11institutetext: INAF Istituto di Astrofisica e Planetologia Spaziali, Via del
Fosso del Cavaliere 100, 00133 Roma, Italy 22institutetext: Dipartimento di
Fisica, Università degli Studi di Roma ”La Sapienza”, Piazzale Aldo Moro 5,
00185 Roma, Italy 33institutetext: Dipartimento di Fisica, Università degli
Studi di Roma ”Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma,
Italy 44institutetext: ASI - Agenzia Spaziale Italiana, Via del Politecnico
snc, 00133 Roma, Italy 55institutetext: Finnish Centre for Astronomy with ESO,
20014 University of Turku, Finland 66institutetext: Institute for
Astrophysical Research, Boston University, 725 Commonwealth Avenue, Boston, MA
02215, USA 77institutetext: Saint Petersburg State University, 7/9
Universitetskaya nab., St. Petersburg, 199034 Russia 88institutetext: Space
Science Data Center, Agenzia Spaziale Italiana, Via del Politecnico snc, 00133
Roma, Italy 99institutetext: INAF Osservatorio Astronomico di Roma, Via
Frascati 33, 00078 Monte Porzio Catone (RM), Italy 1010institutetext: MIT
Kavli Institute for Astrophysics and Space Research, Massachusetts Institute
of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
1111institutetext: Instituto de Astrofísica de Andalucía—CSIC, Glorieta de la
Astronomía s/n, 18008 Granada, Spain 1212institutetext: INAF Osservatorio
Astronomico di Brera, Via E. Bianchi 46, 23807 Merate (LC), Italy
1313institutetext: Dipartimento di Fisica, Università degli Studi di Torino,
Via Pietro Giuria 1, 10125 Torino, Italy 1414institutetext: Istituto Nazionale
di Fisica Nucleare, Sezione di Torino, Via Pietro Giuria 1, 10125 Torino,
Italy 1515institutetext: University of Maryland, Baltimore County, Baltimore,
MD 21250, USA 1616institutetext: NASA Goddard Space Flight Center, Greenbelt,
MD 20771, USA 1717institutetext: Louisiana State University, Baton Rouge, LA
70803, USA 1818institutetext: Istituto Nazionale di Fisica Nucleare, Sezione
di Roma ”Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Roma, Italy
1919institutetext: Department of Astronomy, University of Maryland, College
Park, Maryland 20742, USA 2020institutetext: Hiroshima Astrophysical Science
Center, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima
739-8526, Japan 2121institutetext: Department of Physics, Graduate School of
Advanced Science and Engineering, Hiroshima University Kagamiyama, 1-3-1
Higashi-Hiroshima, Hiroshima 739-8526, Japan 2222institutetext: Core Research
for Energetic Universe (Core-U), Hiroshima University, 1-3-1 Kagamiyama,
Higashi-Hiroshima, Hiroshima 739-8526, Japan 2323institutetext: Department of
Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo
152-8551, Japan 2424institutetext: Planetary Exploration Research Center,
Chiba Institute of Technology 2-17-1 Tsudanuma, Narashino, Chiba 275-0016,
Japan 2525institutetext: Institut de Radioastronomie Millimétrique, Avenida
Divina Pastora 7, Local 20, E-18012 Granada, Spain 2626institutetext: Max-
Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany
2727institutetext: Section of Astrophysics, Astronomy & Mechanics, Department
of Physics, National and Kapodistrian University of Athens, Panepistimiopolis
Zografos 15784, Greece 2828institutetext: Korea Astronomy and Space Science
Institute, Daedeokdae-ro 776, Yuseong-gu, Daejeon 34055, Republic of Korea
2929institutetext: University of Science and Technology, Gajeong-ro 217,
Yuseong-gu, Daejeon 34113, Republic of Korea 3030institutetext: Center for
Astrophysics — Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA
3131institutetext: Department of Physics and Astronomy, 20014 University of
Turku, Finland 3232institutetext: Department of Physics, University of Crete,
70013, Heraklion, Greece 3333institutetext: Institute of Astrophysics,
Foundation for Research and Technology-Hellas, GR-71110 Heraklion, Greece
3434institutetext: Owens Valley Radio Observatory, California Institute of
Technology, MC 249-17, Pasadena, CA 91125, USA 3535institutetext: Special
Astrophysical Observatory, Russian Academy of Sciences, 369167, Nizhnii
Arkhyz, Russia 3636institutetext: Pulkovo Observatory, St.Petersburg, 196140,
Russia 3737institutetext: INAF Osservatorio Astronomico di Cagliari, Via della
Scienza 5, 09047 Selargius (CA), Italy 3838institutetext: Istituto Nazionale
di Fisica Nucleare, Sezione di Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy
3939institutetext: Dipartimento di Fisica, Università di Pisa, Largo B.
Pontecorvo 3, 56127 Pisa, Italy 4040institutetext: NASA Marshall Space Flight
Center, Huntsville, AL 35812, USA 4141institutetext: Dipartimento di
Matematica e Fisica, Università degli Studi Roma Tre, Via della Vasca Navale
84, 00146 Roma, Italy 4242institutetext: INAF Osservatorio Astrofisico di
Arcetri, Largo Enrico Fermi 5, 50125 Firenze, Italy 4343institutetext:
Dipartimento di Fisica e Astronomia, Università degli Studi di Firenze, Via
Sansone 1, 50019 Sesto Fiorentino (FI), Italy 4444institutetext: Istituto
Nazionale di Fisica Nucleare, Sezione di Firenze, Via Sansone 1, 50019 Sesto
Fiorentino (FI), Italy 4545institutetext: Science and Technology Institute,
Universities Space Research Association, Huntsville, AL 35805, USA
4646institutetext: Department of Physics and Kavli Institute for Particle
Astrophysics and Cosmology, Stanford University, Stanford, California 94305,
USA 4747institutetext: Institut für Astronomie und Astrophysik, Universität
Tübingen, Sand 1, 72076 Tübingen, Germany 4848institutetext: Astronomical
Institute of the Czech Academy of Sciences, Boční II 1401/1, 14100 Praha 4,
Czech Republic 4949institutetext: RIKEN Cluster for Pioneering Research, 2-1
Hirosawa, Wako, Saitama 351-0198, Japan 5050institutetext: California
Institute of Technology, Pasadena, CA 91125, USA 5151institutetext: Yamagata
University,1-4-12 Kojirakawa-machi, Yamagata-shi 990-8560, Japan
5252institutetext: Osaka University, 1-1 Yamadaoka, Suita, Osaka 565-0871,
Japan 5353institutetext: University of British Columbia, Vancouver, BC V6T
1Z4, Canada 5454institutetext: Department of Physics, Faculty of Science and
Engineering, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan
5555institutetext: Department of Physics and Astronomy and Space Science
Center, University of New Hampshire, Durham, NH 03824, USA 5656institutetext:
Physics Department and McDonnell Center for the Space Sciences, Washington
University in St. Louis, St. Louis, MO 63130, USA 5757institutetext:
Université de Strasbourg, CNRS, Observatoire Astronomique de Strasbourg, UMR
7550, 67000 Strasbourg, France 5858institutetext: Graduate School of Science,
Division of Particle and Astrophysical Science, Nagoya University, Furo-cho,
Chikusa-ku, Nagoya, Aichi 464-8602, Japan 5959institutetext: Department of
Physics, The University of Hong Kong, Pokfulam, Hong Kong 6060institutetext:
Department of Astronomy and Astrophysics, Pennsylvania State University,
University Park, PA 16802, USA 6161institutetext: Université Grenoble Alpes,
CNRS, IPAG, 38000 Grenoble, France 6262institutetext: Dipartimento di Fisica e
Astronomia, Università degli Studi di Padova, Via Marzolo 8, 35131 Padova,
Italy 6363institutetext: Mullard Space Science Laboratory, University College
London, Holmbury St Mary, Dorking, Surrey RH5 6NT, UK 6464institutetext: Anton
Pannekoek Institute for Astronomy & GRAPPA, University of Amsterdam, Science
Park 904, 1098 XH Amsterdam, The Netherlands 6565institutetext: Guangxi Key
Laboratory for Relativistic Astrophysics, School of Physical Science and
Technology, Guangxi University, Nanning 530004, China

# Magnetic Field Properties inside the Jet of Mrk 421

Multiwavelength Polarimetry Including the Imaging X-ray Polarimetry Explorer

Dawoon E. Kim 0000-0001-5717-3736 corresponding author,
<EMAIL_ADDRESS>Field Properties inside the Jet of Mrk 421Magnetic
Field Properties inside the Jet of Mrk 421Magnetic Field Properties inside the
Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421Magnetic
Field Properties inside the Jet of Mrk 421Magnetic Field Properties inside the
Jet of Mrk 421 Laura Di Gesu 0000-0002-5614-5028 Magnetic Field Properties
inside the Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421
Ioannis Liodakis 0000-0001-9200-4006 Magnetic Field Properties inside the Jet
of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421 Alan P. Marscher
0000-0001-7396-3332 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Svetlana G. Jorstad
0000-0001-9522-5453 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field
Properties inside the Jet of Mrk 421Magnetic Field Properties inside the Jet
of Mrk 421 Riccardo Middei 0000-0001-9815-9092 Magnetic Field Properties
inside the Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field
Properties inside the Jet of Mrk 421 Herman L. Marshall 0000-0002-6492-1293
Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field Properties
inside the Jet of Mrk 421 Luigi Pacciani 0000-0001-6897-5996 Magnetic Field
Properties inside the Jet of Mrk 421Magnetic Field Properties inside the Jet
of Mrk 421 Iván Agudo 0000-0002-3777-6182 Magnetic Field Properties inside the
Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421 Fabrizio
Tavecchio 0000-0003-0256-0995 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Nicolò Cibrario
0000-0003-3842-4493 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field
Properties inside the Jet of Mrk 421Magnetic Field Properties inside the Jet
of Mrk 421 Stefano Tugliani 0000-0002-3318-9036 Magnetic Field Properties
inside the Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421
Raffaella Bonino 0000-0002-4264-1215 Magnetic Field Properties inside the Jet
of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field
Properties inside the Jet of Mrk 421Magnetic Field Properties inside the Jet
of Mrk 421 Michela Negro 0000-0002-6548-5622 Magnetic Field Properties inside
the Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421Magnetic
Field Properties inside the Jet of Mrk 421Magnetic Field Properties inside the
Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421Magnetic
Field Properties inside the Jet of Mrk 421 Simonetta Puccetti
0000-0002-2734-7835 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Francesco Tombesi
0000-0002-6562-8654 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field
Properties inside the Jet of Mrk 421Magnetic Field Properties inside the Jet
of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field
Properties inside the Jet of Mrk 421 Enrico Costa 0000-0003-4925-8523 Magnetic
Field Properties inside the Jet of Mrk 421Magnetic Field Properties inside the
Jet of Mrk 421 Immacolata Donnarumma 0000-0002-4700-4549 Magnetic Field
Properties inside the Jet of Mrk 421Magnetic Field Properties inside the Jet
of Mrk 421 Paolo Soffitta 0000-0001-8916-4156 Magnetic Field Properties inside
the Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421
Tsunefumi Mizuno 0000-0001-7263-0296 Magnetic Field Properties inside the Jet
of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421 Yasushi Fukazawa
Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field Properties
inside the Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field
Properties inside the Jet of Mrk 421Magnetic Field Properties inside the Jet
of Mrk 421 Koji S. Kawabata Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field
Properties inside the Jet of Mrk 421Magnetic Field Properties inside the Jet
of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field
Properties inside the Jet of Mrk 421 Tatsuya Nakaoka Magnetic Field Properties
inside the Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421
Makoto Uemura Magnetic Field Properties inside the Jet of Mrk 421Magnetic
Field Properties inside the Jet of Mrk 421Magnetic Field Properties inside the
Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421Magnetic
Field Properties inside the Jet of Mrk 421Magnetic Field Properties inside the
Jet of Mrk 421 Ryo Imazawa Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Mahito Sasada Magnetic
Field Properties inside the Jet of Mrk 421Magnetic Field Properties inside the
Jet of Mrk 421 Hiroshi Akitaya Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Francisco Josè Aceituno
Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field Properties
inside the Jet of Mrk 421 Giacomo Bonnoli 0000-0003-2464-9077 Magnetic Field
Properties inside the Jet of Mrk 421Magnetic Field Properties inside the Jet
of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field
Properties inside the Jet of Mrk 421 Vìctor Casanova Magnetic Field Properties
inside the Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421
Ioannis Myserlis 0000-0003-3025-9497 Magnetic Field Properties inside the Jet
of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field
Properties inside the Jet of Mrk 421Magnetic Field Properties inside the Jet
of Mrk 421 Albrecht Sievers Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Emmanouil Angelakis
Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field Properties
inside the Jet of Mrk 421 Alexander Kraus Magnetic Field Properties inside the
Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421 Whee Yeon
Cheong 0009-0002-1871-5824 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field
Properties inside the Jet of Mrk 421Magnetic Field Properties inside the Jet
of Mrk 421 Hyeon-Woo Jeong 0009-0005-7629-8450 Magnetic Field Properties
inside the Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field
Properties inside the Jet of Mrk 421 Sincheol Kang 0000-0002-0112-4836
Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field Properties
inside the Jet of Mrk 421 Sang-Hyun Kim 0000-0001-7556-8504 Magnetic Field
Properties inside the Jet of Mrk 421Magnetic Field Properties inside the Jet
of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field
Properties inside the Jet of Mrk 421 Sang-Sung Lee 0000-0002-6269-594X
Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field Properties
inside the Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Beatriz Agìs-Gonzàlez
Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field Properties
inside the Jet of Mrk 421 Alfredo Sota 0000-0002-9404-6952 Magnetic Field
Properties inside the Jet of Mrk 421Magnetic Field Properties inside the Jet
of Mrk 421 Juan Escudero Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Mark Gurwell
0000-0003-0685-3621 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Garrett K. Keating
0000-0002-3490-146X Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Ramprasad Rao Magnetic
Field Properties inside the Jet of Mrk 421Magnetic Field Properties inside the
Jet of Mrk 421 Pouya M. Kouch 0000-0002-9328-2750 Magnetic Field Properties
inside the Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field
Properties inside the Jet of Mrk 421 Elina Lindfors Magnetic Field Properties
inside the Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421
Ioakeim G. Bourbah Magnetic Field Properties inside the Jet of Mrk 421Magnetic
Field Properties inside the Jet of Mrk 421 Sebastian Kiehlmann Magnetic Field
Properties inside the Jet of Mrk 421Magnetic Field Properties inside the Jet
of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field
Properties inside the Jet of Mrk 421 Evangelos Kontopodis Magnetic Field
Properties inside the Jet of Mrk 421Magnetic Field Properties inside the Jet
of Mrk 421 Nikos Mandarakas 0000-0002-2567-2132 Magnetic Field Properties
inside the Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field
Properties inside the Jet of Mrk 421 Stylianos Romanopoulos Magnetic Field
Properties inside the Jet of Mrk 421Magnetic Field Properties inside the Jet
of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field
Properties inside the Jet of Mrk 421 Raphael Skalidis Magnetic Field
Properties inside the Jet of Mrk 421Magnetic Field Properties inside the Jet
of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field
Properties inside the Jet of Mrk 421Magnetic Field Properties inside the Jet
of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421 Anna Vervelaki
0000-0003-0271-9724 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Sergey S. Savchenko
Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field Properties
inside the Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field
Properties inside the Jet of Mrk 421Magnetic Field Properties inside the Jet
of Mrk 421 Lucio A. Antonelli 0000-0002-5037-9034 Magnetic Field Properties
inside the Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field
Properties inside the Jet of Mrk 421 Matteo Bachetti 0000-0002-4576-9337
Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field Properties
inside the Jet of Mrk 421 Luca Baldini 0000-0002-9785-7726 Magnetic Field
Properties inside the Jet of Mrk 421Magnetic Field Properties inside the Jet
of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field
Properties inside the Jet of Mrk 421 Wayne H. Baumgartner 0000-0002-5106-0463
Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field Properties
inside the Jet of Mrk 421 Ronaldo Bellazzini 0000-0002-2469-7063 Magnetic
Field Properties inside the Jet of Mrk 421Magnetic Field Properties inside the
Jet of Mrk 421 Stefano Bianchi 0000-0002-4622-4240 Magnetic Field Properties
inside the Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421
Stephen D. Bongiorno 0000-0002-0901-2097 Magnetic Field Properties inside the
Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421 Alessandro
Brez 0000-0002-9460-1821 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Niccolò Bucciantini
0000-0002-8848-1392 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field
Properties inside the Jet of Mrk 421Magnetic Field Properties inside the Jet
of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field
Properties inside the Jet of Mrk 421 Fiamma Capitanio 0000-0002-6384-3027
Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field Properties
inside the Jet of Mrk 421 Simone Castellano 0000-0003-1111-4292 Magnetic Field
Properties inside the Jet of Mrk 421Magnetic Field Properties inside the Jet
of Mrk 421 Elisabetta Cavazzuti 0000-0001-7150-9638 Magnetic Field Properties
inside the Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421
Chien-Ting Chen 0000-0002-4945-5079 Magnetic Field Properties inside the Jet
of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421 Stefano Ciprini
0000-0002-0712-2479 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field
Properties inside the Jet of Mrk 421Magnetic Field Properties inside the Jet
of Mrk 421 Alessandra De Rosa 0000-0001-5668-6863 Magnetic Field Properties
inside the Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421
Ettore Del Monte 0000-0002-3013-6334 Magnetic Field Properties inside the Jet
of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421 Niccolò Di Lalla
0000-0002-7574-1298 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Alessandro Di Marco
0000-0003-0331-3259 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Victor Doroshenko
0000-0001-8162-1105 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Michal Dovčiak
0000-0003-0079-1239 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Steven R. Ehlert
0000-0003-4420-2838 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Teruaki Enoto
0000-0003-1244-3100 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Yuri Evangelista
0000-0001-6096-6710 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Sergio Fabiani
0000-0003-1533-0283 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Riccardo Ferrazzoli
0000-0003-1074-8605 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Javier A. Garcia
0000-0003-3828-2448 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Shuichi Gunji
0000-0002-5881-2445 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Kiyoshi Hayashida
Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field Properties
inside the Jet of Mrk 421 Jeremy Heyl 0000-0001-9739-367X Magnetic Field
Properties inside the Jet of Mrk 421Magnetic Field Properties inside the Jet
of Mrk 421 Wataru Iwakiri 0000-0002-0207-9010 Magnetic Field Properties inside
the Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421 Philip
Kaaret 0000-0002-3638-0637 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Vladimir Karas
0000-0002-5760-0459 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Fabian Kislat
0000-0001-7477-0380 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Takao Kitaguchi
Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field Properties
inside the Jet of Mrk 421 Jeffery J. Kolodziejczak 0000-0002-0110-6136
Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field Properties
inside the Jet of Mrk 421 Henric Krawczynski 0000-0002-1084-6507 Magnetic
Field Properties inside the Jet of Mrk 421Magnetic Field Properties inside the
Jet of Mrk 421 Fabio La Monaca 0000-0001-8916-4156 Magnetic Field Properties
inside the Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421
Luca Latronico 0000-0002-0984-1856 Magnetic Field Properties inside the Jet of
Mrk 421Magnetic Field Properties inside the Jet of Mrk 421 Simone Maldera
0000-0002-0698-4421 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Alberto Manfreda
0000-0002-0998-4953 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Frédéric Marin
0000-0003-4952-0835 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Andrea Marinucci
0000-0002-2055-4946 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Francesco Massaro
0000-0002-1704-9850 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field
Properties inside the Jet of Mrk 421Magnetic Field Properties inside the Jet
of Mrk 421 Giorgio Matt 0000-0002-2152-0916 Magnetic Field Properties inside
the Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421 Ikuyuki
Mitsuishi Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field
Properties inside the Jet of Mrk 421 Fabio Muleri 0000-0003-3331-3794 Magnetic
Field Properties inside the Jet of Mrk 421Magnetic Field Properties inside the
Jet of Mrk 421 C.-Y. Ng 0000-0002-5847-2612 Magnetic Field Properties inside
the Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421 Stephen
L. O’Dell 0000-0002-1868-8056 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Nicola Omodei
0000-0002-5448-7577 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Chiara Oppedisano
0000-0001-6194-4601 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Alessandro Papitto
0000-0001-6289-7413 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 George G. Pavlov
0000-0002-7481-5259 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Abel L. Peirson
0000-0001-6292-1911 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Matteo Perri
0000-0003-3613-4409 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field
Properties inside the Jet of Mrk 421Magnetic Field Properties inside the Jet
of Mrk 421 Melissa Pesce-Rollins 0000-0003-1790-8018 Magnetic Field Properties
inside the Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421
Pierre-Olivier Petrucci 0000-0001-6061-3480 Magnetic Field Properties inside
the Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421 Maura
Pilia 0000-0001-7397-8091 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Andrea Possenti
0000-0001-5902-3731 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Juri Poutanen
0000-0002-0983-0049 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Brian D. Ramsey
0000-0003-1548-1524 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 John Rankin
0000-0002-9774-0560 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Ajay Ratheesh
0000-0003-0411-4243 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Oliver Roberts
0000-0002-7150-9061 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Roger W. Romani
0000-0001-6711-3286 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Carmelo Sgrò
0000-0001-5676-6214 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Patrick Slane
0000-0002-6986-6756 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Gloria Spandre
0000-0003-0802-3453 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Doug Swartz
0000-0002-2954-4461 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Toru Tamagawa
0000-0002-8801-6263 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Roberto Taverna
0000-0002-1768-618X Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Yuzuru Tawara Magnetic
Field Properties inside the Jet of Mrk 421Magnetic Field Properties inside the
Jet of Mrk 421 Allyn F. Tennant 0000-0002-9443-6774 Magnetic Field Properties
inside the Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421
Nicholas E. Thomas 0000-0003-0411-4606 Magnetic Field Properties inside the
Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421 Alessio
Trois 0000-0002-3180-6002 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Sergey S. Tsygankov
0000-0002-9679-0793 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Roberto Turolla
0000-0003-3977-8760 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field
Properties inside the Jet of Mrk 421Magnetic Field Properties inside the Jet
of Mrk 421 Jacco Vink 0000-0002-4708-4219 Magnetic Field Properties inside the
Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421 Martin C.
Weisskopf 0000-0002-5270-4240 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Kinwah Wu
0000-0002-7568-8765 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421 Fei Xie
0000-0002-0105-5826 Magnetic Field Properties inside the Jet of Mrk
421Magnetic Field Properties inside the Jet of Mrk 421Magnetic Field
Properties inside the Jet of Mrk 421Magnetic Field Properties inside the Jet
of Mrk 421 Silvia Zane 0000-0001-5326-880X Magnetic Field Properties inside
the Jet of Mrk 421Magnetic Field Properties inside the Jet of Mrk 421

###### Abstract

Aims. To probe the magnetic field geometry and particle acceleration mechanism
in the relativistic jets of supermassive black holes.

Methods. We conducted a polarimetry campaign from radio to X-ray wavelengths
of the high-synchrotron-peak (HSP) blazar Mrk 421, including Imaging X-ray
Polarimetry Explorer (IXPE) measurements on 2022 December 6–8. During the IXPE
observation, we also monitored Mrk 421 using Swift-XRT and obtained a single
observation with XMM-Newton to improve the X-ray spectral analysis. The time-
averaged X-ray polarization was determined consistently based on the event-by-
event Stokes parameter analysis, spectropolarimetric fit, and maximum
likelihood methods. We examined the polarization variability over both time
and energy, the latter via analysis of IXPE  data obtained over a time span of
7 months.

Results. We detected X-ray polarization of Mrk 421 with a degree of $\Pi_{\rm
X}$=14$\pm$1$\%$ and an electric-vector position angle $\psi_{\rm
X}$=107$\pm$3$\degr$ in the 2–8 keV band. From the time variability analysis,
we find a significant episodic variation in $\psi_{\rm X}$. During 7 months
from the first IXPE pointing of Mrk 421 in 2022 May, $\psi_{\rm X}$ varied
across the range of 0$\degr$ to 180$\degr$, while $\Pi_{\rm X}$ maintained
similar values within $\sim$10–15$\%$. Furthermore, a swing in $\psi_{\rm X}$
in 2022 June was accompanied by simultaneous spectral variations. The results
of the multiwavelength polarimetry show that the X-ray polarization degree was
generally $\sim$2–3 times greater than that at longer wavelengths, while the
polarization angle fluctuated. Additionally, based on radio, infrared, and
optical polarimetry, we find that rotation of $\psi$ occurred in the opposite
direction with respect to the rotation of $\psi_{\rm X}$ over longer
timescales at similar epochs.

Conclusions. The polarization behavior observed across multiple wavelengths is
consistent with previous IXPE findings for HSP blazars. This result favors the
energy-stratified shock model developed to explain variable emission in
relativistic jets. We consider two versions of the model, one with linear and
the other with radial stratification geometry, to explain the rotation of
$\psi_{\rm X}$. The accompanying spectral variation during the $\psi_{\rm X}$
rotation can be explained by a fluctuation in the physical conditions, e.g.,
in the energy distribution of relativistic electrons. The opposite rotation
direction of $\psi$ between the X-ray and longer wavelength polarization
accentuates the conclusion that the X-ray emitting region is spatially
separated from that at longer wavelengths. Moreover, we identify a highly
polarized knot of radio emission moving down the parsec-scale jet during the
episode of $\psi_{\rm X}$ rotation, although it is unclear whether there is
any connection between the two events.

###### Key Words.:

BL Lacertae objects: HSP, –Galaxies: jets –Polarization –Relativistic
processes –Magnetic fields

##  1 Introduction

Relativistic jets from active galactic nuclei (AGN) are the most luminous
long-lived phenomena across the entire electromagnetic spectrum in the
universe. This characteristic of jets makes them natural laboratories that can
be studied through multiwavelength and multimessenger observations from radio
to $\gamma$-rays.

Blazars are a subclass of AGN in which a relativistic plasma jet, propelled
from the vicinity of a supermassive black hole, is aligned closely with our
line of sight (e.g. Hovatta & Lindfors 2019). The emission from the jet is
relativistically boosted toward us, and therefore dominates the spectral
energy distribution (SED). The SED typically displays two broad nonthermal
radiation components. The lower-frequency component is generally ascribed to
be synchrotron emission from relativistic electrons. On the other hand,
interpretation of the higher-frequency component falls into two different
scenarios: leptonic and hadronic models. In the case of the leptonic model,
the high-energy emission is attributed to Compton scattering of synchrotron
photons (synchrotron self-Compton, or SSC; e.g., Jones et al. 1974; Maraschi
et al. 1992) or photons from outside the jet (e.g., Dermer & Schlickeiser
1993; Sikora et al. 1994). The hadronic scenario explains the high-energy
emission as proton synchrotron radiation (e.g., Aharonian 2000; Böttcher et
al. 2013) or photo-pion production (e.g., Mannheim 1993).

Blazars are sub-classified into three different types, depending on the
frequency of the synchrotron peak of the SED. Low-synchrotron-peaked blazars
(LSP) have a peak frequency below $10^{14}\leavevmode\nobreak\ $Hz,
intermediate-synchrotron peaked blazars (ISP) between
$10^{14}\leavevmode\nobreak\ $Hz and $10^{15}\leavevmode\nobreak\ $Hz, and
high-synchrotron peaked blazars (HSPs) above $10^{15}\leavevmode\nobreak\ $Hz
(Abdo et al. 2010). Thus, among these subclasses, HSPs feature a synchrotron
spectrum that smoothly connects from radio to X-ray bands (Fossati et al.
1998). Studies of HSPs therefore provide an opportunity to probe the geometry
and dynamics of the magnetic field components present in the synchrotron
emission features of the jet.

In this respect, multiwavelength polarimetry can be a prominent diagnostic
tool to investigate the particle acceleration processes and the geometrical
characteristics of the magnetic field inside the jet (e.g., Rybicki & Lightman
1979; Zhang 2019; Böttcher 2019; Tavecchio 2021). Until recently, polarimetric
observations of blazars have been limited to optical and radio bands. However,
the successful launch of the Imaging X-ray Polarimetry Explorer in late 2021
(IXPE; Weisskopf et al. 2022) has extended multiwavelength polarimetry up to
the X-ray energy band. IXPE is a joint mission of NASA and the Italian Space
Agency (Agenzia Spaziale Italiana, ASI). It carries three identical X-ray
telescope systems (detector units; DUs) corresponding to three gas pixel
detectors (GPDs, Costa et al. 2001) and three mirror module assemblies (MMAs).
IXPE measures linear polarization in the 2–8 keV band from the photoelectric
track produced by the interaction of X-ray photons and gas inside each GPD.

Mrk 421 (redshift, $z=0.030$) is the brightest HSP blazar at X-ray energies,
and therefore a prime target for measuring polarization with IXPE (Liodakis et
al. 2019). Indeed, IXPE securely detected linear polarization at a level
exceeding $10\%$ during a two-day pointing of Mrk 421 in May 2022 (Di Gesu et
al. 2022). In addition, during subsequent observations in June 2022, smooth
rotation of the electric-vector position angle $\psi_{\rm X}$ was discovered
(Di Gesu et al. 2023), in contrast to the relatively stable polarization
degree and direction found for a few other HSP blazars (e.g., Liodakis et al.
2022; Middei et al. 2023). This finding can be interpreted as evidence for a
helical magnetic field inside the X-ray emitting portion of the jet.
Furthermore, if the X-ray emission, and therefore helical field, occurs mainly
in the innermost jet, then it is expected that such rotations of $\psi_{\rm
X}$ should be routinely observed. If so, X-ray observations can probe the
section of the jet where the flow is accelerated and collimated (Marscher et
al. 2008, 2010).

Here we report the results of a new IXPE observation of Mrk 421, conducted as
part of a multiwavelength campaign. In §2, we present the analysis of the
time-averaged data and we search for variability of the polarization
properties as a function of time and photon energy. To do this, we compare the
finding with the results of the previous three IXPE observations, and we
investigate the polarization variability on long time scales. Next, in §3, we
report the multiwavelength polarimetry results obtained from simultaneous and
quasi-simultaneous measurements. Based on these results, we propose possible
geometrical and physical interpretations of the magnetic field geometry inside
the jet of Mrk 421 in §4.

##  2 X-ray polarization of Mrk 421

Previously, IXPE observed Mrk 421 three times: 2022 May 4-6 (hereafter Obs.
1), 2022 June 4–6 (Obs. 2), and 2022 June 7–9 (Obs. 3) (Di Gesu et al. 2022,
2023). Here we focus on an additional observation (Obs. ID: 02004401, Obs. 4)
conducted from UTC 15:30 on 2022 December 6 to 04:00 on December 8, with a
$\sim 74$ ks net exposure time. During the IXPE pointing, Mrk 421 was in a
normal X-ray activity state (e.g., Liodakis et al. 2019), with a 2–8 keV flux
of $\sim$2$\times 10^{-10}$ erg s-1 cm-2. From this observation, we have
derived the time-averaged polarization degree $\Pi_{\rm X}$ and electric-
vector position angle $\psi_{\rm X}$ over the entire exposure. In order to
investigate the X-ray polarization variability, we have also performed time-
and energy-resolved analysis.

We also observed Mrk 421 with XMM-Newton on 2022 December 7 for 7.5 ks. In
addition, the Neil Gehrels Swift Observatory (hereby referred to as Swift-XRT)
monitored the blazar to improve the spectral and temporal coverage. Table 1
presents a summary of the X-ray observations. Detailed information on the data
reduction procedures is presented in Appendix A. Throughout the manuscript,
errors are given at the standard 1$\sigma$ (68%) confidence level.

###  2.1 Time-averaged polarization properties

We have derived the linear polarization parameters from Obs. 4 by four
different methods: (1) the Kislat et al. (2015) method that is implemented in
the PCUBE algorithm in ixpeobssim (Baldini et al. 2022), (2) an event-based
maximum likelihood method (Marshall 2021b, ML) that includes simultaneous
background estimation, (3) the spectropolarimetric analysis described in
Strohmayer (2017) using XSPEC, and (4) a maximum likelihood
spectropolarimetric (MLS) fit implemented by the multi nest algorithm. Table 1
summarizes the $\Pi_{\rm X}$ and $\psi_{\rm X}$ values obtained from these
methods.

Figure 1: Polarization contours from Obs. 4. Contours represent the
significance of the time-averaged polarization detected with confidence levels
of 68.27$\%$, 90.00$\%$, and 99.00$\%$, with two degrees of freedom. The blue
contour indicates the values of $\Pi_{\rm X}$ and $\psi_{\rm X}$ derived from
the PCUBE methods, and the red contours show the same properties from
simultaneous IXPE and XMM-Newton spectropolarimetric analysis. The radial and
angular values represent $\Pi_{\rm X}$ and $\psi_{\rm X}$ , respectively, with
the latter measured from north through east.  Table 1: Results of the X-ray
polarimetric and spectral observations of Mrk 421

Telescope (Method11footnotemark: 1) | Band | Dates | X-ray Polarization | X-ray flux 22footnotemark: 2
---|---|---|---|---
| (keV) | (YYYY-MM-DD) |  $\Pi_{\rm X}$(%) $\psi_{\rm X}$ ($\degr$) | ($10^{-10}$erg s-1 cm-2)
IXPE  Obs. 4 (PCUBE) | 2.0 – 8.0 | 2022-12-06–08 | 13 $\pm$2 107 $\pm$3 | –
IXPE  Obs. 4 (ML) | 2.0 – 8.0 | 2022-12-06–08 | 13 $\pm$1 107 $\pm$3 | –
IXPE  Obs. 4 (XSPEC) | 2.0 – 8.0 | 2022-12-06–08 | 14 $\pm$1 107 $\pm$3 | 2.25 $\pm$0.04
IXPE  Obs. 4 (MLS) | 2.0 – 8.0 | 2022-12-06–08 | 13 $\pm$1 109 $\pm$3 | –
Swift-XRT | 0.3 – 10.0 | 2022-12-06 | – – | 3.20 $\pm$0.06
XMM-Newton | 0.3 – 10.0 | 2022-12-07 | – – | 2.04 $\pm$0.04
Swift-XRT | 0.3 – 10.0 | 2022-12-08 | – – | 2.47 $\pm$0.05
IXPE  Obs. 3 (PCUBE) | 2.0 – 8.0 | 2022-06-07–09 | 10 $\pm$1 Rotation | 3.02 $\pm$0.02
IXPE  Obs. 2 (PCUBE) | 2.0 – 8.0 | 2022-06-04–06 | 10 $\pm$1 Rotation | 1.57 $\pm$0.01
IXPE  Obs. 1 (PCUBE) | 2.0 – 8.0 | 2022-05-04–06 | 15 $\pm$2 35 $\pm$4 | 0.87 $\pm$0.003

11111footnotemark: 1 Methods are described in §2.1. 22footnotemark: 2 X-ray
flux in the 2–8 keV band.

First, using the Kislat et al. (2015) method, we estimated $\Pi_{\rm X}$
=13$\pm$2$\%$ and $\psi_{\rm X}$ =107$\pm$3$\degr$ in the 2–8 keV band after
background subtraction from the three combined DUs. The PCUBE measurement
includes the spectral-model-independent polarization properties over a given
energy and time range (Kislat et al. 2015). We derived $\Pi_{\rm X}$ and
$\psi_{\rm X}$ from the normalized Stokes parameters Q
(${\mathcal{Q}}\equiv{\it Q}/{\it I}$) and U (${\mathcal{U}}\equiv{\it U}/{\it
I}$), according to $\Pi_{X}$=$\sqrt{({\mathcal{Q}})^{2}+({\mathcal{U}})^{2}}$
and $\psi_{X}$=$1/2\tan^{-1}({\mathcal{U}}/{\mathcal{Q}})$. Error estimation
was calculated based on Kislat et al. (2015) and Muleri (2022). Figure 1
indicates the detection significance of this measurement in the form of a
polarization contour plot.

The second method (ML) uses an event-based maximum likelihood method to
determine Q and U (Marshall 2021b). The method accounts for background using
data from an annulus about the target by including a term in the likelihood,
as used in the analysis of the IXPE data of Cen A (Ehlert et al. 2022). Source
events were selected from a region of 60″ radius about the source centroid,
while the background events were selected from an annulus with inner and outer
radii of 200″ and 300″. In this case, $\Pi_{\rm X}$ = 13$\pm$1$\%$ and
$\psi_{\rm X}$ =107$\pm$3$\degr$.

Next, the spectropolarimetric analysis examined the X-ray spectra and
polarization properties based on spectral modeling with XSPEC (ver. 12.13.0c;
Arnaud 1996). With this method, we obtained $\Pi_{\rm X}$ =14$\pm$1$\%$ and
$\psi_{\rm X}$ =107$\pm 3\degr$. The detection significance of this
measurement was $\gtrsim$11$\sigma$. The flux was estimated as 2.25
($\pm$0.04) $\times 10^{-10}\,\hbox{erg s${}^{-1}$ cm${}^{-2}$}$ over 2–8 keV,
the same band over which the IXPE I, Q, and U spectra were obtained. In this
analysis, we additionally included the simultaneous XMM-Newton data in order
to refine the constraints on the spectral shape over the 0.3–10 keV energy
range. The cross-calibration factors among all the spectra are accounted for
using the CONSTANT model, normalized to the XMM-Newton spectrum while the
other spectra were varied. In all the fits, we considered the Galactic
absorption along the line of sight of Mrk 421. For this, we used the TBABS
model with weighted average column density values from HI4PI Collaboration et
al. (2016) ($N_{H}=1.34\times 10^{20}\rm{cm^{-2}}$). The WILM model was
applied to take into account metal abundance (Wilms et al. 2000). For the
spectral modeling, we first applied a simple power law model (POWERLAW in
XSPEC) to reproduce the I synchrotron spectrum, but the best-fit result was
poor, with $\chi^{2}/{\mathrm{d.o.f.}}$=3308/651. Hence, we employed the log-
parabolic model (LOGPAR), in which the photon index varies with energy
following a log parabola function (Massaro et al. 2004):

$\begin{split}N(E)=K(E/E_{pivot})^{(\alpha-\beta\log(E/E_{pivot}))},\end{split}$
(1)

where the pivot energy $E_{pivot}$ is a scaling factor, $\alpha$ describes the
slope of the photon spectrum at $E_{pivot}$, $\beta$ expresses the spectral
curvature, and $K$ denotes a normalization constant. This spectral model
generally describes a typical HSP spectrum well, including that of Mrk 421,
both in quiescence and in flaring states (Donnarumma et al. 2009; Baloković et
al. 2016). As the photon index varies with energy in this model, the choice of
reference energy $E_{pivot}$ changes the determined value of $\alpha$. In our
spectral fit, we fixed the pivot energy to 5.0 keV (e.g., Baloković et al.
2016; Middei et al. 2022). In this case, the $\alpha$ parameter approximately
corresponds to the photon index over 3.0–7.0 keV. In our spectral fitting, we
also allowed the values of $\alpha$, $\beta$, and $K$ to vary. These free
parameters are coupled with the reference spectrum, which here is the XMM-
Newton PN spectrum over the same 2-8 keV band to which the 3 IXPE DUs are
sensitive. Finally, the X-ray polarimetric measurements were constrained by
the POLCONST model based on the Stokes parameter fits. Consequently, we
obtained statistically acceptable fits, with
$\chi^{2}/{\mathrm{d.o.f.}}$=740/650, above 99$\%$ confidence level. Figure 9
and Table 3 present the parameter values of our best-fit results. We note that
in the Stokes-spectra-decoupled case, which only fits I spectra without
involving POLCONST, the best-fit results obtained the same values as we
derived from the simultaneous spectropolarimetric fit.

Finally, the MLS method was also applied to determine the polarization
properties for the POWERLAW spectral model. With this procedure, we derived
the polarization and spectral properties to be $\Pi_{\rm X}$ =13$\pm$1$\%$ and
$\psi_{\rm X}$ =109$\pm$3$\degr$.

We have thus derived that the X-ray polarization properties by four different
methods are consistent within the uncertainties (see Table 1). The small
differences can be explained by the fact that the PCUBE analysis estimated
spectral-model-independent polarization properties, while the XSPEC methods
take into account the best fit from the spectral modeling. Furthermore, among
the current choices, only the XSPEC method can improve sensitivity by applying
event weight methods introduced by Di Marco et al. (2022). Overall, we find
that the $\Pi_{\rm X}$ and $\psi_{\rm X}$ results are robust, with modest
statistical errors (see Figure 1).

###  2.2 Polarization variability

Figure 2: The null-hypothesis probability of the $\chi^{2}$ test for time
variability of the $\mathcal{Q}$ (red) and $\mathcal{U}$ (blue) Stokes
parameters with the constant model for different numbers of time bins for Obs.
4. The left and right vertical axes correspond to the probability values in
logarithmic and linear scales, respectively. The green shaded area indicates
that the null-hypothesis probability is above the 1% significance level. The
black dashed and dotted lines located in the middle of the panel represent 1%
and 3$\sigma$ (99.73%) probability, respectively.
Figure 3: IXPE polarization and photon counts versus time during Obs. 4. From
top to bottom: $\Pi_{\rm X}$ , $\psi_{\rm X}$ , and count rates. Polarization
results are from a time-resolved analysis with seven identical 19 ks time
bins. The detection significance for each bin is displayed at the top of the
figure. In $\Pi_{\rm X}$ and $\psi_{\rm X}$ panels, red dashed lines denote a
fit to a constant function; the shaded area corresponds to $\leq 3\sigma$
uncertainty. In addition, the blue and green solid lines indicate the best-fit
result with the sinusoidal and multicomponent model, respectively. The blue
dashed line in the light curve indicates the average value during the
observation.

We have investigated whether the polarization varies as a function of time or
energy. We tested the time dependence using two methods. In the first one, we
determined the null-hypothesis probability with the $\chi^{2}$ test using the
combined PCUBE and XSPEC analysis (e.g., Di Gesu et al. 2022; Gianolli et al.
2023; Di Gesu et al. 2023). For the second method, we used the unbinned event-
based maximum likelihood analysis implemented in Marshall (2021a). With this
method, we can avoid the error that can be caused by subjective selection bias
resulting from the binning criteria.

The first method measured each normalized Stokes parameter value by dividing
the data into identical time spans that depended on the selected number of
bins (e.g., 2 bins= 100 ks/2 bins = 50 ks/bin). In particular, we split Obs. 4
into 2 to 20 time bins. We then compared $\mathcal{Q}$ and $\mathcal{U}$,
which followed a Gaussian error distribution, with the results from fitting
each parameter treated as constant over time (i.e.,
$\mathcal{Q}(t)=\mathcal{Q}_{0}$ and $\mathcal{U}(t)=\mathcal{U}_{0}$). The
result of fitting the constant model was calibrated for each number of bins.
We then calculated the $\chi^{2}$ and the null-hypothesis probability (for the
corresponding degrees of freedom) for each case. Figure 2 indicates the null-
hypothesis probability as a function of the number of bins. In this figure,
the green-shaded area ($P_{\rm Null}>1\%$) corresponds to cases where the data
are statistically consistent with values of $\mathcal{Q}$ and $\mathcal{U}$
that are constant in time. Conversely, if the points lie outside the region
($P_{\rm Null}<1\%$), it implies that the polarization varies with time. As
seen in Figure 2, we found that splitting the $\mathcal{Q}(t)$ light curves
with 5, 7, 8, 9, 11, 12, and 14 time bins does not produce a good fit with the
constant model ($P_{\rm Null}<1\%$). The case of 7 time bins has the smallest
null-hypothesis probability, with $\chi^{2}/{\mathrm{d.o.f.}}$=25.26/6, beyond
the 3$\sigma$ confidence level. In the polarization light curve of Obs. 4 with
seven identical time bins (Figure 3), we can see that $\psi_{\rm X}$ varies
from the second to the fourth bin, after which it returns to a more stable
value near that of the first three bins. We estimate the rotation rate of this
variation as $\dot{\psi}\sim 92$$\degr/\rm day$ (change by $61\degr$ over
$\sim$57 ks from the second to the fourth bin). This result is comparable to
the rotation rate of $\dot{\psi}=85$$\degr/\rm day$ reported in Di Gesu et al.
(2023).

Additionally, in order to examine the possibility of continuous change of
$\psi_{\rm X}$, we have tested whether a sinusoidal function ($\psi_{\rm
X}(t)$=$A\sin(Ct-D)+B$) provides a better fit of $\psi_{\rm X}(t)$ with
respect to a constant function corresponding to the mean value of $\psi_{\rm
X}$ over 7 time bins (blue line in Figure 3). This approach yielded a smaller
$\chi^{2}/{\mathrm{d.o.f.}}$ value of 4.75/3, with a Bayesian information
criterion (BIC) of 12.53, than the constant model
($\chi^{2}/{\mathrm{d.o.f.}}$=16.66/6, BIC=18.60).

Figure 4: The time variability of the $\mathcal{Q}$ and $\mathcal{U}$ Stokes
parameters during Obs. 4. Each circle indicates the 1$\sigma$ uncertainties of
eight identical time intervals, derived using an event-based maximum
likelihood technique as described in Di Gesu et al. (2023). The black circle
shows the time-averaged polarization. Each color represents each time interval
from the start to the end of the Obs. 4.

As a second approach, we employed a maximum likelihood method that allows for
$\psi_{\rm X}$ rotation in each interval, as described elsewhere (Di Gesu et
al. 2023) and shown in Figure 4. Briefly, the ML method is used (see §2.1) for
eight equal time intervals but also allowing for $\psi_{\rm X}$ rotation
during the time interval, as an “uninteresting” parameter. This approach
prevents $\psi_{\rm X}$ rotations during a time interval from reducing the
average polarization of that interval, which is important when there are large
$\psi_{\rm X}$ changes. For example, Figure 4 shows that there appear to be
large, nearly $90\degr$ $\psi_{\rm X}$ shifts between intervals one and two
and between intervals three and four. As a result, we confirmed that
$\psi_{\rm X}$ varied over time during Obs. 4 with $\sim 4.5\sigma$
confidence.

In addition, with a maximum likelihood method, we tested a more complex model
fit — the multicomponent model (Pacciani et al. in preparation), which
involves the convolution of constant and rotating polarization components — to
check the possibility of continuous variation within the Obs. 4 period. The
details of this model are described in Appendix B. From this test, we computed
the significance of the multicomponent model relative to the constant
polarization model based on the delta likelihood, $\Delta
S^{\prime}=S^{\prime}(\Pi_{0},\Psi_{0},0)-S^{\prime}(\Pi_{1},\Psi_{1},\Pi_{2},\Psi_{2},\omega)$.
We obtained a decrease in the likelihood estimator, $\Delta L=-32.4$, which
follows a chi-square distribution with three degrees of freedom. Hence, the
constant polarization model is rejected in favor of the multicomponent model,
at the $\sim 5\sigma$ confidence level that this result occurred by chance.
The parameters estimated with the multicomponent model are reported in Table
2. The observed discrepancy between the rotation rate measured from the
multicomponent model (i.e., $\omega=245$$\degr/\rm day$) and the rate
estimated directly from the light curve (i.e., $\dot{\psi}\sim$92$\degr/\rm
day$) can be explained by differences in physical interpretation. The
multicomponent model assumes a persistent presence of the rotating component
in conjunction with the constant emission component, while the other case
assumes that the emission originates from a single dominant component. The
multicomponent model is indicated with a green line in Figure 3.

Therefore, based on these different methods, we conclude that we have found an
episodic $\psi_{\rm X}$ variation over time during Obs. 4. Furthermore, the
results of sinusoidal and multicomponent models suggest that the variations
may be attributed to continuous changes in stable components. However, due to
differences in the analysis methods employed, these two models were not
compared in this study. Moreover, since these model fit results were derived
as relative outcomes compared to the constant model, we cannot rule out the
possibility of encountering more complex components and potential stochastic
variations.

Table 2: Parameters for the multicomponent model $R_{1}\cdot\Pi_{1}$ | $\Psi_{1}$ | $(1-R_{1})\cdot\Pi_{2}$ | $\Psi_{2}^{0}$ | $\omega$
---|---|---|---|---
($\%$) | ($\degr$) | ($\%$) | ($\degr$) | ($\degr/\rm day$)
12.6$\pm$1.6 | 108$\pm$ 4 | $6.3\pm 1.8$ | $-96\pm 17$ | $246\pm 23$

We also tested whether the X-ray polarization depends on energy by applying
the same null-hypothesis probability test discussed above, but with the IXPE
energy band (2–8 keV) divided into smaller ranges. We did so for two energy
bins (2–4 and 4–8 keV), as well as three energy bins (2–4, 4–6, and 6–8 keV),
etc., up to 12 energy bins. We found no statistically significant differences
with a constant fit, as $P_{\rm Null}\geq 13\%$ from both $\mathcal{Q}$ and
$\mathcal{U}$, hence the data are consistent with energy-independent X-ray
polarization.

###  2.3 X-ray polarization and spectral variability across multiple IXPE
observations

Time-averaged polarization contours

Time-resolved polarization contours

Figure 5: Time-averaged and time-resolved polarization contours of four
multiple IXPE observations. In the top panel, each colored contour represents
the significance of the time-averaged polarization detection for the
corresponding observation (Obs. 1: red, Obs. 2: orange, Obs. 3: green, and
Obs. 4: blue). Contours are shown at confidence levels of 68.27%, 90.00%, and
99.00%, from a $\chi^{2}$ test with two degrees of freedom. Additionally, the
black line and gray shaded areas indicate the jet’s electric vector position
angle (EVPA) in degrees as observed by the VLBA at 43 GHz (Obs. 1:
100$\pm$10$\degr$, Obs. 2: 147$\pm$7$\degr$, Obs. 3: 147$\pm$7$\degr$, and
Obs. 4: 171$\pm$10$\degr$). In the bottom panel, the 99.00% confidence level
contours show the time variation of polarization properties by dividing the
data into three identical time intervals for each observation. In the cases of
Obs. 2 (orange) and Obs. 3 (green), arrows indicate a rotation of $\psi_{\rm
X}$ from the start to the end of the observation.
Figure 6: SWIFT-XRT X-ray spectral parameters and flux versus time of Mrk 421
from a log-parabolic best-fit model with $E_{pivot}$=5 keV. From top to
bottom: $\alpha$, $\beta$, hardness ratio, and flux in the soft (0.3–2 keV,
gray circles) and hard (2–10 keV, black diamonds) energy bands. The time
ranges of the IXPE pointings are indicated by the gray-shaded regions.

Significant X-ray polarization from Mrk 421 has been detected at four epochs
with IXPE (see Table 1). Figure 5 shows error contours for both time-averaged
and time-resolved data of each observation. Over the seven months from IXPE
Obs. 1 to Obs. 4, the value of $\psi_{\rm X}$ varied widely, with a continuous
rotation over 180$\degr$ observed during Obs. 2 and Obs. 3. In contrast,
measurements of $\Pi_{\rm X}$ for all events were similar within a range of
10–15$\%$, even though the time-averaged $\Pi_{\rm X}$ of Obs. 2 and Obs. 3
appears to be clearly lower than in the other cases due to dilution by the
changing of $\psi_{X}$. However, the mean value of $\Pi_{\rm X}$ during
periods of nonrotation of $\psi_{\rm X}$ (Obs. 1 and Obs. 4) was $14\pm 2\%$,
a factor of $1.4\pm 0.2$ higher than the value of $10\pm 1\%$ during the
rotation (Obs. 2 and Obs. 3). This result is similar to the findings of
previous optical polarimetry studies: (1) Blinov et al. (2016b) observed that
the ratio of $\Pi_{\rm O}$ of rotating to that of nonrotating cases is less
than 1 in 18 out of the 27 observed rotations, and (2) Jermak et al. (2016)
reported a $26\%$ reduction in the average $\Pi_{\rm O}$ during rotational
periods compared to nonrotational periods. Nonetheless, because of the limited
amount of X-ray polarization data, this result should be verified with an
increased number of measurements from future X-ray polarimetry monitoring
observations.

The radio electric-vector position angle $\psi_{\mathrm{43GHz}}$ obtained from
the 43 GHz Very Long Baseline Array (VLBA) observations (black line in Figure
5) differs from $\psi_{\rm X}$ and which also changes by $70\degr$ over the 7
months from Obs. 1 to Obs. 4. The pronounced variation of $\psi_{\rm X}$ and
$\psi_{\mathrm{43GHz}}$ contrasts with the steadier X-ray polarization
observed in other sources of the same sub-class of blazars, whose synchrotron
SED peaks at X-ray frequencies (i.e., Mrk 501; Liodakis et al. 2022).

We have also investigated the spectral properties and X-ray activity of Mrk
421 through Swift-XRT monitoring observations. Figure 6 presents the $\alpha$
and $\beta$ spectral parameters obtained from the LOGPAR model fit. We also
derived the X-ray flux in the soft (0.3–2 keV), $F_{\mathrm{soft}}$, and hard
(2–10 keV), $F_{\mathrm{hard}}$, energy bands, and define the hardness ratio
as ($F_{\mathrm{hard}}-F_{\mathrm{soft}}$) /
($F_{\mathrm{hard}}+F_{\mathrm{soft}}$). We have found that the $\alpha$
parameter, representing the slope of the spectrum at the pivot energy,
maintained similar values in Obs. 1 and Obs. 4. However, during Obs. 2 and
Obs. 3, $\alpha$ decreased with time. In addition, the hardness ratio also
varied during the $\psi_{\rm X}$ rotation, while its value in Obs. 1 was
consistent with that in Obs. 4. Nonetheless, in the case of Obs. 4, Swift-XRT
observations were conducted only at the beginning and shortly after the IXPE
pointing, thus we cannot exclude the possibility of variations of spectral
properties during Obs. 4.

The X-ray fluxes of Mrk 421 during all epochs of IXPE  observations were
within $1\sigma$ of the median historical value (e.g., Liodakis et al. 2019).
Nonetheless, we found significant fluctuations during Obs. 2 and Obs. 3.
Although the flux variations occurred during all of the IXPE  epochs (see,
e.g., Figure 3), the discrepancy between the minimum and maximum counting
rates was larger in Obs. 2 and Obs. 3. Therefore, we suggest that the smooth
$\psi_{\rm X}$ rotation behavior may accompany more pronounced spectral and
flux fluctuations.

##  3 Multiwavelength polarization analysis

Figure 7: VLBA total (contours) and polarized (color scale) intensity images
of Mrk 421 at 43 GHz. The peak of the total intensity is 295 mJy/beam; yellow
linear segments within the images indicate the direction of polarization;
black horizontal line marks the position of the core, $A0$; colored circles
indicate locations of jet features $A1$ (blue), $A2$ (gray), and $P$ (red).
Images are convolved with the same elliptical Gaussian beam, which is shown in
the bottom right corner by a black ellipse.

Multiwavelength campaigns for the first three observations of Mrk 421 are
reported in Di Gesu et al. (2022, 2023). During Obs. 4, Mrk 421 was observed
with the VLBA, the Effelsberg 100-m Radio Telescope, the Korean VLBI Network
(KVN), the Submillimeter Array (SMA), the Kanata telescope, the Perkins
telescope, and the Sierra Nevada Observatory (SNO, T90 telescope). Details of
the observations and observatories can be found in Appendix A.3.

We have analyzed VLBA data obtained for Mrk 421 within the BEAM-ME (Blazars
Entering the Astrophysical Multimessenger Era)
program222www.bu.edu/blazars/BEAM-ME.html during the period from MJD 59616
(2022 February 5) to MJD 59986 (2023 February 11) to investigate the parsec-
scale jet behavior during the IXPE observations. The data include total and
polarized intensity images at 43 GHz at 12 epochs. The data reduction and
modeling are described in §A.4. The parsec-scale jet of Mrk 421 is strongly
core-dominated, with the extended jet contributing an average of only $17\%$
of the core flux density at 43 GHz. We consider the core to be a stationary
feature of the jet and designate it as $A0$. During the one-year period
analyzed here, we have detected two stationary features in the jet in addition
to the core, $A1$ and $A2$, located at distances of $\sim$0.4 and $\sim$0.8
mas from $A0$, respectively (see Table 5). Dominance by stationary structures
is a well-documented property of the Mrk 421 jet (Lico et al. 2012; Jorstad et
al. 2017; Lister et al. 2021; Weaver et al. 2022). Nonetheless, we have found
motion in the jet, which we identify with a prominent polarized feature, $P$,
which has a subluminal apparent speed $\beta_{app}$=0.7$\pm$0.1 in units of
$c$. Figure 7 presents a sequence of images at epochs where the knot is
prominent. Figure 13 shows that knot $P$ had a high degree of polarization,
with $\psi_{\mathrm{P}}$ almost aligned with the jet direction, as expected if
feature $P$ was a transverse shock in the jet. According to its kinematics,
the knot passed through the core on 2022 March 14 (MJD 59653$\pm$80, Figure
11). If one considers the sizes of $A0$ and $P$ listed in Table 5, plus the
proper motion of $P$ (§A.4), it should have taken 107$\pm$19 days for knot $P$
to leave the core completely, which should have occurred within 2022 June 10 –
July 8. This means that during IXPE Obs. 1, Obs. 2, and Obs. 3, knot $P$ was
within the millimeter-wave core.

The Effelsberg 100-m Telescope observation was obtained within the framework
of the QUIVER program (Monitoring the Stokes Q, U, I and V Emission of AGN
jets in Radio) on 2022 December 2 (MJD 59915.11). The linear polarization was
measured at 3 bands: 4.85 GHz, 8.35 GHz and 10.45 GHz. Its polarization degree
was found to be 0.9$\pm$0.2$\%$, 1.2$\pm$0.4$\%$, and 1.2$\pm$0.2$\%$. and its
polarization angle 96.3$\pm$3.5$\degr$, 134.8$\pm$4.9$\degr$, and
159.2$\pm$7.2$\degr$, respectively. The KVN observations, with the antennas
combined to act as a single dish, were performed on 2022 December 9 (MJD
59922.73) at 22, 43, 86, and 129 GHz. The linear polarization was measured as
$\Pi_{\mathrm{22GHz}}$=2.4$\pm$0.7$\%$ along
$\psi_{\mathrm{22GHz}}$=30$\pm$12$\degr$ at 22 GHz and
$\Pi_{\mathrm{43GHz}}$=2.7$\pm$0.5$\%$ along
$\psi_{\mathrm{43GHz}}$=142$\pm$9$\degr$ at 43 GHz. The measured values of
$\Pi$ at 86 GHz (3.1$\pm$2.3$\%$) and 129 GHz (4.4$\pm$2.0$\%$) are not
significant detections. The polarization was measured with the SMA at 225.5
GHz as $\Pi_{\mathrm{225GHz}}$=2.0$\pm$0.3$\%$ along position angle
$\psi_{\mathrm{225GHz}}$=163$\pm$3$\degr$. At the same time, the intrinsic
polarization degree (after subtraction of the host-galaxy flux) in the optical
R-band from SNO was $\Pi_{O}$=4.6$\pm$1.3$\%$ along
$\psi_{O}$=206$\pm$9$\degr$. The Perkins telescope obtained optical R-band
polarization covering Obs. 4 from 2022 November 26 (MJD 59909) to December 17
(MJD 59930) to give the average optical polarization parameters just before
and after Obs. 4, which can be compared with the SNO measurement. In the
infrared J and optical R bands, the Kanata telescope data (not corrected for
the host-galaxy depolarization) yield $\Pi_{J}$=2.41$\pm$0.02$\%$ and
$\Pi_{O}$=2.1$\pm$0.03$\%$ along $\psi_{J}$=176$\pm$0.2$\degr$ and
$\psi_{O}$=167.0$\pm$0.3$\degr$, respectively. The latter is similar to
previous Mrk 421 R-band observations with the Kanata telescope, for which an
intrinsic value of $\Pi_{O}$$\approx$3.3$\%$ was derived (Di Gesu et al.
2023), which is consistent with the SNO measurement. All of the new
multiwavelength polarimetry results conducted during Obs. 4 are summarized in
Table 4.

Figure 10 exhibits the evolution of polarization properties of Mrk 421
obtained from the contemporaneous multiwavelength polarimetry campaign from
just before IXPE Obs. 1 to shortly after Obs. 4. Throughout this time period,
we find a similar strong chromatic behavior, with $\Pi$ 2–3 times higher at
X-ray rather than at longer wavelengths. In contrast, the infrared, optical,
and radio degrees of polarization were similar. Meanwhile, the polarization
position angle exhibited marked changes, with different values during the
various IXPE pointings. The range of $\psi$ observed in Obs. 1 across
millimeter to X-ray wavelengths was $\sim$30$\degr$, whereas a larger range of
$\sim$90$\degr$ was evident in Obs. 4. Moreover, (Di Gesu et al. 2023) have
reported that, during the rotation of $\psi_{\rm X}$ of Obs. 2 and Obs. 3,
$\psi$ values at other wavelengths were consistent with each other, with a
weak temporal variation. Hence, we conclude that the region where the
polarized X-rays are emitted is mostly or completely distinct from that at
longer wavelengths.

Furthermore, Mrk 421 exhibited a clockwise $\psi_{\rm O}$ rotation of
approximately 120$\degr$ over $\sim$40 days during the first three IXPE
observation periods. Similar behavior has been reported in previous optical
polarimetry monitoring studies, where the polarization angle changed by
$\sim$180$\degr$ over $\sim$50 days (Blinov et al. 2015; Jermak et al. 2016;
Blinov et al. 2016a; Fraija et al. 2017; Blinov et al. 2018). The direction of
the $\psi_{\rm O}$ rotation was opposite to the counter-clockwise $\psi_{\rm
X}$ rotation observed by IXPE during Obs. 2 and Obs. 3.

##  4 Discussion

Previous IXPE observations of HSPs Mrk 421 (Di Gesu et al. 2022, 2023) and Mrk
501 (Liodakis et al. 2022) suggest that an energy-stratified shock can most
readily explain the $\sim$2–3 times higher degree of polarization at X-ray
rather than at longer wavelengths. In this work, we have found consistent
multiwavelength measurements from Obs. 4. In this energy-stratified shock
model, the relativistic electrons convert their energy to radiation as they
move farther from the shock front (Marscher & Gear 1985; Tavecchio et al.
2018). The particles are efficiently accelerated at the shock front, where the
magnetic field is relatively well ordered, and hence emit X-ray synchrotron
radiation with relatively high polarization. Conversely, electrons lose energy
as they propagate away from the shock, causing them to radiate at longer
wavelengths, and the degree of polarization decreases as they encounter
increasingly turbulent magnetic fields (Di Gesu et al. 2022, 2023; Liodakis et
al. 2022). Hence, higher $\Pi$ is predicted at higher frequencies. This effect
ceases at wavelengths long enough that the electrons radiating at those
wavelengths can cross the entire shocked region before losing most of their
energy (Marscher & Gear 1985). This limitation can explain the similar
polarization from millimeter to optical wavelengths observed in Mrk 421. It is
also possible that the longer wavelength emission occurred mainly in a
relatively slow sheath surrounding a much faster X-ray emitting spine of the
jet (e.g., Di Gesu et al. 2023). This is consistent with the subluminal speed
we have measured for knot $P$. In this case, the X-ray and longer wavelength
polarization properties may not be related, consistent with their different
position angles.

We therefore consider two possible geometries for an energy-stratified jet in
Mrk 421: linear and radial models, as suggested by Di Gesu et al. (2022,
2023). In the case of linear geometry, the energy stratification and the
$\psi$ rotation can be explained by emission features propagating downstream
in the jet, following the helical magnetic field. On the other hand, the
radial structure can correspond to a helical, rotating innermost region and a
surrounding layer, similar to the spine-sheath jet model (Chhotray et al.
2017). The currently available observational data are insufficient to discern
between the linear and radial geometries. Nevertheless, the episodic variation
in polarization observed during Obs. 4 offers further insights into the
internal geometry of the jet. For instance, it implies alternative
perspectives on the geometric structure within the jet, including the
intricate interactions between coexisting stable and rotating magnetic field
structures, as well as stochastic transitions within the dominant magnetic
field structure responsible for particle acceleration. Therefore, future
observations of polarization variability are expected to yield further
evidence about the geometric structure inside the jet.

On the other hand, despite the comparable multiwavelength results reported for
Mrk 421 and Mrk 501, we have found a difference in the behavior of $\psi_{\rm
X}$ between them. In the case of Mrk 421, $\psi_{\mathrm{43GHz}}$ and
$\psi_{\rm X}$ exhibited significant variations without any consistent
alignment with either each other or the direction of the jet axis. However, in
the case of Mrk 501, an alignment was observed (within the uncertainties)
between measurements of the position angle of the jet axis, $\psi_{X}$, and
$\psi_{\mathrm{43GHz}}$ conducted within a month of each other (Liodakis et
al. 2022). Further IXPE and VLBA observations of HSPs are needed to confirm
whether, and if so, why, Mrk 421 is different in this regard.

We have found that the X-ray flux and hardness ratio of Mrk 421 were less
variable during Obs. 1 and Obs. 4, when $\psi_{\rm X}$ was essentially
constant, than during the $\psi_{\rm X}$ rotation of Obs. 2 and Obs. 3. This
implies different physical conditions within the jet between the rotating and
nonrotating states. Instead, the similar values of $\Pi_{\rm X}$ across all
observations suggest that the basic particle acceleration scenario remained
roughly independent of the magnetic field geometry. This can be accommodated
within the energy-stratified shock scenario, since the degree of order of the
magnetic field could be similar whether the shock moves along a straight or
helical trajectory. The flux and spectral variations during the rotation event
could have been caused by changes in the Doppler factor as the shock executed
helical motion, although the data are too sparsely sampled to test this.
Future X-ray spectroscopic and polarimetric observations with improved time
resolution can potentially test whether spiral motion along a helical magnetic
field causes cyclical Doppler factor variations that lead to observed
variations in flux, hardness ratio, and polarization.

Our examination of a sequence of 43 GHz VLBA images has revealed the presence
of a prominent, highly polarized knot moving away from the “core” at a speed
of $0.7c$ during a time span that includes Obs. 1 \- Obs. 3 (see Figure 11).

This finding suggests a potential connection between the morphological changes
near the jet core region and the variability in polarization, as discussed in
Di Gesu et al. (2023). The knot could represent a shock containing
relativistic electrons accelerated up to Lorentz factor $\sim$106 that radiate
at X-ray energies. However, in this scenario, it is difficult to explain the
difference in the behavior of $\psi_{\mathrm{X}}$ between Obs. 1 and Obs. 2 \-
Obs. 3, unless the geometry of the magnetic field varies across the core, with
a tighter helical structure toward the downstream end. However, our analysis
does not indicate a direct connection between the X-ray polarization position
angle and that of either the core or knot $P$ (see Figure 13), although the
degree of polarization of $P$, 15-20%, is comparable to $\Pi_{\rm X}$ . This
apparent lack of connection supports the conclusion, drawn above, that
emission regions at longer wavelengths (millimeter, IR, and optical) were
separate from, or only partially coincident with, that of the X-ray emission.

In the case of Obs. 4, there is no apparent connection between the values of
$\psi$ of the polarized features in the jet observed on 2022 December 6 (MJD
59919) and $\psi_{\rm X}$ (Fig. 13). However, Figure 12 indicates an increase
of the core flux density in 2023 February (MJD 59986) that could be a
signature of an emerging moving feature that would have been upstream of the
43 GHz VLBI core during IXPE  Obs. 4. Further combined IXPE and VLBI
monitoring could help to clarify whether there is any relation between the
X-ray emission regions and features seen in the jet at millimeter wavelength.

Figure 8: Schematic diagram of double helical magnetic field components inside
the jet. The downstream direction of the jet is to the left. The arrows
indicate each helical magnetic field component involved in the emission at
different wavelengths (blue: X-ray, black: longer wavelengths).

We note that, from Obs. 1 to Obs. 3, the polarization angle at optical, IR,
and radio wavelengths rotated in the opposite direction relative to the 5-day
rotation of $\psi_{\rm X}$ during Obs. 2 and Obs. 3. This finding supports the
conclusion that the X-ray emission region is separate from that at longer
wavelengths. The observed similarity in radio to optical polarization
properties implies that the emission at these wavelengths originates from
spatially interconnected regions. The longer (relative to X-ray) rotation time
scale suggests that the X-ray emission region is smaller than that of these
other wavelengths. Although the polarization vector rotation at longer
wavelengths could be explained by propagation of a larger emission feature
down a helical magnetic field (as in Figure 8), the long-term behavior of the
optical polarization of Mrk 421 implies stochastic, rather than systematic,
variations of both $\psi_{O}$ and $\Pi_{O}$ (Marscher & Jorstad 2022).

##  5 Conclusions

We have reported X-ray, optical, IR, and radio polarization and flux
measurements of the HSP blazar Mrk 421, including four IXPE pointings between
2022 May and December. Such observations probe the magnetic field structure
and particle acceleration mechanisms inside the jets of blazars. The combined
7 months of observations sample the time and energy dependence of the X-ray
polarization. Over this time span, $\psi_{\rm X}$ varied over the full range
of $\sim 180\degr$, including a 5-day episode of rotation, while the degree of
polarization maintained a value between $\sim$10–15$\%$ across all IXPE
pointings. The X-ray flux varied by a higher fraction during the rotation than
during the two IXPE observations without rotations.

The simultaneous multiwavelength polarimetry results over four IXPE
observations provide evidence useful for constraining the physics of the jet.
The degree of X-ray polarization was typically $\sim$2–3 times greater than
that at longer wavelengths at all epochs sampled, while the polarization
angles fluctuated. The discrepancy between the results of X-ray compared with
radio, IR, and optical polarimetry supports the previous conclusion that the
X-ray emission region is distinct from that at longer wavelengths in HSP
blazars (Liodakis et al. 2022; Di Gesu et al. 2022, 2023). As with these
previous studies, we conclude that the observations are consistent with the
energy stratified shock model, with the level of turbulence increasing with
distance from the shock front.

One difference between Mrk 421 and Mrk 501 is that there is no apparent
correlation between the direction of the jet from VLBA and $\psi_{X}$ in the
former. While this could be due to the bending of the jet from the X-ray to
the radio emitting region, amplified by the narrow angle to the line of sight,
the optical polarization angle is also much more highly variable in Mrk 421
than in Mrk 501 (Marscher & Jorstad 2022). This implies an intrinsic
difference between the two objects that should be explored with further
observations.

Following Di Gesu et al. (2023), we have proposed a linear and a radial
stratification scenarios to explain the rotation behavior of $\psi_{X}$. The
accompanying spectral variation during the $\psi_{\rm X}$ rotation suggests
that the physical conditions of the jet, such as the energy distribution of
relativistic electrons, differed between the periods of rotation and
nonrotation. In addition, we reported rotation in the opposite direction of
the $\psi$ between the X-ray and other wavelengths, with the latter occurring
over a much longer time scale. This could potentially be interpreted by the
presence of multi helical magnetic field structures inside the jet.
Morphological changes in the parsec-scale jet, possibly associated with the
contemporaneous emergence of a new knot of emission observed to move down the
jet in 43 GHz VLBA images, may be linked to the $\psi_{\rm X}$ rotation,
although differences in the radio and X-ray polarization angles argue against
such a connection.

In conclusion, the present study continues to develop a new perspective on the
physical and geometrical features of the magnetic field inside the jets of
blazars by employing polarimetry that extends from radio to X-ray wavelengths.
The IXPE observations, incorporating data at other wavelengths, have played a
significant role in constraining the emission arising from the innermost
regions of the jet. The polarization properties, sampled over different time
scales and energy regimes, suggest a possible connection between spectral and
polarization variations. The connection may include morphological changes in
radio images, which coincided with a period of $\psi_{\rm X}$ rotation.
However, due to infrequent data sampling, there remain uncertainties regarding
apparent correlations, which can be chance coincidences. In addition, thus far
the IXPE observations of Mrk 421 have been obtained when the blazar was in
average activity states. It is of great interest to determine whether the
polarization and physical properties change during strong flaring events. As
the IXPE mission continues, further studies of Mrk 421 and other blazars are
expected to provide the data needed to improve our understanding of the
magnetic field geometry and particle acceleration processes in relativistic
jets.

###### Acknowledgements.

The authors thank the anonymous referee for comments that improved this
manuscript. 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
QUIVER data are based on observations with the 100-m telescope of the MPIfR
(Max-Planck-Institut für Radioastronomie) at Effelsberg. Observations with the
100-m radio telescope at Effelsberg have received funding from the European
Union’s Horizon 2020 research and innovation programme under grant agreement
No 101004719 (ORP). The POLAMI observations were carried out at the IRAM 30m
Telescope. IRAM is supported by INSU/CNRS (France), MPG (Germany), and IGN
(Spain). 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). 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 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). 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 study was based in part on observations
conducted using the Perkins Telescope Observatory (PTO) in Arizona, USA, which
is owned and operated by Boston University. 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 toward 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. 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. 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. 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 a cooperative agreement by Associated Universities,
Inc. S. Kang, S.-S. Lee, W. Y. Cheong, S.-H. Kim, and H.-W. Jeong were
supported by the National Research Foundation of Korea (NRF) grant funded by
the Korea government (MIST) (2020R1A2C2009003). The KVN is a facility operated
by the Korea Astronomy and Space Science Institute. The KVN operations are
supported by KREONET (Korea Research Environment Open NETwork) which is
managed and operated by KISTI (Korea Institute of Science and Technology
Information). The VLBA 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.

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##  Appendix A Observation and data reduction

###  A.1 IXPE data

We obtained cleaned level 2 IXPE data processed by a standard IXPE pipeline
from the Science Operation Center
(SOC)333https://heasarc.gsfc.nasa.gov/docs/ixpe/analysis/IXPE-SOC-
DOC-009-UserGuide-Software.pdf. The pipeline includes the photoelectron events
correction in the Gas Pixel Detector (GPD, Costa et al. 2001; Bellazzini et
al. 2007; Fabiani et al. 2012; Baldini et al. 2021), as well as the photo-
electron track reconstruction process based on standard moments analysis
(Bellazzini et al. 2003; Fabiani & Muleri 2014; Di Marco et al. 2022). In
particular, the pipeline rectifies the fluctuations in gain properties
affected by gas conditions (e.g., temperature and pressure), nonuniform
charging of the Gas Electron Multiplier (GEM) material, and polarization
artifacts induced by spurious modulation (Rankin et al. 2022). The level 2
data of IXPE contain the time of arrival, position, and energy of each photon
event, along with polarization information represented by the Q and U Stokes
parameters.

The scientific data analysis was performed using the public ixpeobssim
software version 30.2.1 (Baldini et al. 2022). First, we extracted the source
and background data using an optimized region selection criterion as suggested
by Di Marco et al. (2023). The source data were extracted from a circular
region with a 60″radius, while the background data were extracted from an
annular source-free region with inner and outer radii of 150″and 300″,
respectively. Both regions were centered on the source position in the
detector frame. Next, we created the polarization cube (PCUBE) and the Stokes
parameter spectra (I , Q , and U ) using xpbin. To eliminate the influence due
to the background, we applied the background subtraction technique (Baldini et
al. 2022) and created PCUBEs of the source and background for each detector
unit. The final source polarization properties were recalibrated by
considering the BACKSCALE ratio between the source and background ($\sim
0.05$). In a similar way, we produced three Stokes parameter spectra for three
detectors, totaling 9 spectra, using the PHA1, PHAQ, and PHAU algorithms in
xpbin. We grouped the I spectra with a minimum of 30 counts in each energy
bin, as required for the $\chi^{2}$ statistics in the fits, and applied
constant energy binning for the Q and U spectra for every 0.2 keV interval. We
employed the recent version of the IXPE calibration database files (CALDB
20221020) contained in both ixpeobssim and the heasoft package (v6.31.1) for
both the PCUBE and the Stokes parameter spectra. In particular, we utilized
the weighted analysis method (Di Marco et al. 2022) on the I , Q , and U
spectra to improve the significance of our measurements with alpha075 response
matrices. The flux variability was analyzed using a light curve created with
the LC algorithms in xpbin of ixpeobssim.

The polarization error contour (Figure 1) was drawn from calibrated normalized
${\mathcal{Q}}\,(={\it Q}/{\it I})$ and ${\mathcal{U}}\,(={\it U}/{\it I})$ at
specific confidence levels (68.27%, 90.00%, and 99.0%) according to:

$\begin{split}&{\mathcal{Q}}_{C.L.}={\mathcal{Q}}_{0}+\epsilon\cos(\zeta),\\\
&{\mathcal{U}}_{C.L.}={\mathcal{U}}_{0}+\epsilon\sin(\zeta),\\\ \end{split}$
(2)

where ${\mathcal{Q}}_{0}$ and ${\mathcal{U}}_{0}$ represent the averaged
Stokes parameter values, and ${\mathcal{Q}}_{C.L.}$ and ${\mathcal{U}}_{C.L.}$
denote the ${\mathcal{Q}}$ and ${\mathcal{U}}$ values of a given confidence
level calculated with two degrees of freedom, respectively. We note that,
since we are considering two dependent variables, $\Pi$ and $\psi$, at the
same time, the error should be recalculated based on the $\chi^{2}$
distribution with 2 degrees of freedom
($\epsilon=\sigma$$\sqrt{\chi^{2}_{\mathrm{2\,d.o.f.}}}$,
$\sigma=\sigma_{\mathcal{Q}}=\sigma_{\mathcal{U}}$). The variable $\zeta$
follows the angle distribution from 0 to $2\pi$.

###  A.2 Spectroscopic X-ray data

During IXPE Obs. 4, a snapshot of about 5 ks was performed with XMM-Newton in
timing mode to limit photon pile-up. To extract the science products, we used
the XMM-Newton Science Analysis Software (SAS), version 21, with the most
updated CCF (current calibration files). Two boxed regions were adopted to
extract the source spectra and background. In particular, we chose the source
box to have a width of 27 pixels, as this size was found to maximize the
signal-to-noise ratio. (Details of this procedure can be found in Piconcelli
et al. 2004.) Then, the resulting spectrum was binned to achieve at least 30
counts in each energy bin. To allow simultaneous spectropolarimetric fitting
within XSPEC, the keyword “XFLT0001 Stokes:0” was added to the headers of the
corresponding PHA files.

The Swift-XRT monitoring consisted of $\sim$1 ks long observations performed
in Windowed Timing (WT) mode. Raw data were reduced using the standard
commands within the XRT Data Analysis Software (XRTDAS, v. 3.6.1) and adopting
the latest calibration files available in the Swift-XRT CALDB (version
20210915). The cleaned event files were used to extract the source spectrum
from a circular region with a radius of 47$\arcsec$, while the background was
extracted using a blank sky WT observation available in the Swift archive,
also within a circular region of radius 47$\arcsec$. The resulting spectra
were subsequently binned, requiring at least 25 counts in each energy bin.

Figure 9: Spectropolarimetric fit of XMM-Newton (blue) and IXPE I (red), Q
(green), and U (orange) spectra, with residuals of the best-fit model. The
left panel displays the I spectrum with a log-parabolic model (Massaro et al.
2004), while the right panel presents the constant polarization model fit with
the POLCONST model for Q and U spectra. The black line in both panels
indicates the model fit result.  Table 3: Best-fit parameters from our
simultaneous spectropolarimetric analysis.

MODEL = CONSTANT $\times$ TBABS $\times$ LOGPAR $\times$ POLCONST
---
Component | Parameter | Units | Value ($\pm 1\sigma$)
LOGPAR | $\alpha$ |  | 2.82 $\pm$0.005
| $\beta$ |  | 0.20 $\pm$0.004
| $pivot\;E$ | $\rm{keV}$ | 5\. (f)
| $Norm$ |  | 2.96e-3 $\pm$1.e-5
POLCONST | $\Pi_{\rm X}$ | ($\%$) | 14 $\pm$1
| $\psi_{\rm X}$ | ($\degr$) | 107 $\pm$3
TBABS | $N_{H}$ | $10^{22}{\rm cm^{-2}}$ | 1.34e-2 (f)
CONSTANT | XMM-Newton |  | 1\. (f)
|  IXPE  DU 1 |  | 1.11 $\pm$ 0.005
|  IXPE  DU 2 |  | 1.06 $\pm$ 0.005
|  IXPE  DU 3 |  | 1.01 $\pm$ 0.005
$\chi^{2}$ / ${\rm d.o.f.}$= 740/650
444Errors are $1\sigma$ confidence and (f) denotes a fixed parameter.

###  A.3 Radio, Infrared and optical data

During all of the IXPE pointings, we observed Mrk 421 at millimeter (radio),
infrared (IR), and optical wavelengths, measuring both flux density and linear
polarization. The observations and data analysis for the first three
observations can be found in detail in Di Gesu et al. (2022, 2023); here we
provide only a short description. For Obs. 4, radio, IR, and optical
observations were provided by QUIVER at the Effelsberg telescope, KVN, SMA (Ho
et al. 2004), Hiroshima Optical and Near-InfraRed camera (HONIR, Akitaya et
al. 2014) at the Kanata telescope, the Perkins telescope, and T90 at the
Sierra Nevada Observatory (SNO).

The QUIVER observations were performed at several radio bands (depending on
receiver availability and weather conditions) from 2.6 GHz to 44 GHz (11 cm to
7 mm wavelength) using six receivers located at the secondary focus of the
100-m Effelsberg Radio Telescope (S110mm, S45mm, S28mm, S20mm, S14mm, and
S7mm). The receivers are equipped with two orthogonally polarized feeds
(either circular or linear) that can deliver polarimetric parameters using
either conventional polarimeters or by connecting the SpecPol
spectropolarimetric backend. Instrumental polarization is calibrated via
observations of both polarized and unpolarized calibrators performed in each
session, and then removed from the data (e.g., Myserlis et al. 2018; Kraus et
al. 2003). The polarized intensity, degree, and position angle were derived
from the Stokes I, Q, and U cross-scans. The total flux density was
successfully recovered at 13 bands between 4.85 GHz and 43.75 GHz. The
calibrators 3C 286, 3C 48, and NGC 7027 were used for the total flux and
polarization calibration (e.g., Myserlis et al. 2018; Kraus et al. 2003).

Observations were conducted with the KVN simultaneously at 4 frequencies from
22 to 129 GHz in single-dish mode. This was achieved by using the Tamna (22,
43 GHz) and Yonsei (86,129 GHz) antennas with circularly polarized feed horns
to conduct two-frequency dual-polarization observations. The polarization
angle was calibrated using the Crab nebula (152∘; Aumont et al. 2010), and the
polarization degree using Jupiter (unpolarized) and 3C286 (polarized, Agudo et
al. 2012) following Kang et al. (2015).

The SMA observations were taken at 225.538 GHz (1.3 mm) through the SMAPOL
program (SMA Monitoring of AGN with POLarization). The SMAPOL observations
were taken on 2022 December 7 (MJD 59920) in full polarization mode (Marrone &
Rao 2008) using the SWARM correlator (Primiani et al. 2016), and calibrated
with the MIR software package
555https://lweb.cfa.harvard.edu/~cqi/mircook.html.

Observations using HONIR were taken on 2022 December 6 (MJD 59919.7537) in the
R and J
bands666http://hasc.hiroshima-u.ac.jp/instruments/honir/filters-e.html. Each
observation comprised four exposures in different positions of the half-wave
plate (Kawabata et al. 1999), which were then used to estimate the Stokes
parameters and calculate the polarization degree and angle. The host galaxy of
Mrk 421 contributes a significant fraction of unpolarized flux to the total
emission, thereby reducing the polarization. None of the polarization degree
estimates from Kanata have been corrected for the host-galaxy contribution due
to the lack of the photometry data needed for such a correction, and hence
they should be considered as lower limits to the intrinsic polarization degree
of the blazar component.

The optical photometric and polarimetric observations were also obtained at
the 1.8 m Perkins telescope (Flagstaff, AZ, USA) with the PRISM
camera777https://www.bu.edu/prism/ in R band performed before and after Obs.
4. The camera includes a polarimeter with a rotating half-wave plate. Each
polarization observation has consisted of four consecutive exposures of 60 s
at instrumental position angles 0$\degr$, 90$\degr$, 45$\degr$, and 135$\degr$
of the waveplate to calculate the normalized Stokes parameters Q and U.

The SNO observations were performed using polarized filters oriented to
represent different positions of a half-wave plate, similar to a conventional
polarimeter. The data are then analyzed following standard photometric
procedures. Mrk 421 was observed on 2022 December 7 in the R band. In this
case, following Nilsson et al. (2007) and Hovatta et al. (2016), we have
corrected the host-galaxy contribution. Several observations were taken within
the same night. We report the weighted average and uncertainty after
considering all of the intra-night observations to account for variations of
the system and the observing conditions during the rotation of the filter
wheel.

Table 4: Multiwavelength polarization properties of Mrk 421

Telescope | Band | Dates | Radio Polarization | Radio Flux Density
---|---|---|---|---
| (GHz) | (YYYY-MM-DD) |  $\Pi_{\rm R}$(%) $\psi_{\rm R}$ ($\degr$) | ($10^{-25}$erg s-1 cm-2 Hz-1)
KVN | 22 | 2022-12-09 | 2.4 $\pm$ 0.7 210 $\pm$ 12 | 45 $\pm$ 4
KVN | 43 | 2022-12-09 | 2.7 $\pm$ 0.5 142 $\pm$ 9 | 40 $\pm$ 2
KVN | 86 | 2022-12-09 | 3.1 $\pm$ 2.3 153 $\pm$ 15 | 34 $\pm$ 7
KVN | 129 | 2022-12-09 | 4.4 $\pm$ 2.0 191 $\pm$ 16 | 71 $\pm$ 19
SMA | 230 | 2022-12-07 | 2.0 $\pm$ 0.3 163 $\pm$ 3 | 20 $\pm$ 2
Effelsberg | 4.85 | 2022-12-02 | 0.9 $\pm$ 0.2 96 $\pm$ 4 | 59.4 $\pm$ 1.0
Effelsberg | 8.35 | 2022-12-02 | 1.2 $\pm$ 0.4 135 $\pm$ 5 | 50.1 $\pm$ 0.5
Effelsberg | 10.45 | 2022-12-02 | 1.2 $\pm$ 0.2 159 $\pm$ 7 | 48.7 $\pm$ 1.1
VLBA | 43 | 2022-12-06 | 1.7 $\pm$ 0.6 171 $\pm$ 10 | 25.9 $\pm$ 3.3
VLBA | 43 | 2022-11-20 | 2.1 $\pm$ 0.4 178 $\pm$ 7 | 22.9 $\pm$ 3.3
VLBA | 43 | 2022-11-01 | 1.5 $\pm$ 0.3 181 $\pm$ 8 | 27.5 $\pm$ 3.5
VLBA | 43 | 2022-08-21 | 2.0 $\pm$ 0.6 166 $\pm$ 9 | 29.4 $\pm$ 3.6
VLBA | 43 | 2022-07-22 | 1.3 $\pm$ 0.5 215 $\pm$ 13 | 28.8 $\pm$ 3.6
VLBA | 43 | 2022-07-15 | ¡ 1.1 Undetermined | 29.0 $\pm$ 4.1
Telescope | Band | Dates | Infrared Polarization | Infrared flux density
| ($\mu m$) | (YYYY-MM-DD) |  $\Pi_{\rm IR}$(%) $\psi_{\rm IR}$ ($\degr$) | ($10^{-25}$erg s-1 cm-2 Hz-1)
Kanata 11footnotemark: 1 | J:1.1 | 2022-12-07 | 2.1 $\pm$ 0.03 167.0 $\pm$ 0.3 | 1.65 $\pm$ 0.03
Telescope | Band | Dates | Optical Polarization | Optical flux density
| (nm) | (YYYY-MM-DD) |  $\Pi_{\rm O}$(%) $\psi_{\rm O}$ ($\degr$) | ($10^{-25}$erg s-1 cm-2 Hz-1)
SNO | R:653 | 2022-12-07 | 4.6 $\pm$ 1.3 206 $\pm$ 9 | 2.11 $\pm$ 0.03
Kanata 11footnotemark: 1 | R:653 | 2022-12-07 | 2.41 $\pm$ 0.02 176 $\pm$ 0.2 | –
Perkins | R:653 | 2022-12-17 | 2.65 $\pm$ 0.21 151.9 $\pm$ 2.0 | 2.75 $\pm$ 0.03
Perkins | R:653 | 2022-12-17 | 2.68 $\pm$ 0.21 156.2 $\pm$ 2.3 | 2.74 $\pm$ 0.03
Perkins | R:653 | 2022-12-17 | 2.51 $\pm$ 0.21 155.9 $\pm$ 2.5 | 2.75 $\pm$ 0.03
Perkins | R:653 | 2022-12-16 | 1.12 $\pm$ 0.27 156.2 $\pm$ 7.2 | 2.87 $\pm$ 0.03
Perkins | R:653 | 2022-12-15 | 1.48 $\pm$ 0.28 160.6 $\pm$ 5.7 | 2.77 $\pm$ 0.03
Perkins | R:653 | 2022-12-14 | 1.74 $\pm$ 0.30 125.1 $\pm$ 5.3 | 2.72 $\pm$ 0.02
Perkins | R:653 | 2022-12-01 | 6.89 $\pm$ 0.39 161.7 $\pm$ 1.8 | 2.98 $\pm$ 0.04
Perkins | R:653 | 2022-12-01 | 7.11 $\pm$ 0.37 164.5 $\pm$ 1.7 | 2.99 $\pm$ 0.03
Perkins | R:653 | 2022-12-01 | 6.91 $\pm$ 0.37 159.0 $\pm$ 1.6 | 3.00 $\pm$ 0.03
Perkins | R:653 | 2022-11-28 | 4.76 $\pm$ 0.25 179.1 $\pm$ 1.6 | 2.99 $\pm$ 0.03
Perkins | R:653 | 2022-11-28 | 4.39 $\pm$ 0.28 164.4 $\pm$ 1.9 | 3.00 $\pm$ 0.04
Perkins | R:653 | 2022-11-28 | 5.22 $\pm$ 0.22 164.7 $\pm$ 1.3 | 3.00 $\pm$ 0.03
Perkins | R:653 | 2022-11-28 | 4.22 $\pm$ 0.22 168.6 $\pm$ 1.6 | 2.99 $\pm$ 0.03
Perkins | R:653 | 2022-11-27 | 5.79 $\pm$ 0.26 162.3 $\pm$ 1.3 | 2.98 $\pm$ 0.03
Perkins | R:653 | 2022-11-27 | 5.79 $\pm$ 0.21 165.3 $\pm$ 0.8 | 2.99 $\pm$ 0.02
Perkins | R:653 | 2022-11-27 | 5.02 $\pm$ 0.21 163.7 $\pm$ 0.8 | 2.98 $\pm$ 0.02
Perkins | R:653 | 2022-11-27 | 5.29 $\pm$ 0.21 163.3 $\pm$ 0.4 | 2.99 $\pm$ 0.02
Perkins | R:653 | 2022-11-26 | 6.42 $\pm$ 0.21 151.5 $\pm$ 0.8 | 2.86 $\pm$ 0.02
Perkins | R:653 | 2022-11-26 | 5.71 $\pm$ 0.23 152.0 $\pm$ 1.1 | 2.87 $\pm$ 0.02
Perkins | R:653 | 2022-11-26 | 5.44 $\pm$ 0.21 155.0 $\pm$ 0.9 | 2.86 $\pm$ 0.02
888 11footnotemark: 1Not corrected for dilution of polarization by unpolarized
starlight from the host galaxy.

Figure 10: Multiwavelength polarization versus time of Mrk 421. Each row
presents the polarization degree (circle) and polarization angle (triangle),
with 1$\sigma$ errors measured by radio (red: $f$¡50 GHz, yellow: 50¡$f$¡100
GHz, green: 100¡$f$¡200 GHz, blue: $f$¡200 GHz), infrared (green), and optical
(red), and X-ray (orange) facilities, from top to bottom. The gray shaded
areas denote the durations of the four IXPE observations.

###  A.4 VLBA observations and analysis

The VLBA data were reduced with the Astronomical Image Process System (AIPS)
and Difmap software packages in the manner described by Jorstad et al. (2017).
The total intensity images are modeled by a number of components with circular
Gaussian brightness distributions, with the minimum number of components
determined by the best fit between the data and model at each epoch according
to a $\chi^{2}$ test. We identify the brightest feature located at the
southeastern end of the jet as the “core,” designated as $A0$, which we assume
to be a stationary feature. To obtain polarized intensity images, the data
were corrected for the instrumental-polarization “D-terms,” calculated by
averaging the D-terms obtained individually from a number of sources (usually
15) observed along with Mrk 421 in the BEAM-ME program. The electric-vector
position angle ($\psi_{\mathrm{43GHz}}$) calibration was obtained by different
methods, as discussed in Jorstad et al. (2017). The modeling of total
intensity images provides the following parameters of each component: flux
density, $S$, distance from the core, $R$, position angle with respect to the
core, $\Theta$, and angular size of the component, $a$ (FWHM of the best-fit
Gaussian). For polarized features in the jet, we have calculated the degree of
polarization, $\Pi_{\mathrm{knot}}$, and $\psi_{\mathrm{knot}}$, by
integrating the $Q$, and $U$ Stokes parameter models obtained with Difmap
during imaging within the size of the corresponding feature from the total
flux density modeling. As a result of modeling, we have detected three
stationary features, $A0$, $A1$, and $A2$, and one moving knot, $P$ in the jet
of Mrk 421 during 12 epochs from 2022 February 5 to 2023 February 11. Table 5
gives the average (over epochs) parameters of the jet features, while Figure
11 plots the distance of all knots detected in the jet versus time, and Figure
12 presents the light curve of the core, $A0$.

Table 5: Average parameters of the main features in the jet of Mrk 421 at 43 GHz Parameter | A0 | A1 | A2 | P
---|---|---|---|---
Number of epochs | 12 | 11 | 12 | 8
Average flux, $S$, (Jy) | 0.31 $\pm$ 0.03 | 0.023 $\pm$ 0.014 | 0.017 $\pm$ 0.008 | 0.014 $\pm$ 0.010
Average distance, $R$ (mas) | – | 0.41 $\pm$ 0.05 | 0.78 $\pm$ 0.03 | 0.23 $\pm$ 0.08
Average PA, $\Theta$, ($\degr$) | – |  $-$31 $\pm$ 10 |  $-$45 $\pm$ 5 |  $-$10 $\pm$ 12
Average Size, $a$, (mas) | 0.06 $\pm$ 0.02 | 0.24 $\pm$ 0.07 | 0.46 $\pm$ 0.12 | 0.14 $\pm$ 0.07
Figure 11: Distance from the core of jet components $A1$ (blue circles), $A2$
(gray circles), and $P$ (red circles) vs. time; black dotted line indicates
position of the core, $A0$; the red line approximates the motion of knot $P$
with the uncertainty of its ejection time shown by the red area on the bottom
left; yellow areas mark epochs of IXPE observations. Figure 12: Light curve of
the core, $A0$; the red vertical line marks the time of the ejection of knot
$P$, with its 1$\sigma$ uncertainty indicated by the red horizontal line;
yellow areas mark epochs of IXPE observations. Figure 13: Polarization
parameters of component $A0$ (black circles) and $P$ (red circles): top panel
- degree of polarization, bottom panel - position angle of polarization; the
yellow areas mark IXPE observations, while blue triangles plot
$\Pi_{\mathrm{X}}$, and $\psi_{\mathrm{X}}$; the dashed gray line in the
bottom panel shows the direction of the jet according to Weaver et al. (2022),
with dotted gray lines indicating its 1$\sigma$ uncertainty.

We have approximated the motion of knot $P$ with a ballistic trajectory with
proper motion $\mu$=0.34$\pm$0.06 mas yr-1, corresponding to a subluminal
apparent speed (in units of $c$) $\beta_{app}=0.7\pm 0.1$. The MOJAVE survey
999www.cv.nrao.edu/MOJAVE/sourcepages/1101+384.shtml at 15 GHz finds an
historical maximum $\beta_{app}\sim 0.2$ in the Mrk 421 jet. At 43 GHz, with 3
times higher angular resolution, Weaver et al. (2022) reported three knots
with $\beta_{app}>1$, with the fastest having an apparent speed of $2.41\pm
0.14\leavevmode\nobreak\ c$, based on monitoring within the VLBA-BU-BLAZAR
program. We have extrapolated the motion of $P$ back to the core (as shown in
Figure 11 by a red solid line), which yields the time of ejection of $P$ from
the core 101010The time of passage of the centroid of $P$ through the centroid
of $A0$ as MJD 59653$\pm$80 (2022 March 14). Figure 12 plots the light curve
of the core ($A0$), which has a local maximum on MJD 59699 (2022, April 30),
the epoch of VLBA observation closest to the ejection time of $P$; this
supports the derived time of ejection. Knot $P$ possesses the highest degree
of polarization in the jet, with $\Pi_{P}$ reaching $\sim$20%. Another
polarized feature in the jet is the core $A0$, with an average $\Pi_{A0}\sim
2\%$. Figure 13 shows the evolution of the polarization parameters
$\Pi_{\mathrm{A0}}$, $\psi_{\mathrm{A0}}$, and $\Pi_{\mathrm{P}}$,
$\psi_{\mathrm{P}}$. Figures 11 and 13 also indicate the times of the four
IXPE observations by yellow areas, with the widest area covering two pointings
(Obs. 2 and Obs. 3). Figure 13 plots the values of $\Pi_{\rm X}$ and
$\psi_{X}$, except during observations Obs. 2 & Obs. 3, when $\psi_{\rm X}$
rotated at a fast rate of $\sim$85$\degr/\rm day$ (Di Gesu et al. 2023).

##  Appendix B Maximum likelihood method of multicomponent model

The multicomponent model (Pacciani et al., in preparation) comprises (1) a
constant component with a steady polarization degree and angle, and (2) a
rotating component with a constant polarization degree and a rotating
polarization angle. The likelihood estimator for this model is:

$S(\Pi_{1},\Psi_{1},\Pi_{2},\Psi^{0}_{2},R_{1},\omega)=\\\
-2\sum_{i}\ln\biggr{[}1+\mu_{i}R_{1}\Pi_{1}\left(\cos 2\Psi_{1}\cos
2\psi_{i}+\sin 2\Psi_{1}\sin 2\psi_{i}\right)+\\\
\mu_{i}(1-R_{1})\Pi_{2}\left(\cos 2\Psi^{0}_{2}\cos 2(\psi_{i}-\omega t)+\sin
2\Psi^{0}_{2}\sin 2(\psi_{i}-\omega t)\right)\biggr{]}$ (3)

where $\Pi_{1}$ and $\Psi_{1}$ are the polarization degree and angle of the
constant component; $\Pi_{2}$ is the polarization degree of the rotating
component, $\Psi_{2}^{0}$ is the initial polarization angle of the rotating
component; $\omega$ is the rotation rate of the rotating component; $R_{1}$ is
the relative intensity of the constant component; and $\mu_{i}$ is modulation
factor of each event. We note that the fit with this model allows only
extraction of the products of $R_{1}\Pi_{1}$ and of $(1-R_{1})\Pi_{2}$. The
constant polarization model can be obtained from Equation 3 by setting the
ratio parameter to $R_{1}=1$. Similarly, the rotation model with constant
polarization degree can be obtained by setting $R_{1}=0$.