST, the
establishment of university fellowships towards the creation of science and
technology innovation, Grant Number JPMJFS2129. This work was supported by
Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Numbers
JP21H01137. This work was also partially supported by the Optical and Near-
Infrared Astronomy Inter-University Cooperation Program from the Ministry of
Education, Culture, Sports, Science and Technology (MEXT) of Japan. We are
grateful to the observation and operating members of the Kanata Telescope.
Some of the data are based on observations collected at the Observatorio de
Sierra Nevada, owned and operated by the Instituto de Astrofísica de Andalucía
(IAA-CSIC). Further data are based on observations collected at the Centro
Astronómico Hispano en Andalucía (CAHA), operated jointly by Junta de
Andalucía and Consejo Superior de Investigaciones Científicas (IAA-CSIC). This
research has made use of data from the RoboPol program, a collaboration
between Caltech, the University of Crete, IA-FORTH, IUCAA, the MPIfR, and the
Nicolaus Copernicus University, which was conducted at Skinakas Observatory in
Crete, Greece. D.B., S.K., R.S., N. M., acknowledge support from the European
Research Council (ERC) under the European Unions Horizon 2020 research and
innovation program under grant agreement No. 771282. CC acknowledges support
from the European Research Council (ERC) under the HORIZON ERC Grants 2021
program under grant agreement No. 101040021. The research at Boston University
was supported in part by National Science Foundation grant AST-2108622, NASA
Fermi Guest Investigator grant 80NSSC21K1917 and 80NSSC22K1571, and NASA Swift
Guest Investigator grant 80NSSC22K0537. This work was supported by NSF grant
AST-2109127. We acknowledge the use of public data from the Swift data
archive. Based on observations obtained with XMM-Newton, an ESA science
mission with instruments and contributions directly funded by ESA Member
States and NASA. Data from the Steward Observatory spectropolarimetric
monitoring project were used. This program is supported by Fermi Guest
Investigator grants NNX08AW56G, NNX09AU10G, NNX12AO93G, and NNX15AU81G. We
acknowledge funding to support our NOT observations from the Finnish Centre
for Astronomy with ESO (FINCA), University of Turku, Finland (Academy of
Finland grant nr 306531). This work has made use of data from the Asteroid
Terrestrial-impact Last Alert System (ATLAS) project. The Asteroid
Terrestrial-impact Last Alert System (ATLAS) project is primarily funded to
search for near earth asteroids through NASA grants NN12AR55G, 80NSSC18K0284,
and 80NSSC18K1575; byproducts of the NEO search include images and catalogs
from the survey area. This work was partially funded by Kepler/K2 grant
J1944/80NSSC19K0112 and HST GO-15889, and STFC grants ST/T000198/1 and
ST/S006109/1. The ATLAS science products have been made possible through the
contributions of the University of Hawaii Institute for Astronomy, the Queen’s
University Belfast, the Space Telescope Science Institute, the South African
Astronomical Observatory, and The Millennium Institute of Astrophysics (MAS),
Chile. The Very Long Baseline Array is an instrument of the National Radio
Astronomy Observatory. The National Radio Astronomy Observatory is a facility
of the National Science Foundation operated under cooperative agreement by
Associated Universities, Inc.

Author contributions

L. Di Gesu performed the analysis and led the writing of the paper. H.
Marshall, S. R Ehlert, D. E. Kim, F. Muleri and A. Di Marco contributed to the
IXPE data analysis. I. Donnarumma, F. Tavecchio, R.W. Romani A. P. Marscher,
and S. G. Jorstad contributed to the discussion and other parts of the paper,
and the latter two provided the VLBA images. I. Liodakis coordinated the
multiwavelength observations. I. Liodakis and S. Kiehlmann contributed the
random walk simulations. I. Agudo, C. Casadio, J. Escudero, M. Gurwell, I.
Myserlis, R. Rao and G. Keating contributed the radio observations. S.G.
Jorstad contributed the infrared observations. S.S. Savchenko performed
modeling of the host galaxy in H band. B. Agís-González, F. J. Aceituno, I.
Agudo, H. Akitaya, M. I. Bernardos, D. Blinov, G. Bonnoli, I. G. Bourbah, V.
Casanova, Y. Fukazawa, M. García-Comas, C. Husillos, R. Imazawa, K. S.
Kawabata, S. Kielhmann, E. Kontopodis, P. M. Kouch, E. Lindfors, N.
Mandarakas, A. Marchini, T. Mizuno, T. Nakaoka, S. Romanopoulos, M. Sasada, R.
Skalidis, A. Sota, M. Uemura, and A. Vervelaki, contributed the optical
observations. A.A. Vasilyev has contributed to the paper discussion. R.
Middei, M. Perri, and S. Puccetti contributed to the Swift data analysis. L.
Pacciani and M. Negro contributed the Fermi data. The remaining authors are
part of the IXPE team whose significant contribution made the X-ray
polarization observations possible.

Competing interest The authors declare that they have no competing interests.

Data availability The data that support the findings of this study are freely
available in the HEASARC IXPE Data Archive
(https://heasarc.gsfc.nasa.gov/docs/ixpe/archive/). The multiwavelength data
are available on request from the individual observatories.

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## Methods

Table 1: Log of polarization measurements of Mrk 421 used in this work. Telescope | Band | Dates | Radio Flux Density | Radio Polarization
---|---|---|---|---
| (eV) | (YYYY-MM-DD) | ($10^{-25}$erg s-1 cm-2 Hz-1) |  $\Pi_{\rm R}$(%) $\Psi_{\rm R}$ (∘)
SMA | $9.5\times 10^{-4}$ | 2022-06-04 | $28\pm 2$ |  $2.3\pm 0.2$ $130\pm 2$
SMA | $9.5\times 10^{-4}$ | 2022-06-16 | $38\pm 2$ |  $2.3\pm 0.2$ $133\pm 4$
IRAM | $3.5\times 10^{-4}$ | 2022-05-04 | $28\pm 1$ |  $3.0\pm 0.7$ $237\pm 7$
IRAM | $3.5\times 10^{-4}$ | 2022-05-05 | $38\pm 2$ |  $3.8\pm 0.7$ $232\pm 4$
IRAM | $3.5\times 10^{-4}$ | 2022-05-31 | $32\pm 2$ |  $7.9\pm 0.8$ $128\pm 2$
IRAM | $3.5\times 10^{-4}$ | 2022-05-06 | $36\pm 1$ |  $1.8\pm 0.6$ $116\pm 9$
VLBA | $1.7\times 10^{-4}$ | 2022-04-30 | $34.6\pm 0.1$ |  $1.6\pm 0.3$ $100\pm 10$
VLBA | $1.7\times 10^{-4}$ | 2022-06-05 | $27.6\pm 0.1$ |  $2.6\pm 0.3$ $147\pm 10$
VLBA | $1.7\times 10^{-4}$ | 2022-06-24 | $25.6\pm 0.1$ |  $2.2\pm 0.3$ $132\pm 10$
Telescope | Band | Dates | Infrared flux density | Infrared Polarization
| (eV) | (YYYY-MM-DD) | ($10^{-25}$erg s-1 cm-2 Hz-1) |  $\Pi_{\rm IR}$(%) $\Psi_{\rm IR}$ (∘)
Perkins | H:0.9 | 2022-05-04 | $3.92\pm 0.05$ |  $1.3\pm 0.5$ $220\pm 11$
Perkins | H:0.9 | 2022-05-05 | $1.77\pm 0.03$ |  $2.9\pm 0.8$ $212\pm 8$
Perkins | H:0.9 | 2022-05-09 | $4.19\pm 0.05$ |  $3.1\pm 0.8$ $181\pm 7$
Perkins | H:0.9 | 2022-05-14 | $3.55\pm 0.04$ |  $1.7\pm 1.0$ $175\pm 17$
Perkins | H:0.9 | 2022-06-06 | $4.87\pm 0.05$ |  $3.9\pm 0.5$ $128\pm 3$
Perkins | H:0.9 | 2022-06-07 | $4.52\pm 0.05$ |  $4.8\pm 0.9$ $147\pm 5$
Perkins | H:0.9 | 2022-06-08 | $4.14\pm 0.06$ |  $6.2\pm 1.3$ $145\pm 6$
Kanata 111Not corrected for dilution of polarization by unpolarized starlight from the host galaxy. | H:0.9 | 2022-05-04 | $-$ |  $3.1\pm 0.2$ $197\pm 2$
Kanata 111Not corrected for dilution of polarization by unpolarized starlight from the host galaxy. | H:0.9 | 2022-05-05 | $-$ |  $3.4\pm 0.3$ $187\pm 4$
Telescope | Band | Dates | Optical flux density | Optical Polarization
| (eV) | (YYYY-MM-DD) | ($10^{-25}$erg s-1 cm-2 Hz-1) |  $\Pi_{\rm O}$(%) $\Psi_{\rm O}$ (∘)
Skinakas | R:1.9 | 2022-05-13 | $1.60\pm 0.04$ |  $2.7\pm 0.2$ $192\pm 2$
Skinakas | R:1.9 | 2022-05-13 | $1.64\pm 0.04$ |  $3.2\pm 0.2$ $195\pm 2$
Skinakas | R:1.9 | 2022-05-13 | $1.61\pm 0.04$ |  $2.7\pm 0.2$ $193\pm 2$
Skinakas | R:1.9 | 2022-05-14 | $1.48\pm 0.04$ |  $3.7\pm 0.2$ $179\pm 2$
Skinakas | R:1.9 | 2022-05-16 | $1.41\pm 0.04$ |  $3.5\pm 0.3$ $176\pm 2$
Skinakas | R:1.9 | 2022-05-18 | $1.41\pm 0.04$ |  $2.7\pm 0.4$ $179\pm 5$
Skinakas | R:1.9 | 2022-05-22 | $1.53\pm 0.04$ |  $3.3\pm 0.2$ $118\pm 2$
Skinakas | R:1.9 | 2022-05-30 | $2.02\pm 0.04$ |  $5.6\pm 0.2$ $149\pm 1$
Skinakas | R:1.9 | 2022-05-31 | $1.98\pm 0.04$ |  $5.4\pm 0.2$ $150\pm 1$
Skinakas | R:1.9 | 2022-06-01 | $2.05\pm 0.04$ |  $5.3\pm 0.2$ $142\pm 1$
Skinakas | R:1.9 | 2022-06-02 | $1.93\pm 0.04$ |  $5.1\pm 0.2$ $152\pm 1$
Skinakas | R:1.9 | 2022-06-07 | $2.12\pm 0.04$ |  $5.4\pm 0.3$ $145\pm 1$
Skinakas | R:1.9 | 2022-06-17 | $2.05\pm 0.04$ |  $4.7\pm 0.3$ $152\pm 2$
NOT | R:1.9 | 2022-05-04 | $1.89\pm 0.04$ |  $2.9\pm 0.1$ $202\pm 1$
NOT | R:1.9 | 2022-06-04 | $2.02\pm 0.04$ |  $4.8\pm 0.1$ $153\pm 1$
NOT | R:1.9 | 2022-06-05 | $2.25\pm 0.04$ |  $3.9\pm 0.1$ $133\pm 1$
NOT | R:1.9 | 2022-06-06 | $2.27\pm 0.07$ |  $5.6\pm 0.2$ $146\pm 1$
SNO | R:1.9 | 2022-01-16 | $2.34\pm 0.01$ |  $3.8\pm 0.4$ $136\pm 3$
SNO | R:1.9 | 2022-04-19 | $1.95\pm 0.01$ |  $4.6\pm 0.7$ $264\pm 3$
SNO | R:1.9 | 2022-06-16 | $2.40\pm 0.50$ |  $9.5\pm 1.1$ $137\pm 4$
Kanata 111Not corrected for dilution of polarization by unpolarized starlight from the host galaxy. | R:1.9 | 2022-05-04 | $2.50\pm 0.40$ |  $3.1\pm 0.2$ $200\pm 1$
Kanata 111Not corrected for dilution of polarization by unpolarized starlight from the host galaxy. | R:1.9 | 2022-05-05 | $2.20\pm 0.02$ |  $3.4\pm 0.3$ $201\pm 1$
Telescope | Energy range | Dates | X-ray flux | X-ray Polarization222X-ray polarization degrees are for the time-averaged IXPE data.
| (keV) | (YYYY-MM-DD) | ($10^{-11}$erg s-1 cm-2) |  $\Pi_{\rm X}$(%) $\Psi_{\rm X}$ (∘)
IXPE | 2.0-8.0 | 2022-05-04 | $8.67\pm 0.03$ |  $15\pm 2$ $35\pm 4$
IXPE | 2.0-8.0 | 2022-06-04 | $15.69\pm 0.09$ | NOT DETECTED
IXPE | 2.0-8.0 | 2022-06-07 | $30.2\pm 0.2$ | NOT DETECTED
|  |  |  |
Table 2: Log of Swift observations of Mrk 421 in May/June 2022. Swift-Obs ID | Dates | 0.3-10.0 keV flux
---|---|---
| (YYYY-MM-DD) | ($10^{-11}$erg s-1 cm-2)
31540031 | 2022-05-01 | $46\pm 1$
31540032 | 2022-05-03 | $38\pm 1$
31540033 | 2022-05-05 | $44\pm 1$
31540034 | 2022-05-07 | $64\pm 1$
96557003 | 2022-05-02 | $46\pm 1$
96557004 | 2022-05-03 | $42\pm 1$
96557005 | 2022-05-04 | $39\pm 1$
96557006 | 2022-05-05 | $48\pm 1$
96557008 | 2022-05-08 | $84\pm 1$
31540051 | 2022-06-01 | $93\pm 1$
31540052 | 2022-06-02 | $90\pm 1$
96557009 | 2022-06-03 | $88\pm 1$
31540053 | 2022-06-04 | $75\pm 1$
31540054 | 2022-06-04 | $79\pm 1$
31540055 | 2022-06-05 | $69\pm 1$
96557010 | 2022-06-06 | $116\pm 1$
89326001 | 2022-06-07 | $141\pm 1$
96557011 | 2022-06-08 | $104\pm 2$
96557012 | 2022-06-12 | $73\pm 2$
31540056 | 2022-06-16 | $52\pm 1$
31540057 | 2022-06-17 | $69\pm 1$
31540059 | 2022-06-19 | $110\pm 1$
31540060 | 2022-06-20 | $74\pm 1$
31540061 | 2022-06-21 | $59\pm 1$
31540062 | 2022-06-22 | $61\pm 1$
31540063 | 2022-06-23 | $62\pm 1$
31540064 | 2022-06-24 | $56\pm 1$
|  |

### IXPE and multi-wavelength data

#### IXPE data

The log of all of our observations of Mrk 421 in May-June 2022, including
radio/millimetre, optical, infrared, and X-ray polarization measurements,
along with Swift monitoring in the X-ray band, is listed in Table 1 and 2.
Here we report the analysis of the second and third IXPE observations of Mrk
421 that were performed from 11:20 UTC on 4 June 2022 to 11:00 UTC on 6 June
(hereafter Obs. 2), and from 09:00 UTC on 7 June to 11:10 UTC on 9 June
(hereafter Obs. 3). The livetimes for these pointings are $\sim$96 ks and
$\sim$86 ks, respectively.

The raw IXPE data were processed using a standard
pipeline111https://heasarc.gsfc.nasa.gov/docs/ixpe/analysis/IXPE-SOC-
DOC-009-UserGuide-Software.pdf that estimates the photo-electron emission
direction, correcting for charging effects, detector temperature, Gas Electron
Multiplier (GEM) gain non-uniformity, and instrumental spurious modulation
(Rankin et al., 2022). The level-2 event files (one for each of the three
detector units (DU) on-board IXPE) from the pipeline contain all the
information typical of an imaging X-ray astronomy mission (e.g., photon
arrival time, detector coordinates, and energy), with the addition of the
polarization information in the form of event-by-event Stokes parameters.
Before proceeding with the scientific analysis, when necessary, we adjusted
the data for small time-dependent changes to the gain correction obtained from
data taken with the on-board calibration sources close to the actual time of
observation (Ferrazzoli et al., 2020).

We performed the scientific analysis of the IXPE data using the ixpeobssim
software (Pesce-Rollins et al., 2019; Baldini et al., 2022). At the angular
resolution of IXPE ($\sim 30^{\prime\prime}$), blazars like Mrk 421 are point-
like sources. Using the xpselect tool, we selected the source events using a
circular 60′′ radius region, while we used an annulus centered on the source
(with inner and outer radii of 2.5′ and 5.0′) to estimate the background.
Within the ixpeobssim suite, the method of Kislat et al. (2015) for estimating
the polarization of a set of events is implemented in the PCUBE algorithm. We
created the Stokes-parameter spectra of the source, using the PHA1, PHAQ, PHAU
algorithms. The PHA1, PHAQ, and PHAU algorithms map the $I$, $Q$, and $U$
Stokes parameters of the photons into OGIP-compliant PHA spectral files (3
Stokes-parameter spectra per 3 detector units). We prepared the spectra for
the spectro-polarimetric fit by binning the $I$ spectrum, requiring a minimum
of 30 counts in each energy bin, as needed for the $\chi^{2}$ statistics in
the fits. Finally, we binned the $Q$ and $U$ spectra by grouping the channels
by a constant factor of 10. To search for variability of the flux and
polarization properties within the time span of the IXPE pointings, we created
light curves with different time binning, using the LC algorithm in
ixpeobssim. We applied the time binning of these light-curves to time-slice
each of the event files and computed a PCUBE in each time interval. We thereby
created light curves of the polarization properties as a function of time,
including normalized Stokes parameters ($q(t)$,$u(t)$), polarization degree
($\Pi(t)$), and polarization angle ($\Psi(t)$).

We applied to the $\Psi(t)$ light-curves a standard procedure (Blinov et al.,
2015) to solve the 180∘ ambiguity of $\Psi$. By assuming that temporal
variations of $\Psi$ are gradual, we minimized the variation $\Delta_{\rm
n}\Psi$ between consecutive points $\Psi_{\rm n}$ and $\Psi_{\rm n+1}$ as
follows. For two consecutive data points $\Delta_{\rm n}\Psi$ was defined as
$\lvert\Psi_{\rm n+1}-\Psi_{\rm n}\rvert-\sqrt{\sigma^{2}(\Psi_{\rm
n+1})+\sigma^{2}(\Psi_{\rm n})^{2}}$. where $\sigma(\Psi_{\rm n+1})$ and
$\sigma(\Psi_{\rm n})$ are the errors for the consecutive angles considered.
If $\Delta_{\rm n}\Psi$ was larger than $90\hbox{${}^{\circ}$}$, we shifted
$\Psi_{\rm n+1}$ by multiples of $\pm 180\hbox{${}^{\circ}$}$ as needed to
minimize $\Delta_{\rm n}\Psi$. Otherwise, when $\Delta_{n}\leq
90\hbox{${}^{\circ}$}$, we left $\Psi_{\rm n+1}$ unshifted. This procedure
serves only for clarity of displaying the data, and is not critical for our
estimation of the rotation rate, which was performed using unbinned event data
or Stokes-parameter time series (see below).

####  $\gamma$-ray data

We studied the $\gamma$-ray activity of Mrk 421, analyzing data collected with
the Fermi-LAT in the energy range 0.1-100 GeV. We made use of the 3-day
cadence light curve (within the years 2021 and 2022), which was extracted from
the LAT Light Curve Repository (LCR), as displayed in Figure 3. We refer to
(Soheila et al., 2023) for the detailed description of the unbinned maximum-
likelihood data analysis. For this work, we adopted the light curve that was
obtained using the option of leaving the spectral slope free to vary in the
fit.

Figure 3: Fermi-LAT light curve of Mrk 421, integrated over the 0.1 – 100 GeV
energy range. The gray band illustrates the time range of the two IXPE
observations of Mrk 421 discussed in this work. The red dashed line indicates
the average photon flux of Mrk 421 over the LAT mission duration, while the
blue solid line marks the highest photon flux recorded by the LAT, which
corresponds to the Mrk 421 flare of 2012. The plots include only points for
which the test statistic for source detection is above 9. Figure 4: Swift-XRT
light curves of Mrk 421 in May-June 2022. From top to bottom: soft (S band:
0.3-2.0 keV, red stars) and hard (H band: 2.0-10.0 keV, blue squares) fluxes,
hardness ratio, $\alpha$ and $\beta$ parameters for a best-fit log-parabolic
model. The dashed vertical lines indicate the beginning and the end of each
IXPE observation. Figure 5: Time evolution of the flux (top panel),
polarization angle (middle panel), and polarization degree (bottom panel) of
Mrk 421 at radio/millimetre (IRAM: black stars, SMA: black diamonds), infrared
(Perkins: blue triangles), and optical (NOT: magenta circles, RoboPol: magenta
squares, SNO: magenta triangles, Kanata: magenta diamonds) wavelengths. The
dashed vertical lines mark the beginning and the end of each of the three IXPE
observations.

#### X-ray flux and continuum spectrum

We monitored the X-ray (0.3-10.0 keV) flux and spectral variations of Mrk 421
both near the time of, and simultaneously with, the IXPE pointings of May-June
2022 using the Neil Gehrels Swift X-Ray Telescope (XRT) (Table 2). The Swift-
XRT observations each had an exposure time of $\sim 1$ ks and were performed
in Windowed Timing (WT) mode. The data were processed using the XRT Data
Analysis Software (XRTDAS, v. 3.6.1). In the analysis, we used the latest
calibration files available in the Swift-XRT CALDB (version 20210915). The
X-ray source spectrum was extracted from the cleaned event file using a
circular region with a radius of 47′′. The background was extracted using a
circular region with the same radius from a blank sky WT observation available
in the Swift archive. As a final step, we binned the 0.3-10 keV data to
achieve at least 25 counts in each energy bin.

For our spectral analysis, we used the Xspec program Arnaud et al. (1999). In
all our fits we included Galactic absorption along the line of sight (
$N{\rm{}_{H}}$ =$1.34\times 10^{20}$ cm-2; HI4PI Collaboration et al. (2016))
using the TBABS model, with Fe abundances set according to Wilms et al.
(2000). For all the datasets, we found that a log-parabolic model, where the
photon index varies as a log-parabola in energy (i.e., $N(E)=K(E/E_{\rm
p})^{(\alpha-\beta\log{(E/E_{p}}))}$ Massaro et al. (2004)), fits the spectra
better than a simple power law (with $\Delta\chi^{2}$ in the range between -23
and -233, for one additional degree of freedom). In the log-parabolic model,
the pivot energy $E_{\rm p}$ is a scaling factor, $\alpha$ is the spectral
slope at the pivot energy, $\beta$ corresponds to the spectral curvature, and
$K$ is a normalization constant. In our fits, we set the pivot $E_{\rm p}$ to
5.0 keV (Baloković et al., 2016; Di Gesu et al., 2022), so that $\alpha$
approximates the photon index in the 3.0-7.0 keV range. In this way, we
extracted the X-ray spectral evolution of Mrk 421 in May/June 2022 (Fig. 4),
including soft (S band: 0.3-2.0 keV) and hard (H band: 2.0-10.0 keV) X-ray
fluxes, hardness ratio (H-S/H+S), and $\alpha$ and $\beta$ parameters.

Within the IXPE pointing of May 2022, the X-ray flux rose by a factor of
$\sim$2 with no significant change in spectral shape Di Gesu et al. (2022).
Conversely, during the two IXPE pointings of June 2022, the flux increased
from $\sim 7.5\times 10^{-10}$ erg s-1 cm-2 at the beginning of Obs. 2 to the
brightest point of $\sim 1.4\times 10^{-9}$ erg s-1 cm-2 at the beginning of
Obs. 3, and then decreased back to the previous level. The maximum variation
of the soft flux was by a factor of $\sim 1.2$, while the hard flux varied by
a factor of $\sim 3$. This is in agreement with the IXPE measurement in the
2.0-8.0 keV band, where the flux of Obs. 3 was $\sim$twice as high as during
Obs. 2 (see Table 1). Therefore, the spectrum became harder and flatter (i.e.,
$\alpha$ decreased while the hardness ratio increased), with the flattest
spectral shape coinciding with the brightest point.

#### Optical & infrared data

All the optical, infrared, and radio/millimetre polarization measurements of
Mrk 421 in the time frame as the IXPE pointings are displayed in Fig. 5.
Polarization observations in the optical were performed simultaneously with
the IXPE pointings using the Hiroshima Optical and Near-InfraRed camera
(HONIR) at the KANATA telescope, RoboPol at the Skinakas observatory 1.3-m
telescope (Ramaprakash et al., 2019), the Alhambra Faint Object Spectrograph
and Camera (ALFOSC) mounted at the 2.5m Nordic Optical Telescope (NOT), and
the T90 telescope at the Sierra Nevada Observatory (SNO) during May/June 2022
in the R-band ($\lambda$=6500Å/$E$=1.9 eV, FWHM=1300 Å/0.4 eV). The HONIR
observations were performed in three optical
bands222http://hasc.hiroshima-u.ac.jp/instruments/honir/filters-e.html. The
Stokes parameters are estimated by rotating the half-wave plate to four
positions angles and combining the exposures. The instrument and data analysis
are described in detail in Kawabata et al. (1999); Akitaya et al. (2014).
RoboPol is a novel 4-channel opto-polarimeter with no moving parts and with
the ability to measure the normalized Stokes $q$ and $u$ parameters with a
single exposure simultaneously. A detailed description of the semi-automated
analysis and data reduction pipeline can be found in Panopoulou et al. (2015);
Blinov et al. (2021). The NOT data were reduced using the pipeline developed
at the Tuorla Observatory, which follows standard photometric procedures
(Hovatta et al., 2016; Nilsson et al., 2018). We performed the SNO
observations with polarized $R_{c}$ filters, and calibrated the data using
polarized and unpolarized standard stars. For the RoboPol, HONIR, and NOT
observations in the R-band, we corrected for the host-galaxy depolarization by
subtracting the (presumed unpolarized) host-galaxy flux density estimated for
the individual apertures used in each observation Nilsson et al. (2007)):
$\Pi_{\mathrm{intr}}=\Pi_{\mathrm{obs}}\times{I}/(I-I_{\mathrm{host}})$, where
$\Pi_{\mathrm{intr}}$ and $\Pi_{\mathrm{obs}}$ are the intrinsic and observed
polarization degree, respectively, $I$ is the total flux density in mJy, and
$I_{\mathrm{host}}$ is the host-galaxy flux density in mJy for the aperture of
the observation (Hovatta et al., 2016).

The IR observations were performed in H-band with the 1.8m Boston University
Perkins telescope (Flagstaff, AZ) using the IR camera MIMIR in May/June 2022,
supplemented by H-band HONIR observations in May 2022. The MIMIR instrument
and data reduction procedures are described in detail in Clemens et al.
(2012). The Perkins H-band observations were corrected for the host-galaxy
contribution following the prescription above. To find the contribution of the
host galaxy for the aperture necessary for the correction, we performed a
photometric decomposition of the galaxy light using the IMFIT package (Erwin,
2015). The model for the image decomposition consisted of a Sersic profile for
the host galaxy plus a point source for the blazar, determined by fitting a
Moffat function (Moffat, 1969) to several isolated stars in the image to
create an average point-source function. We determined the parameters of the
host galaxy as $n=1.18\pm 0.35$ for the Sersic index, an ellipticity of
$e=0.238\pm 0.03$, an effective radius of $r_{e}=2.1^{\prime\prime}\pm
0.14^{\prime\prime}$, and a surface brightness of $\mu_{e}=16.59\pm 0.17$. For
the average aperture diameter of $5.2^{\prime\prime}$ used for the
polarization observations, we estimated the contribution of the host-galaxy to
be $12.41\pm 0.06$ mag. Any additional infrared observations have not been
corrected for the host-galaxy contribution, and hence should be considered as
lower limits to the intrinsic polarization degree. All the optical/infrared
observations are listed in Table 1.

Figure 6: VLBA images of Mrk 421 at 7 mm at three epochs near the times of the
IXPE observations. Contours represent total intensity, in factors of 2
starting at 2% (left image) or 1% (middle and right images) of the peak
intensity of (left to right) 0.35, 0.28, and 0.26 Jy per beam. Colour coding
indicates relative linearly polarized intensity, with maxima of (left to
right) 5.7, 7.2, and 5.7 mJy per beam. The yellow line segments correspond to
the position angle of polarization. The cross-hatched ellipse in the lower
left of each panel gives the FWHM of the elliptical restoring beam, which
corresponds to the direction-dependent angular resolution. Marked positions
are relative to the locations of the maximum intensity.

#### Radio/millimetre data

Images of Mrk 421 at 7 mm wavelength with a resolution of $\sim 0.2$
milliarcseconds (mas), corresponding to 0.12 pc, were produced with data from
Very Long Baseline Array (VLBA) observations at epochs 30 April, 5 June, and
24 June 2022. The observations were from the Blazars Entering the
Astrophysical Multi-Messenger Era (BEAM-ME) program (see website
www.bu.edu/blazars/BEAM-ME.html). The data acquisition and analysis are
described in Jorstad et al. (2017). Figure 6 presents the images, in which
total intensity is indicated by contours, polarized intensity by colour
coding, and polarization direction by yellow line segments. (Note that the 30
April image has lower dynamic range than the others, hence the jet is not as
apparent.) The images reveal a change in the polarization direction of the
“core” (feature of peak intensity) between 30 April and the June epochs. The 5
June image exhibits some fanning out of the polarization vectors on the
southeast side of the core, indicating an azimuthal component to the magnetic
field, as one would expect if that component were helical. However, at the
location of the 7 mm core, this component is weak relative to a more randomly
oriented magnetic field component.

Polarization at the shorter wavelengths of 3.5 mm (86.24 GHz) and 1.3 mm (230
GHz) was measured with the 30 m Telescope of the Institut de Radioastronomie
Millimétrique (IRAM), located at the Pico Veleta Observatory (Sierra Nevada,
Granada, Spain), on several nights in May/June 2022, as listed in Table 1. The
observations were obtained within the Polarimetric Monitoring of AGN at
Millimetre Wavelengths (POLAMI) program (Agudo et al., 2018a, b) following the
reduction and calibration pipeline described in detail in Agudo et al.
(2018a).

No significant polarization at 1.3 mm from Mrk 421 was detected by IRAM either
in May or June 2022, with an upper limit (at a confidence level of 95%) of
$<6$% and $<4$%, respectively. The early May 3.5 mm observations are
consistent within uncertainties with an average $\Pi_{R}=3.4\pm 0.7\%$ at an
average $\psi_{R}=54.5\pm 6^{\circ}$. The 30 May 2022 observation shows an
increase in the polarization degree to $\Pi_{R}=7.9\pm 0.9$, with a
polarization angle almost perpendicular to the early May observations
($\psi=128\pm 2.4^{\circ}$). The subsequent observation shows a similar
$\Pi_{R}$ to the early May observations ($1.76\pm 0.6\%$) at a roughly
consistent angle of $\psi=116\pm 9^{\circ}$. We obtained additional
observations at 1.3 mm (225.538 GHz) using the SubMillimeter Array (SMA) Ho et
al. (2004) within the SMAPOL (SMA Monitoring of AGNs with POLarization)
program. SMAPOL follows the polarization evolution of 40 $\gamma$-ray bright
blazars, including Mrk 421, on a bi-weekly cadence, as well as other sources
in a target-of-opportunity (ToO) mode, including objects observed by IXPE. The
Mrk 421 observations reported here were conducted on 4 June and 16 June 2022.
The SMA observations employed two orthogonally polarized receivers, tuned to
the same frequency range in full polarization mode, and used the SWARM
correlator Primiani et al. (2016). These receivers are inherently linearly
polarized, but are converted to circular polarization by using the quarter-
wave plates of the SMA polarimeter Marrone and Rao (2008). The lower sideband
(LSB) and upper sideband (USB) covered 209-221 and 229-241 GHz, respectively.
Each sideband was divided into six sub-bands with bandwidths of 2 GHz and a
fixed channel width of 140 kHz. The SMA data were calibrated with the MIR
software package 333https://lweb.cfa.harvard.edu/ cqi/mircook.html.
Instrumental polarization leakage was calibrated independently for USB and LSB
using the MIRIAD task gpcal Sault et al. (1995) and removed from the data. The
polarized intensity, position angle, and percentage were derived from the
Stokes I, Q, and U visibilities. Between 4 and 16 June 2022, Mrk 421 decreased
slightly in total flux, with values of 0.20$\pm$0.02 Jy and 0.14$\pm$0.02 Jy
on 4 and 16 June, respectively. The linear polarization remained stable within
the uncertainties between the two epochs, with a linear polarization degree of
2.3$\pm$0.2$\%$ and 2.3$\pm$0.3$\%$ and a polarization angle of 129.6$\pm$2.6∘
and 132.6$\pm$3.9∘ on 4 and 16 June, respectively. MWC 349 A was used for the
total flux calibration, and the calibrator 3C 286, which has a high linear
polarization degree and stable polarization angle, was observed in both
sessions as a cross-check of the polarization calibration.

### X-ray Polarization Analysis

#### Time-averaged IXPE data

The analysis of the first IXPE observation of Mrk 421, which discovered an
X-ray polarization $\Pi_{X}=15\%\pm 2\%$ at a polarization angle
$\Psi_{X}=35\hbox{${}^{\circ}$}\pm 4\hbox{${}^{\circ}$}$, is reported in Di
Gesu et al. (2022). Conversely, the X-ray polarization of Mrk 421 was
undetected in the IXPE data of Obs. 2 and 3 when time-averaged over the entire
length of the observations. Using the procedure of Kislat et al. (2015) within
ixpeobssim, we found for the 2.0-8.0 keV band data a nominal minimum
detectable polarization (equivalent to a 99% confidence upper limit) of 5% and
4% for Obs. 2 and 3, respectively. Fig. 7 displays the two-dimensional
confidence contours for the X-ray polarization degree and angle obtained with
the same method, to visually illustrate the lack of constraints on the X-ray
polarization angle in these time-averaged data. A spectropolarimetric fit
(Strohmayer, 2017) of the data gives similar upper bounds for the polarization
properties, assuming that the polarization degree and angle are constant with
energy (using the polconst model in Xspec). A log-parabolic model best fits
the I spectra, where we used the same fixed pivot energy as for the Swift-XRT
data, $E_{\rm p}$=5.0 keV (see above). The IXPE data follow the same harder-
when-brighter trend as for the Swift-XRT data: $\alpha=3.17\pm 0.06$,
$\beta=0.7\pm 0.1$ and $\alpha=2.81\pm 0.03$, $\beta=0.61\pm 0.03$, for Obs. 2
and 3, respectively.

In the following, we investigate the cause of the change in the polarization
properties of Mrk 421. The non-detection of the X-ray polarization of Mrk 421
in time-averaged IXPE Obs. 2 and 3 data can be due to time variability of the
polarization angle within the observation (see, e.g., Di Gesu et al. (2022)).
A rotation of the polarization angle, as often observed for blazars in the
optical band, can in principle cancel the polarization degree in time-averaged
data. In the following, we test this hypothesis using a simple model where the
polarization angle rotates at a constant rate while the polarization degree
remains constant. This assumption is sufficient to our aim, which is to assess
the reality of the polarization angle rotation. Consideration of more complex
models for the time variability is beyond the scope of this paper. In order to
make this assessment, we employ three methods: (1) a maximum-likelihood,
unbinned event-based method, as in Marshall (2021), (2) a fit of the binned
light-curves of the Stokes parameters using $\chi^{2}$ statistics, and (3) a
fit of a rotation pattern in the Q-U plane using $\chi^{2}$ statistics. A by-
product of this analysis is an estimation of the rotation rate, for which all
three methods return consistent results.

Figure 7: Polar plot of the $\Pi_{\rm x}$-$\psi_{\rm x}$ plane. Confidence
contours of X-ray polarization derived within ixpeobssim for IXPE are shown
for Obs. 2 (left panel) and 3 (right panel) of Mrk 421. The polarization
degree is displayed in percentage units. For each case, the contours at
68.26%, 95.45%, 99.73%, and 99.994% confidence level (considering two degrees
of freedom) are displayed, colour-coded in red, yellow, green, blue. Figure 8:
Log-likelihood difference as a function of the polarization angle rotation
rate according to the maximum likelihood analysis method (see section
Searching for the optimal rotation rate via a likelihood analysis and Marshall
(2021)). For each observation set, the difference is relative to the minimum
at the best fit value of the rotation rate, $\dot{\Psi}$. Results of the
search for Obs. 1, 2, 3, and for Obs. 2 and 3 combined are colour-coded in
black, green, red and blue, respectively. In Obs. 2, for example,
$\dot{\Psi}=0$ can be ruled out at the 7 $\sigma$ level because $\Delta S>50$
with respect to the best fit.

Figure 9: Results of fitting the Stokes parameters vs. time ($q(t)$: circles
$u(t)$: stars), with a constant model (black symbols) or with a model
including a constant rotation of the polarization angle (red symbols), for
Obs. 2 (left panel) and 3 (right panel). The null hypothesis probability for
all the fits is displayed as a function of the number of time bins involved.

Figure 10: Stokes parameters vs. time (($q(t)$: circles, top panel, $u(t)$:
stars, middle panel) for Obs. 2 (red symbols) and 3 (blue symbols) for a
number of time bins between 3 (i.e $\sim$ 17 hours) and 20 (i.e. $\sim$ one