# Probing the connection between IceCube neutrinos and MOJAVE AGN

R. Abbasi Department of Physics, Loyola University Chicago, Chicago, IL 60660,
USA  M. Ackermann Deutsches Elektronen-Synchrotron DESY, Platanenallee 6,
D-15738 Zeuthen, Germany  J. Adams  Dept. of Physics and Astronomy, University
of Canterbury, Private Bag 4800, Christchurch, New Zealand  S. K. Agarwalla
also at Institute of Physics, Sachivalaya Marg, Sainik School Post,
Bhubaneswar 751005, India  Dept. of Physics and Wisconsin IceCube Particle
Astrophysics Center, University of Wisconsin—Madison, Madison, WI 53706, USA
J. A. Aguilar Université Libre de Bruxelles, Science Faculty CP230, B-1050
Brussels, Belgium  M. Ahlers Niels Bohr Institute, University of Copenhagen,
DK-2100 Copenhagen, Denmark  J.M. Alameddine Dept. of Physics, TU Dortmund
University, D-44221 Dortmund, Germany  N. M. Amin  Bartol Research Institute
and Dept. of Physics and Astronomy, University of Delaware, Newark, DE 19716,
USA  K. Andeen Department of Physics, Marquette University, Milwaukee, WI
53201, USA  C. Argüelles Department of Physics and Laboratory for Particle
Physics and Cosmology, Harvard University, Cambridge, MA 02138, USA  Y. Ashida
Department of Physics and Astronomy, University of Utah, Salt Lake City, UT
84112, USA  S. Athanasiadou  Deutsches Elektronen-Synchrotron DESY,
Platanenallee 6, D-15738 Zeuthen, Germany  L. Ausborm  III. Physikalisches
Institut, RWTH Aachen University, D-52056 Aachen, Germany  S. N. Axani Bartol
Research Institute and Dept. of Physics and Astronomy, University of Delaware,
Newark, DE 19716, USA  X. Bai Physics Department, South Dakota School of Mines
and Technology, Rapid City, SD 57701, USA  A. Balagopal V. Dept. of Physics
and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin—Madison, Madison, WI 53706, USA  M. Baricevic  Dept. of Physics and
Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin—Madison, Madison, WI 53706, USA  S. W. Barwick Dept. of Physics and
Astronomy, University of California, Irvine, CA 92697, USA  S. Bash  Physik-
department, Technische Universität München, D-85748 Garching, Germany  V. Basu
Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center,
University of Wisconsin—Madison, Madison, WI 53706, USA  R. Bay  Dept. of
Physics, University of California, Berkeley, CA 94720, USA  J. J. Beatty Dept.
of Astronomy, Ohio State University, Columbus, OH 43210, USA  Dept. of Physics
and Center for Cosmology and Astro-Particle Physics, Ohio State University,
Columbus, OH 43210, USA  J. Becker Tjus also at Department of Space, Earth and
Environment, Chalmers University of Technology, 412 96 Gothenburg, Sweden
Fakultät für Physik & Astronomie, Ruhr-Universität Bochum, D-44780 Bochum,
Germany  J. Beise Dept. of Physics and Astronomy, Uppsala University, Box 516,
SE-75120 Uppsala, Sweden  C. Bellenghi Physik-department, Technische
Universität München, D-85748 Garching, Germany  C. Benning  III.
Physikalisches Institut, RWTH Aachen University, D-52056 Aachen, Germany  S.
BenZvi Dept. of Physics and Astronomy, University of Rochester, Rochester, NY
14627, USA  D. Berley  Dept. of Physics, University of Maryland, College Park,
MD 20742, USA  E. Bernardini Dipartimento di Fisica e Astronomia Galileo
Galilei, Università Degli Studi di Padova, I-35122 Padova PD, Italy  D. Z.
Besson  Dept. of Physics and Astronomy, University of Kansas, Lawrence, KS
66045, USA  E. Blaufuss Dept. of Physics, University of Maryland, College
Park, MD 20742, USA  L. Bloom Dept. of Physics and Astronomy, University of
Alabama, Tuscaloosa, AL 35487, USA  S. Blot Deutsches Elektronen-Synchrotron
DESY, Platanenallee 6, D-15738 Zeuthen, Germany  F. Bontempo  Karlsruhe
Institute of Technology, Institute for Astroparticle Physics, D-76021
Karlsruhe, Germany  J. Y. Book Motzkin Department of Physics and Laboratory
for Particle Physics and Cosmology, Harvard University, Cambridge, MA 02138,
USA  C. Boscolo Meneguolo Dipartimento di Fisica e Astronomia Galileo Galilei,
Università Degli Studi di Padova, I-35122 Padova PD, Italy  S. Böser Institute
of Physics, University of Mainz, Staudinger Weg 7, D-55099 Mainz, Germany  O.
Botner Dept. of Physics and Astronomy, Uppsala University, Box 516, SE-75120
Uppsala, Sweden  J. Böttcher III. Physikalisches Institut, RWTH Aachen
University, D-52056 Aachen, Germany  J. Braun  Dept. of Physics and Wisconsin
IceCube Particle Astrophysics Center, University of Wisconsin—Madison,
Madison, WI 53706, USA  B. Brinson School of Physics and Center for
Relativistic Astrophysics, Georgia Institute of Technology, Atlanta, GA 30332,
USA  J. Brostean-Kaiser  Deutsches Elektronen-Synchrotron DESY, Platanenallee
6, D-15738 Zeuthen, Germany  L. Brusa  III. Physikalisches Institut, RWTH
Aachen University, D-52056 Aachen, Germany  R. T. Burley  Department of
Physics, University of Adelaide, Adelaide, 5005, Australia  D. Butterfield
Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center,
University of Wisconsin—Madison, Madison, WI 53706, USA  M. A. Campana Dept.
of Physics, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104,
USA  I. Caracas  Institute of Physics, University of Mainz, Staudinger Weg 7,
D-55099 Mainz, Germany  K. Carloni  Department of Physics and Laboratory for
Particle Physics and Cosmology, Harvard University, Cambridge, MA 02138, USA
J. Carpio Department of Physics & Astronomy, University of Nevada, Las Vegas,
NV 89154, USA  Nevada Center for Astrophysics, University of Nevada, Las
Vegas, NV 89154, USA  S. Chattopadhyay  also at Institute of Physics,
Sachivalaya Marg, Sainik School Post, Bhubaneswar 751005, India  Dept. of
Physics and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin—Madison, Madison, WI 53706, USA  N. Chau  Université Libre de
Bruxelles, Science Faculty CP230, B-1050 Brussels, Belgium  Z. Chen  Dept. of
Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794-3800, USA
D. Chirkin Dept. of Physics and Wisconsin IceCube Particle Astrophysics
Center, University of Wisconsin—Madison, Madison, WI 53706, USA  S. Choi
Dept. of Physics, Sungkyunkwan University, Suwon 16419, Republic of Korea
Institute of Basic Science, Sungkyunkwan University, Suwon 16419, Republic of
Korea  B. A. Clark Dept. of Physics, University of Maryland, College Park, MD
20742, USA  A. Coleman Dept. of Physics and Astronomy, Uppsala University, Box
516, SE-75120 Uppsala, Sweden  G. H. Collin  Dept. of Physics, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA  A. Connolly  Dept. of
Astronomy, Ohio State University, Columbus, OH 43210, USA  Dept. of Physics
and Center for Cosmology and Astro-Particle Physics, Ohio State University,
Columbus, OH 43210, USA  J. M. Conrad Dept. of Physics, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA  R. Corley  Department of
Physics and Astronomy, University of Utah, Salt Lake City, UT 84112, USA  D.
F. Cowen Dept. of Astronomy and Astrophysics, Pennsylvania State University,
University Park, PA 16802, USA  Dept. of Physics, Pennsylvania State
University, University Park, PA 16802, USA  P. Dave School of Physics and
Center for Relativistic Astrophysics, Georgia Institute of Technology,
Atlanta, GA 30332, USA  C. De Clercq Vrije Universiteit Brussel (VUB), Dienst
ELEM, B-1050 Brussels, Belgium  J. J. DeLaunay Dept. of Physics and Astronomy,
University of Alabama, Tuscaloosa, AL 35487, USA  D. Delgado Department of
Physics and Laboratory for Particle Physics and Cosmology, Harvard University,
Cambridge, MA 02138, USA  S. Deng  III. Physikalisches Institut, RWTH Aachen
University, D-52056 Aachen, Germany  A. Desai Dept. of Physics and Wisconsin
IceCube Particle Astrophysics Center, University of Wisconsin—Madison,
Madison, WI 53706, USA  P. Desiati Dept. of Physics and Wisconsin IceCube
Particle Astrophysics Center, University of Wisconsin—Madison, Madison, WI
53706, USA  K. D. de Vries Vrije Universiteit Brussel (VUB), Dienst ELEM,
B-1050 Brussels, Belgium  G. de Wasseige Centre for Cosmology, Particle
Physics and Phenomenology - CP3, Université catholique de Louvain, Louvain-la-
Neuve, Belgium  T. DeYoung Dept. of Physics and Astronomy, Michigan State
University, East Lansing, MI 48824, USA  A. Diaz Dept. of Physics,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  J. C. Díaz-
Vélez Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center,
University of Wisconsin—Madison, Madison, WI 53706, USA  P. Dierichs  III.
Physikalisches Institut, RWTH Aachen University, D-52056 Aachen, Germany  M.
Dittmer  Institut für Kernphysik, Westfälische Wilhelms-Universität Münster,
D-48149 Münster, Germany  A. Domi  Erlangen Centre for Astroparticle Physics,
Friedrich-Alexander-Universität Erlangen-Nürnberg, D-91058 Erlangen, Germany
L. Draper  Department of Physics and Astronomy, University of Utah, Salt Lake
City, UT 84112, USA  H. Dujmovic Dept. of Physics and Wisconsin IceCube
Particle Astrophysics Center, University of Wisconsin—Madison, Madison, WI
53706, USA  D. Durnford Dept. of Physics, University of Alberta, Edmonton,
Alberta, T6G 2E1, Canada  K. Dutta  Institute of Physics, University of Mainz,
Staudinger Weg 7, D-55099 Mainz, Germany  M. A. DuVernois Dept. of Physics and
Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin—Madison, Madison, WI 53706, USA  T. Ehrhardt  Institute of Physics,
University of Mainz, Staudinger Weg 7, D-55099 Mainz, Germany  L. Eidenschink
Physik-department, Technische Universität München, D-85748 Garching, Germany
A. Eimer  Erlangen Centre for Astroparticle Physics, Friedrich-Alexander-
Universität Erlangen-Nürnberg, D-91058 Erlangen, Germany  P. Eller Physik-
department, Technische Universität München, D-85748 Garching, Germany  E.
Ellinger  Dept. of Physics, University of Wuppertal, D-42119 Wuppertal,
Germany  S. El Mentawi  III. Physikalisches Institut, RWTH Aachen University,
D-52056 Aachen, Germany  D. Elsässer Dept. of Physics, TU Dortmund University,
D-44221 Dortmund, Germany  R. Engel  Karlsruhe Institute of Technology,
Institute for Astroparticle Physics, D-76021 Karlsruhe, Germany  Karlsruhe
Institute of Technology, Institute of Experimental Particle Physics, D-76021
Karlsruhe, Germany  H. Erpenbeck Dept. of Physics and Wisconsin IceCube
Particle Astrophysics Center, University of Wisconsin—Madison, Madison, WI
53706, USA  J. Evans  Dept. of Physics, University of Maryland, College Park,
MD 20742, USA  P. A. Evenson  Bartol Research Institute and Dept. of Physics
and Astronomy, University of Delaware, Newark, DE 19716, USA  K. L. Fan  Dept.
of Physics, University of Maryland, College Park, MD 20742, USA  K. Fang
Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center,
University of Wisconsin—Madison, Madison, WI 53706, USA  K. Farrag  Dept. of
Physics and The International Center for Hadron Astrophysics, Chiba
University, Chiba 263-8522, Japan  A. R. Fazely Dept. of Physics, Southern
University, Baton Rouge, LA 70813, USA  A. Fedynitch Institute of Physics,
Academia Sinica, Taipei, 11529, Taiwan  N. Feigl  Institut für Physik,
Humboldt-Universität zu Berlin, D-12489 Berlin, Germany  S. Fiedlschuster
Erlangen Centre for Astroparticle Physics, Friedrich-Alexander-Universität
Erlangen-Nürnberg, D-91058 Erlangen, Germany  C. Finley Oskar Klein Centre and
Dept. of Physics, Stockholm University, SE-10691 Stockholm, Sweden  L. Fischer
Deutsches Elektronen-Synchrotron DESY, Platanenallee 6, D-15738 Zeuthen,
Germany  D. Fox Dept. of Astronomy and Astrophysics, Pennsylvania State
University, University Park, PA 16802, USA  A. Franckowiak Fakultät für Physik
& Astronomie, Ruhr-Universität Bochum, D-44780 Bochum, Germany  S. Fukami
Deutsches Elektronen-Synchrotron DESY, Platanenallee 6, D-15738 Zeuthen,
Germany  P. Fürst III. Physikalisches Institut, RWTH Aachen University,
D-52056 Aachen, Germany  J. Gallagher Dept. of Astronomy, University of
Wisconsin—Madison, Madison, WI 53706, USA  E. Ganster III. Physikalisches
Institut, RWTH Aachen University, D-52056 Aachen, Germany  A. Garcia
Department of Physics and Laboratory for Particle Physics and Cosmology,
Harvard University, Cambridge, MA 02138, USA  M. Garcia  Bartol Research
Institute and Dept. of Physics and Astronomy, University of Delaware, Newark,
DE 19716, USA  G. Garg  also at Institute of Physics, Sachivalaya Marg, Sainik
School Post, Bhubaneswar 751005, India  Dept. of Physics and Wisconsin IceCube
Particle Astrophysics Center, University of Wisconsin—Madison, Madison, WI
53706, USA  E. Genton  Department of Physics and Laboratory for Particle
Physics and Cosmology, Harvard University, Cambridge, MA 02138, USA  Centre
for Cosmology, Particle Physics and Phenomenology - CP3, Université catholique
de Louvain, Louvain-la-Neuve, Belgium  L. Gerhardt  Lawrence Berkeley National
Laboratory, Berkeley, CA 94720, USA  A. Ghadimi Dept. of Physics and
Astronomy, University of Alabama, Tuscaloosa, AL 35487, USA  C. Girard-Carillo
Institute of Physics, University of Mainz, Staudinger Weg 7, D-55099 Mainz,
Germany  C. Glaser Dept. of Physics and Astronomy, Uppsala University, Box
516, SE-75120 Uppsala, Sweden  T. Glüsenkamp Erlangen Centre for Astroparticle
Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg, D-91058 Erlangen,
Germany  Dept. of Physics and Astronomy, Uppsala University, Box 516, SE-75120
Uppsala, Sweden  J. G. Gonzalez  Bartol Research Institute and Dept. of
Physics and Astronomy, University of Delaware, Newark, DE 19716, USA  S.
Goswami  Department of Physics & Astronomy, University of Nevada, Las Vegas,
NV 89154, USA  Nevada Center for Astrophysics, University of Nevada, Las
Vegas, NV 89154, USA  A. Granados  Dept. of Physics and Astronomy, Michigan
State University, East Lansing, MI 48824, USA  D. Grant  Dept. of Physics and
Astronomy, Michigan State University, East Lansing, MI 48824, USA  S. J. Gray
Dept. of Physics, University of Maryland, College Park, MD 20742, USA  O.
Gries  III. Physikalisches Institut, RWTH Aachen University, D-52056 Aachen,
Germany  S. Griffin Dept. of Physics and Wisconsin IceCube Particle
Astrophysics Center, University of Wisconsin—Madison, Madison, WI 53706, USA
S. Griswold Dept. of Physics and Astronomy, University of Rochester,
Rochester, NY 14627, USA  K. M. Groth Niels Bohr Institute, University of
Copenhagen, DK-2100 Copenhagen, Denmark  D. Guevel  Dept. of Physics and
Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin—Madison, Madison, WI 53706, USA  C. Günther  III. Physikalisches
Institut, RWTH Aachen University, D-52056 Aachen, Germany  P. Gutjahr Dept. of
Physics, TU Dortmund University, D-44221 Dortmund, Germany  C. Ha  Dept. of
Physics, Chung-Ang University, Seoul 06974, Republic of Korea  C. Haack
Erlangen Centre for Astroparticle Physics, Friedrich-Alexander-Universität
Erlangen-Nürnberg, D-91058 Erlangen, Germany  A. Hallgren Dept. of Physics and
Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden  L. Halve
III. Physikalisches Institut, RWTH Aachen University, D-52056 Aachen, Germany
F. Halzen Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center,
University of Wisconsin—Madison, Madison, WI 53706, USA  H. Hamdaoui Dept. of
Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794-3800, USA
M. Ha Minh  Physik-department, Technische Universität München, D-85748
Garching, Germany  M. Handt  III. Physikalisches Institut, RWTH Aachen
University, D-52056 Aachen, Germany  K. Hanson  Dept. of Physics and Wisconsin
IceCube Particle Astrophysics Center, University of Wisconsin—Madison,
Madison, WI 53706, USA  J. Hardin  Dept. of Physics, Massachusetts Institute
of Technology, Cambridge, MA 02139, USA  A. A. Harnisch  Dept. of Physics and
Astronomy, Michigan State University, East Lansing, MI 48824, USA  P. Hatch
Dept. of Physics, Engineering Physics, and Astronomy, Queen’s University,
Kingston, ON K7L 3N6, Canada  A. Haungs Karlsruhe Institute of Technology,
Institute for Astroparticle Physics, D-76021 Karlsruhe, Germany  J. Häußler
III. Physikalisches Institut, RWTH Aachen University, D-52056 Aachen, Germany
K. Helbing Dept. of Physics, University of Wuppertal, D-42119 Wuppertal,
Germany  J. Hellrung Fakultät für Physik & Astronomie, Ruhr-Universität
Bochum, D-44780 Bochum, Germany  J. Hermannsgabner  III. Physikalisches
Institut, RWTH Aachen University, D-52056 Aachen, Germany  L. Heuermann  III.
Physikalisches Institut, RWTH Aachen University, D-52056 Aachen, Germany  N.
Heyer Dept. of Physics and Astronomy, Uppsala University, Box 516, SE-75120
Uppsala, Sweden  S. Hickford  Dept. of Physics, University of Wuppertal,
D-42119 Wuppertal, Germany  A. Hidvegi  Oskar Klein Centre and Dept. of
Physics, Stockholm University, SE-10691 Stockholm, Sweden  C. Hill Dept. of
Physics and The International Center for Hadron Astrophysics, Chiba
University, Chiba 263-8522, Japan  G. C. Hill  Department of Physics,
University of Adelaide, Adelaide, 5005, Australia  K. D. Hoffman  Dept. of
Physics, University of Maryland, College Park, MD 20742, USA  S. Hori Dept. of
Physics and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin—Madison, Madison, WI 53706, USA  K. Hoshina  also at Earthquake
Research Institute, University of Tokyo, Bunkyo, Tokyo 113-0032, Japan  Dept.
of Physics and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin—Madison, Madison, WI 53706, USA  M. Hostert Department of Physics
and Laboratory for Particle Physics and Cosmology, Harvard University,
Cambridge, MA 02138, USA  W. Hou Karlsruhe Institute of Technology, Institute
for Astroparticle Physics, D-76021 Karlsruhe, Germany  T. Huber Karlsruhe
Institute of Technology, Institute for Astroparticle Physics, D-76021
Karlsruhe, Germany  K. Hultqvist Oskar Klein Centre and Dept. of Physics,
Stockholm University, SE-10691 Stockholm, Sweden  M. Hünnefeld Dept. of
Physics, TU Dortmund University, D-44221 Dortmund, Germany  R. Hussain  Dept.
of Physics and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin—Madison, Madison, WI 53706, USA  K. Hymon  Dept. of Physics, TU
Dortmund University, D-44221 Dortmund, Germany  Institute of Physics, Academia
Sinica, Taipei, 11529, Taiwan  A. Ishihara  Dept. of Physics and The
International Center for Hadron Astrophysics, Chiba University, Chiba
263-8522, Japan  W. Iwakiri Dept. of Physics and The International Center for
Hadron Astrophysics, Chiba University, Chiba 263-8522, Japan  M. Jacquart
Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center,
University of Wisconsin—Madison, Madison, WI 53706, USA  S. Jain  Institute of
Physics, University of Mainz, Staudinger Weg 7, D-55099 Mainz, Germany  O.
Janik Erlangen Centre for Astroparticle Physics, Friedrich-Alexander-
Universität Erlangen-Nürnberg, D-91058 Erlangen, Germany  M. Jansson  Oskar
Klein Centre and Dept. of Physics, Stockholm University, SE-10691 Stockholm,
Sweden  G. S. Japaridze CTSPS, Clark-Atlanta University, Atlanta, GA 30314,
USA  M. Jeong Department of Physics and Astronomy, University of Utah, Salt
Lake City, UT 84112, USA  M. Jin Department of Physics and Laboratory for
Particle Physics and Cosmology, Harvard University, Cambridge, MA 02138, USA
B. J. P. Jones Dept. of Physics, University of Texas at Arlington, 502 Yates
St., Science Hall Rm 108, Box 19059, Arlington, TX 76019, USA  N. Kamp
Department of Physics and Laboratory for Particle Physics and Cosmology,
Harvard University, Cambridge, MA 02138, USA  D. Kang Karlsruhe Institute of
Technology, Institute for Astroparticle Physics, D-76021 Karlsruhe, Germany
W. Kang Dept. of Physics, Sungkyunkwan University, Suwon 16419, Republic of
Korea  X. Kang  Dept. of Physics, Drexel University, 3141 Chestnut Street,
Philadelphia, PA 19104, USA  A. Kappes Institut für Kernphysik, Westfälische
Wilhelms-Universität Münster, D-48149 Münster, Germany  D. Kappesser
Institute of Physics, University of Mainz, Staudinger Weg 7, D-55099 Mainz,
Germany  L. Kardum  Dept. of Physics, TU Dortmund University, D-44221
Dortmund, Germany  T. Karg Deutsches Elektronen-Synchrotron DESY,
Platanenallee 6, D-15738 Zeuthen, Germany  M. Karl Physik-department,
Technische Universität München, D-85748 Garching, Germany  A. Karle Dept. of
Physics and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin—Madison, Madison, WI 53706, USA  A. Katil  Dept. of Physics,
University of Alberta, Edmonton, Alberta, T6G 2E1, Canada  U. Katz Erlangen
Centre for Astroparticle Physics, Friedrich-Alexander-Universität Erlangen-
Nürnberg, D-91058 Erlangen, Germany  M. Kauer Dept. of Physics and Wisconsin
IceCube Particle Astrophysics Center, University of Wisconsin—Madison,
Madison, WI 53706, USA  J. L. Kelley Dept. of Physics and Wisconsin IceCube
Particle Astrophysics Center, University of Wisconsin—Madison, Madison, WI
53706, USA  M. Khanal  Department of Physics and Astronomy, University of
Utah, Salt Lake City, UT 84112, USA  A. Khatee Zathul Dept. of Physics and
Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin—Madison, Madison, WI 53706, USA  A. Kheirandish Department of
Physics & Astronomy, University of Nevada, Las Vegas, NV 89154, USA  Nevada
Center for Astrophysics, University of Nevada, Las Vegas, NV 89154, USA  J.
Kiryluk Dept. of Physics and Astronomy, Stony Brook University, Stony Brook,
NY 11794-3800, USA  S. R. Klein Dept. of Physics, University of California,
Berkeley, CA 94720, USA  Lawrence Berkeley National Laboratory, Berkeley, CA
94720, USA  A. Kochocki Dept. of Physics and Astronomy, Michigan State
University, East Lansing, MI 48824, USA  R. Koirala Bartol Research Institute
and Dept. of Physics and Astronomy, University of Delaware, Newark, DE 19716,
USA  H. Kolanoski Institut für Physik, Humboldt-Universität zu Berlin, D-12489
Berlin, Germany  T. Kontrimas Physik-department, Technische Universität
München, D-85748 Garching, Germany  L. Köpke  Institute of Physics, University
of Mainz, Staudinger Weg 7, D-55099 Mainz, Germany  C. Kopper Erlangen Centre
for Astroparticle Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg,
D-91058 Erlangen, Germany  D. J. Koskinen Niels Bohr Institute, University of
Copenhagen, DK-2100 Copenhagen, Denmark  P. Koundal Bartol Research Institute
and Dept. of Physics and Astronomy, University of Delaware, Newark, DE 19716,
USA  M. Kovacevich Dept. of Physics, Drexel University, 3141 Chestnut Street,
Philadelphia, PA 19104, USA  M. Kowalski Institut für Physik, Humboldt-
Universität zu Berlin, D-12489 Berlin, Germany  Deutsches Elektronen-
Synchrotron DESY, Platanenallee 6, D-15738 Zeuthen, Germany  T. Kozynets
Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark
J. Krishnamoorthi also at Institute of Physics, Sachivalaya Marg, Sainik
School Post, Bhubaneswar 751005, India  Dept. of Physics and Wisconsin IceCube
Particle Astrophysics Center, University of Wisconsin—Madison, Madison, WI
53706, USA  K. Kruiswijk Centre for Cosmology, Particle Physics and
Phenomenology - CP3, Université catholique de Louvain, Louvain-la-Neuve,
Belgium  E. Krupczak  Dept. of Physics and Astronomy, Michigan State
University, East Lansing, MI 48824, USA  A. Kumar Deutsches Elektronen-
Synchrotron DESY, Platanenallee 6, D-15738 Zeuthen, Germany  E. Kun  Fakultät
für Physik & Astronomie, Ruhr-Universität Bochum, D-44780 Bochum, Germany  N.
Kurahashi Dept. of Physics, Drexel University, 3141 Chestnut Street,
Philadelphia, PA 19104, USA  N. Lad Deutsches Elektronen-Synchrotron DESY,
Platanenallee 6, D-15738 Zeuthen, Germany  C. Lagunas Gualda Deutsches
Elektronen-Synchrotron DESY, Platanenallee 6, D-15738 Zeuthen, Germany  M.
Lamoureux Centre for Cosmology, Particle Physics and Phenomenology - CP3,
Université catholique de Louvain, Louvain-la-Neuve, Belgium  M. J. Larson
Dept. of Physics, University of Maryland, College Park, MD 20742, USA  S.
Latseva  III. Physikalisches Institut, RWTH Aachen University, D-52056 Aachen,
Germany  F. Lauber Dept. of Physics, University of Wuppertal, D-42119
Wuppertal, Germany  J. P. Lazar Centre for Cosmology, Particle Physics and
Phenomenology - CP3, Université catholique de Louvain, Louvain-la-Neuve,
Belgium  J. W. Lee Dept. of Physics, Sungkyunkwan University, Suwon 16419,
Republic of Korea  K. Leonard DeHolton Dept. of Physics, Pennsylvania State
University, University Park, PA 16802, USA  A. Leszczyńska Bartol Research
Institute and Dept. of Physics and Astronomy, University of Delaware, Newark,
DE 19716, USA  J. Liao School of Physics and Center for Relativistic
Astrophysics, Georgia Institute of Technology, Atlanta, GA 30332, USA  M.
Lincetto Fakultät für Physik & Astronomie, Ruhr-Universität Bochum, D-44780
Bochum, Germany  Y. T. Liu  Dept. of Physics, Pennsylvania State University,
University Park, PA 16802, USA  M. Liubarska  Dept. of Physics, University of
Alberta, Edmonton, Alberta, T6G 2E1, Canada  C. Love  Dept. of Physics, Drexel
University, 3141 Chestnut Street, Philadelphia, PA 19104, USA  L. Lu Dept. of
Physics and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin—Madison, Madison, WI 53706, USA  F. Lucarelli Département de
physique nucléaire et corpusculaire, Université de Genève, CH-1211 Genève,
Switzerland  W. Luszczak Dept. of Astronomy, Ohio State University, Columbus,
OH 43210, USA  Dept. of Physics and Center for Cosmology and Astro-Particle
Physics, Ohio State University, Columbus, OH 43210, USA  Y. Lyu Dept. of
Physics, University of California, Berkeley, CA 94720, USA  Lawrence Berkeley
National Laboratory, Berkeley, CA 94720, USA  J. Madsen Dept. of Physics and
Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin—Madison, Madison, WI 53706, USA  E. Magnus Vrije Universiteit
Brussel (VUB), Dienst ELEM, B-1050 Brussels, Belgium  K. B. M. Mahn  Dept. of
Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA
Y. Makino  Dept. of Physics and Wisconsin IceCube Particle Astrophysics
Center, University of Wisconsin—Madison, Madison, WI 53706, USA  E. Manao
Physik-department, Technische Universität München, D-85748 Garching, Germany
S. Mancina Dipartimento di Fisica e Astronomia Galileo Galilei, Università
Degli Studi di Padova, I-35122 Padova PD, Italy  W. Marie Sainte  Dept. of
Physics and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin—Madison, Madison, WI 53706, USA  I. C. Mariş Université Libre de
Bruxelles, Science Faculty CP230, B-1050 Brussels, Belgium  S. Marka Columbia
Astrophysics and Nevis Laboratories, Columbia University, New York, NY 10027,
USA  Z. Marka Columbia Astrophysics and Nevis Laboratories, Columbia
University, New York, NY 10027, USA  M. Marsee  Dept. of Physics and
Astronomy, University of Alabama, Tuscaloosa, AL 35487, USA  I. Martinez-Soler
Department of Physics and Laboratory for Particle Physics and Cosmology,
Harvard University, Cambridge, MA 02138, USA  R. Maruyama Dept. of Physics,
Yale University, New Haven, CT 06520, USA  F. Mayhew Dept. of Physics and
Astronomy, Michigan State University, East Lansing, MI 48824, USA  F. McNally
Department of Physics, Mercer University, Macon, GA 31207-0001, USA  J. V.
Mead  Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen,
Denmark  K. Meagher Dept. of Physics and Wisconsin IceCube Particle
Astrophysics Center, University of Wisconsin—Madison, Madison, WI 53706, USA
S. Mechbal  Deutsches Elektronen-Synchrotron DESY, Platanenallee 6, D-15738
Zeuthen, Germany  A. Medina  Dept. of Physics and Center for Cosmology and
Astro-Particle Physics, Ohio State University, Columbus, OH 43210, USA  M.
Meier Dept. of Physics and The International Center for Hadron Astrophysics,
Chiba University, Chiba 263-8522, Japan  Y. Merckx  Vrije Universiteit Brussel
(VUB), Dienst ELEM, B-1050 Brussels, Belgium  L. Merten Fakultät für Physik &
Astronomie, Ruhr-Universität Bochum, D-44780 Bochum, Germany  J. Micallef
Dept. of Physics and Astronomy, Michigan State University, East Lansing, MI
48824, USA  J. Mitchell  Dept. of Physics, Southern University, Baton Rouge,
LA 70813, USA  T. Montaruli Département de physique nucléaire et
corpusculaire, Université de Genève, CH-1211 Genève, Switzerland  R. W. Moore
Dept. of Physics, University of Alberta, Edmonton, Alberta, T6G 2E1, Canada
Y. Morii  Dept. of Physics and The International Center for Hadron
Astrophysics, Chiba University, Chiba 263-8522, Japan  R. Morse  Dept. of
Physics and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin—Madison, Madison, WI 53706, USA  M. Moulai Dept. of Physics and
Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin—Madison, Madison, WI 53706, USA  T. Mukherjee Karlsruhe Institute of
Technology, Institute for Astroparticle Physics, D-76021 Karlsruhe, Germany
R. Naab Deutsches Elektronen-Synchrotron DESY, Platanenallee 6, D-15738
Zeuthen, Germany  R. Nagai Dept. of Physics and The International Center for
Hadron Astrophysics, Chiba University, Chiba 263-8522, Japan  M. Nakos  Dept.
of Physics and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin—Madison, Madison, WI 53706, USA  U. Naumann  Dept. of Physics,
University of Wuppertal, D-42119 Wuppertal, Germany  J. Necker Deutsches
Elektronen-Synchrotron DESY, Platanenallee 6, D-15738 Zeuthen, Germany  A.
Negi  Dept. of Physics, University of Texas at Arlington, 502 Yates St.,
Science Hall Rm 108, Box 19059, Arlington, TX 76019, USA  L. Neste Oskar Klein
Centre and Dept. of Physics, Stockholm University, SE-10691 Stockholm, Sweden
M. Neumann  Institut für Kernphysik, Westfälische Wilhelms-Universität
Münster, D-48149 Münster, Germany  H. Niederhausen Dept. of Physics and
Astronomy, Michigan State University, East Lansing, MI 48824, USA  M. U. Nisa
Dept. of Physics and Astronomy, Michigan State University, East Lansing, MI
48824, USA  K. Noda Dept. of Physics and The International Center for Hadron
Astrophysics, Chiba University, Chiba 263-8522, Japan  A. Noell  III.
Physikalisches Institut, RWTH Aachen University, D-52056 Aachen, Germany  A.
Novikov  Bartol Research Institute and Dept. of Physics and Astronomy,
University of Delaware, Newark, DE 19716, USA  A. Obertacke Pollmann Dept. of
Physics and The International Center for Hadron Astrophysics, Chiba
University, Chiba 263-8522, Japan  V. O’Dell Dept. of Physics and Wisconsin
IceCube Particle Astrophysics Center, University of Wisconsin—Madison,
Madison, WI 53706, USA  B. Oeyen Dept. of Physics and Astronomy, University of
Gent, B-9000 Gent, Belgium  A. Olivas  Dept. of Physics, University of
Maryland, College Park, MD 20742, USA  R. Orsoe  Physik-department, Technische
Universität München, D-85748 Garching, Germany  J. Osborn  Dept. of Physics
and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin—Madison, Madison, WI 53706, USA  E. O’Sullivan Dept. of Physics and
Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden  V. Palusova
Institute of Physics, University of Mainz, Staudinger Weg 7, D-55099 Mainz,
Germany  H. Pandya Bartol Research Institute and Dept. of Physics and
Astronomy, University of Delaware, Newark, DE 19716, USA  N. Park Dept. of
Physics, Engineering Physics, and Astronomy, Queen’s University, Kingston, ON
K7L 3N6, Canada  G. K. Parker  Dept. of Physics, University of Texas at
Arlington, 502 Yates St., Science Hall Rm 108, Box 19059, Arlington, TX 76019,
USA  E. N. Paudel Bartol Research Institute and Dept. of Physics and
Astronomy, University of Delaware, Newark, DE 19716, USA  L. Paul Physics
Department, South Dakota School of Mines and Technology, Rapid City, SD 57701,
USA  C. Pérez de los Heros Dept. of Physics and Astronomy, Uppsala University,
Box 516, SE-75120 Uppsala, Sweden  T. Pernice  Deutsches Elektronen-
Synchrotron DESY, Platanenallee 6, D-15738 Zeuthen, Germany  J. Peterson
Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center,
University of Wisconsin—Madison, Madison, WI 53706, USA  A. Pizzuto Dept. of
Physics and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin—Madison, Madison, WI 53706, USA  M. Plum Physics Department, South
Dakota School of Mines and Technology, Rapid City, SD 57701, USA  A. Pontén
Dept. of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala,
Sweden  Y. Popovych  Institute of Physics, University of Mainz, Staudinger Weg
7, D-55099 Mainz, Germany  M. Prado Rodriguez  Dept. of Physics and Wisconsin
IceCube Particle Astrophysics Center, University of Wisconsin—Madison,
Madison, WI 53706, USA  B. Pries Dept. of Physics and Astronomy, Michigan
State University, East Lansing, MI 48824, USA  R. Procter-Murphy  Dept. of
Physics, University of Maryland, College Park, MD 20742, USA  G. T. Przybylski
Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA  C. Raab Centre
for Cosmology, Particle Physics and Phenomenology - CP3, Université catholique
de Louvain, Louvain-la-Neuve, Belgium  J. Rack-Helleis  Institute of Physics,
University of Mainz, Staudinger Weg 7, D-55099 Mainz, Germany  M. Ravn  Dept.
of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala,
Sweden  K. Rawlins  Dept. of Physics and Astronomy, University of Alaska
Anchorage, 3211 Providence Dr., Anchorage, AK 99508, USA  Z. Rechav  Dept. of
Physics and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin—Madison, Madison, WI 53706, USA  A. Rehman Bartol Research Institute
and Dept. of Physics and Astronomy, University of Delaware, Newark, DE 19716,
USA  P. Reichherzer  Fakultät für Physik & Astronomie, Ruhr-Universität
Bochum, D-44780 Bochum, Germany  E. Resconi Physik-department, Technische
Universität München, D-85748 Garching, Germany  S. Reusch  Deutsches
Elektronen-Synchrotron DESY, Platanenallee 6, D-15738 Zeuthen, Germany  W.
Rhode Dept. of Physics, TU Dortmund University, D-44221 Dortmund, Germany  B.
Riedel Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center,
University of Wisconsin—Madison, Madison, WI 53706, USA  A. Rifaie  III.
Physikalisches Institut, RWTH Aachen University, D-52056 Aachen, Germany  E.
J. Roberts  Department of Physics, University of Adelaide, Adelaide, 5005,
Australia  S. Robertson  Dept. of Physics, University of California, Berkeley,
CA 94720, USA  Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA
S. Rodan  Dept. of Physics, Sungkyunkwan University, Suwon 16419, Republic of
Korea  Institute of Basic Science, Sungkyunkwan University, Suwon 16419,
Republic of Korea  G. Roellinghoff  Dept. of Physics, Sungkyunkwan University,
Suwon 16419, Republic of Korea  M. Rongen Erlangen Centre for Astroparticle
Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg, D-91058 Erlangen,
Germany  A. Rosted Dept. of Physics and The International Center for Hadron
Astrophysics, Chiba University, Chiba 263-8522, Japan  C. Rott Department of
Physics and Astronomy, University of Utah, Salt Lake City, UT 84112, USA
Dept. of Physics, Sungkyunkwan University, Suwon 16419, Republic of Korea  T.
Ruhe Dept. of Physics, TU Dortmund University, D-44221 Dortmund, Germany  L.
Ruohan  Physik-department, Technische Universität München, D-85748 Garching,
Germany  D. Ryckbosch  Dept. of Physics and Astronomy, University of Gent,
B-9000 Gent, Belgium  I. Safa Dept. of Physics and Wisconsin IceCube Particle
Astrophysics Center, University of Wisconsin—Madison, Madison, WI 53706, USA
J. Saffer  Karlsruhe Institute of Technology, Institute of Experimental
Particle Physics, D-76021 Karlsruhe, Germany  D. Salazar-Gallegos Dept. of
Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA
P. Sampathkumar  Karlsruhe Institute of Technology, Institute for
Astroparticle Physics, D-76021 Karlsruhe, Germany  A. Sandrock Dept. of
Physics, University of Wuppertal, D-42119 Wuppertal, Germany  M. Santander
Dept. of Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487,
USA  S. Sarkar Dept. of Physics, University of Alberta, Edmonton, Alberta, T6G
2E1, Canada  S. Sarkar Dept. of Physics, University of Oxford, Parks Road,
Oxford OX1 3PU, United Kingdom  J. Savelberg  III. Physikalisches Institut,
RWTH Aachen University, D-52056 Aachen, Germany  P. Savina  Dept. of Physics
and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin—Madison, Madison, WI 53706, USA  P. Schaile  Physik-department,
Technische Universität München, D-85748 Garching, Germany  M. Schaufel  III.
Physikalisches Institut, RWTH Aachen University, D-52056 Aachen, Germany  H.
Schieler Karlsruhe Institute of Technology, Institute for Astroparticle
Physics, D-76021 Karlsruhe, Germany  S. Schindler Erlangen Centre for
Astroparticle Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg,
D-91058 Erlangen, Germany  L. Schlickmann Institute of Physics, University of
Mainz, Staudinger Weg 7, D-55099 Mainz, Germany  B. Schlüter  Institut für
Kernphysik, Westfälische Wilhelms-Universität Münster, D-48149 Münster,
Germany  F. Schlüter Université Libre de Bruxelles, Science Faculty CP230,
B-1050 Brussels, Belgium  N. Schmeisser  Dept. of Physics, University of
Wuppertal, D-42119 Wuppertal, Germany  T. Schmidt  Dept. of Physics,
University of Maryland, College Park, MD 20742, USA  J. Schneider Erlangen
Centre for Astroparticle Physics, Friedrich-Alexander-Universität Erlangen-
Nürnberg, D-91058 Erlangen, Germany  F. G. Schröder Karlsruhe Institute of
Technology, Institute for Astroparticle Physics, D-76021 Karlsruhe, Germany
Bartol Research Institute and Dept. of Physics and Astronomy, University of
Delaware, Newark, DE 19716, USA  L. Schumacher Erlangen Centre for
Astroparticle Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg,
D-91058 Erlangen, Germany  S. Sclafani Dept. of Physics, University of
Maryland, College Park, MD 20742, USA  D. Seckel  Bartol Research Institute
and Dept. of Physics and Astronomy, University of Delaware, Newark, DE 19716,
USA  M. Seikh Dept. of Physics and Astronomy, University of Kansas, Lawrence,
KS 66045, USA  M. Seo  Dept. of Physics, Sungkyunkwan University, Suwon 16419,
Republic of Korea  S. Seunarine Dept. of Physics, University of Wisconsin,
River Falls, WI 54022, USA  P. Sevle Myhr Centre for Cosmology, Particle
Physics and Phenomenology - CP3, Université catholique de Louvain, Louvain-la-
Neuve, Belgium  R. Shah  Dept. of Physics, Drexel University, 3141 Chestnut
Street, Philadelphia, PA 19104, USA  S. Shefali  Karlsruhe Institute of
Technology, Institute of Experimental Particle Physics, D-76021 Karlsruhe,
Germany  N. Shimizu Dept. of Physics and The International Center for Hadron
Astrophysics, Chiba University, Chiba 263-8522, Japan  M. Silva Dept. of
Physics and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin—Madison, Madison, WI 53706, USA  B. Skrzypek Dept. of Physics,
University of California, Berkeley, CA 94720, USA  B. Smithers Dept. of
Physics, University of Texas at Arlington, 502 Yates St., Science Hall Rm 108,
Box 19059, Arlington, TX 76019, USA  R. Snihur  Dept. of Physics and Wisconsin
IceCube Particle Astrophysics Center, University of Wisconsin—Madison,
Madison, WI 53706, USA  J. Soedingrekso  Dept. of Physics, TU Dortmund
University, D-44221 Dortmund, Germany  A. Søgaard  Niels Bohr Institute,
University of Copenhagen, DK-2100 Copenhagen, Denmark  D. Soldin Department of
Physics and Astronomy, University of Utah, Salt Lake City, UT 84112, USA  P.
Soldin III. Physikalisches Institut, RWTH Aachen University, D-52056 Aachen,
Germany  G. Sommani Fakultät für Physik & Astronomie, Ruhr-Universität Bochum,
D-44780 Bochum, Germany  C. Spannfellner  Physik-department, Technische
Universität München, D-85748 Garching, Germany  G. M. Spiczak Dept. of
Physics, University of Wisconsin, River Falls, WI 54022, USA  C. Spiering
Deutsches Elektronen-Synchrotron DESY, Platanenallee 6, D-15738 Zeuthen,
Germany  M. Stamatikos  Dept. of Physics and Center for Cosmology and Astro-
Particle Physics, Ohio State University, Columbus, OH 43210, USA  T. Stanev
Bartol Research Institute and Dept. of Physics and Astronomy, University of
Delaware, Newark, DE 19716, USA  T. Stezelberger Lawrence Berkeley National
Laboratory, Berkeley, CA 94720, USA  T. Stürwald  Dept. of Physics, University
of Wuppertal, D-42119 Wuppertal, Germany  T. Stuttard Niels Bohr Institute,
University of Copenhagen, DK-2100 Copenhagen, Denmark  G. W. Sullivan Dept. of
Physics, University of Maryland, College Park, MD 20742, USA  I. Taboada
School of Physics and Center for Relativistic Astrophysics, Georgia Institute
of Technology, Atlanta, GA 30332, USA  S. Ter-Antonyan Dept. of Physics,
Southern University, Baton Rouge, LA 70813, USA  A. Terliuk  Physik-
department, Technische Universität München, D-85748 Garching, Germany  M.
Thiesmeyer  III. Physikalisches Institut, RWTH Aachen University, D-52056
Aachen, Germany  W. G. Thompson Department of Physics and Laboratory for
Particle Physics and Cosmology, Harvard University, Cambridge, MA 02138, USA
J. Thwaites Dept. of Physics and Wisconsin IceCube Particle Astrophysics
Center, University of Wisconsin—Madison, Madison, WI 53706, USA  S. Tilav
Bartol Research Institute and Dept. of Physics and Astronomy, University of
Delaware, Newark, DE 19716, USA  K. Tollefson Dept. of Physics and Astronomy,
Michigan State University, East Lansing, MI 48824, USA  C. Tönnis  Dept. of
Physics, Sungkyunkwan University, Suwon 16419, Republic of Korea  S. Toscano
Université Libre de Bruxelles, Science Faculty CP230, B-1050 Brussels, Belgium
D. Tosi  Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center,
University of Wisconsin—Madison, Madison, WI 53706, USA  A. Trettin  Deutsches
Elektronen-Synchrotron DESY, Platanenallee 6, D-15738 Zeuthen, Germany  R.
Turcotte  Karlsruhe Institute of Technology, Institute for Astroparticle
Physics, D-76021 Karlsruhe, Germany  J. P. Twagirayezu  Dept. of Physics and
Astronomy, Michigan State University, East Lansing, MI 48824, USA  M. A.
Unland Elorrieta Institut für Kernphysik, Westfälische Wilhelms-Universität
Münster, D-48149 Münster, Germany  A. K. Upadhyay also at Institute of
Physics, Sachivalaya Marg, Sainik School Post, Bhubaneswar 751005, India
Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center,
University of Wisconsin—Madison, Madison, WI 53706, USA  K. Upshaw  Dept. of
Physics, Southern University, Baton Rouge, LA 70813, USA  A. Vaidyanathan
Department of Physics, Marquette University, Milwaukee, WI 53201, USA  N.
Valtonen-Mattila Dept. of Physics and Astronomy, Uppsala University, Box 516,
SE-75120 Uppsala, Sweden  J. Vandenbroucke Dept. of Physics and Wisconsin
IceCube Particle Astrophysics Center, University of Wisconsin—Madison,
Madison, WI 53706, USA  N. van Eijndhoven Vrije Universiteit Brussel (VUB),
Dienst ELEM, B-1050 Brussels, Belgium  D. Vannerom  Dept. of Physics,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  J. van Santen
Deutsches Elektronen-Synchrotron DESY, Platanenallee 6, D-15738 Zeuthen,
Germany  J. Vara  Institut für Kernphysik, Westfälische Wilhelms-Universität
Münster, D-48149 Münster, Germany  F. Varsi  Karlsruhe Institute of
Technology, Institute of Experimental Particle Physics, D-76021 Karlsruhe,
Germany  J. Veitch-Michaelis  Dept. of Physics and Wisconsin IceCube Particle
Astrophysics Center, University of Wisconsin—Madison, Madison, WI 53706, USA
M. Venugopal  Karlsruhe Institute of Technology, Institute for Astroparticle
Physics, D-76021 Karlsruhe, Germany  M. Vereecken  Centre for Cosmology,
Particle Physics and Phenomenology - CP3, Université catholique de Louvain,
Louvain-la-Neuve, Belgium  S. Vergara Carrasco  Dept. of Physics and
Astronomy, University of Canterbury, Private Bag 4800, Christchurch, New
Zealand  S. Verpoest Bartol Research Institute and Dept. of Physics and
Astronomy, University of Delaware, Newark, DE 19716, USA  D. Veske  Columbia
Astrophysics and Nevis Laboratories, Columbia University, New York, NY 10027,
USA  A. Vijai  Dept. of Physics, University of Maryland, College Park, MD
20742, USA  C. Walck  Oskar Klein Centre and Dept. of Physics, Stockholm
University, SE-10691 Stockholm, Sweden  A. Wang School of Physics and Center
for Relativistic Astrophysics, Georgia Institute of Technology, Atlanta, GA
30332, USA  C. Weaver Dept. of Physics and Astronomy, Michigan State
University, East Lansing, MI 48824, USA  P. Weigel  Dept. of Physics,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  A. Weindl
Karlsruhe Institute of Technology, Institute for Astroparticle Physics,
D-76021 Karlsruhe, Germany  J. Weldert  Dept. of Physics, Pennsylvania State
University, University Park, PA 16802, USA  A. Y. Wen  Department of Physics
and Laboratory for Particle Physics and Cosmology, Harvard University,
Cambridge, MA 02138, USA  C. Wendt Dept. of Physics and Wisconsin IceCube
Particle Astrophysics Center, University of Wisconsin—Madison, Madison, WI
53706, USA  J. Werthebach  Dept. of Physics, TU Dortmund University, D-44221
Dortmund, Germany  M. Weyrauch  Karlsruhe Institute of Technology, Institute
for Astroparticle Physics, D-76021 Karlsruhe, Germany  N. Whitehorn Dept. of
Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA
C. H. Wiebusch III. Physikalisches Institut, RWTH Aachen University, D-52056
Aachen, Germany  D. R. Williams  Dept. of Physics and Astronomy, University of
Alabama, Tuscaloosa, AL 35487, USA  L. Witthaus Dept. of Physics, TU Dortmund
University, D-44221 Dortmund, Germany  A. Wolf  III. Physikalisches Institut,
RWTH Aachen University, D-52056 Aachen, Germany  M. Wolf Physik-department,
Technische Universität München, D-85748 Garching, Germany  G. Wrede  Erlangen
Centre for Astroparticle Physics, Friedrich-Alexander-Universität Erlangen-
Nürnberg, D-91058 Erlangen, Germany  X. W. Xu  Dept. of Physics, Southern
University, Baton Rouge, LA 70813, USA  J. P. Yanez  Dept. of Physics,
University of Alberta, Edmonton, Alberta, T6G 2E1, Canada  E. Yildizci  Dept.
of Physics and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin—Madison, Madison, WI 53706, USA  S. Yoshida Dept. of Physics and The
International Center for Hadron Astrophysics, Chiba University, Chiba
263-8522, Japan  R. Young  Dept. of Physics and Astronomy, University of
Kansas, Lawrence, KS 66045, USA  S. Yu Department of Physics and Astronomy,
University of Utah, Salt Lake City, UT 84112, USA  T. Yuan Dept. of Physics
and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin—Madison, Madison, WI 53706, USA  Z. Zhang  Dept. of Physics and
Astronomy, Stony Brook University, Stony Brook, NY 11794-3800, USA  P. Zhelnin
Department of Physics and Laboratory for Particle Physics and Cosmology,
Harvard University, Cambridge, MA 02138, USA  P. Zilberman  Dept. of Physics
and Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin—Madison, Madison, WI 53706, USA  M. Zimmerman  Dept. of Physics and
Wisconsin IceCube Particle Astrophysics Center, University of
Wisconsin—Madison, Madison, WI 53706, USA

###### Abstract

Active Galactic Nuclei (AGN) are prime candidate sources of the high-energy,
astrophysical neutrinos detected by IceCube. This is demonstrated by the real-
time multi-messenger detection of the blazar TXS 0506+056 and the recent
evidence of neutrino emission from NGC 1068 from a separate time-averaged
study. However, the production mechanism of the astrophysical neutrinos in AGN
is not well established which can be resolved via correlation studies with
photon observations. For neutrinos produced due to photohadronic interactions
in AGN, in addition to a correlation of neutrinos with high-energy photons,
there would also be a correlation of neutrinos with photons emitted at radio
wavelengths. In this work, we perform an in-depth stacking study of the
correlation between 15 GHz radio observations of AGN reported in the MOJAVE XV
catalog, and ten years of neutrino data from IceCube. We also use a time-
dependent approach which improves the statistical power of the stacking
analysis. No significant correlation was found for both analyses and upper
limits are reported. When compared to the IceCube diffuse flux, at 100 TeV and
for a spectral index of 2.5, the upper limits derived are $\sim 3\%$ and $\sim
9\%$ for the time-averaged and time-dependent case, respectively.

##  1 Introduction

Neutrinos are a valuable complementary messenger to photons, however, their
elusive nature adds complexities to their detection leading to uncertainties
about the exact sources producing them. Understanding the neutrino production
mechanism and the concurrent detection of photons can help pinpoint their
sources. This can be done using a hypothesis that the observed neutrinos and
photons follow a certain correlation. This correlation supports the theory
that both particles originate from similar or related processes within or
around extragalactic sources. A positive correlation will help us find the
sources that may be producing neutrinos and understand the processes that lead
to their creation. However, these correlation studies are limited by a lack of
coincident photon observations with neutrino data reducing their
sensitivities. One of the theories involving neutrino production in Active
Galactic Nuclei (AGN) involves the possibility of a correlation with photon
detected at radio wavelengths (see Sec 1.1). In this work, we perform a
stacking analysis using 10 years of data collected by the IceCube Neutrino
Observatory (2008-2018) along with the time-dependent photon observations
published in the MOJAVE XV catalog (Lister et al., 2018) to test for
correlation between the radio and neutrino observations. The time-dependent
stacking study makes use of additional coincident information and improves the
statistical power of traditional stacking analyses.

The IceCube Neutrino Observatory, located at the geographic South Pole, is a
cubic kilometer in-ice neutrino detector that has collected $\sim$18 years of
neutrino data (Aartsen et al., 2017). When high-energy neutrinos pass through
the Earth, they may interact with the ice or surrounding bedrock, creating
secondary charged particles. These particles produce Cherenkov light which is
detected and used to reconstruct the high-energy neutrino interaction energy
and direction. While the first evidence of astrophysical neutrino diffuse flux
detection was reported in 2013 (Aartsen et al., 2013), the origin of these
elusive particles and the sources producing them remains uncertain. One of the
prime candidates for the origin of these particles are AGN, which are active
supermassive black holes, some of which have jets of extremely high-energy
particles originating from the center (Eichler, 1979; Berezinsky & Ginzburg,
1981).

In 2017, IceCube detected a high-energy neutrino event in a direction
coincident with the AGN TXS
0506+056111https://gcn.gsfc.nasa.gov/gcn3/21916.gcn3 which was found to be
flaring in gamma rays (Aartsen et al., 2018a). A follow-up analysis of
archival IceCube neutrino data revealed an earlier burst of neutrino events
from the same source in 2014/2015 without an accompanying flare of gamma rays
(Aartsen et al., 2018b). This source is a blazar which is a type of AGN with
the jet pointed in the direction of the observer. Recently, Abbasi et al.
(2022) reported significant evidence of neutrino emission from NGC 1068, a
nearby Seyfert II type of AGN. Seyfert II sources are galaxies that are
observed with narrow emission lines in their spectrum and a variable radio
emission. Some Seyfert II AGN also have jets radiating outward, however, they
are relatively dim gamma-ray sources, unlike blazars which have relativistic
jets oriented in Earth’s direction. These detections motivated studies
involving all AGNs as potential neutrino sources. To better explain neutrino
production in these sources, theoretical modeling efforts and correlation
studies are performed with photon observations in a particular energy regime.
Multiple independent analyses from both: the IceCube Collaboration (see for
example Huber, 2019; Desai et al., 2021; Abbasi et al., 2022c, 2023b) and
other researchers (see for example Plavin et al., 2020, 2021; Kun et al.,
2022; Rodrigues et al., 2021; Zhou et al., 2021) which used different datasets
and analysis methodology were performed. Based on these studies and AGN model
predictions, one of the leading theories is that the neutrinos observed by
IceCube may be correlated with the photons observed in the radio regime as the
radio variability traces environment conditions ideal for neutrino production
(See Jacobsen et al., 2015; Murase & Stecker, 2023, and Sec 1.1 of this work
for more details). Additionally, as blazar jets point in the direction of the
observer, due to Doppler boosting the broadband SED of radio-loud AGN is not
clear with the jet emission dominating over the electromagnetic signal. As the
neutrino signal from these AGNs will not be affected by the jets, a positive
correlation will help identification of radio-loud AGN accelerators and
understand their relevant neutrino production processes.

###  1.1 Neutrino-radio correlation in AGN

Theoretical predictions of neutrino production in AGNs depend on the the type
of interaction and the location where it can occur. Two types of interaction
that can lead to the production of these neutrinos from AGN are the
photohadronic (nucleon-photon or $p\gamma$) and hadronuclear (nucleon-nucleon
or $pp$) processes, which can occur close to the core of the AGN or in the jet
of a jetted-AGN (Sikora et al., 1987; Stecker et al., 1991; Murase & Stecker,
2023). Depending on where these processes take place, there may be a
correlation of the neutrino signal with photon observations at a particular
wavelength. Understanding this correlation, or a lack of it, can help
researchers pinpoint the neutrino production mechanism.

Here we focus on testing the theory behind the neutrino and photon
correlations in AGN. The synchrotron radiation resulting from accelerated
electrons leads to the emission of photons, observable at radio frequencies,
which in turn undergo Inverse Compton scattering to form the X-ray photons. In
the case of neutrino production due to photohadronic interactions, these X-ray
photons interact with protons to give pions that decay to give gamma rays and
neutrinos. For opaque or obscured AGN, the resultant gamma-ray photons will
cascade down to lower energies (e.g. X-rays) before escaping from the core of
the AGN (Murase, 2022). Neutrinos, on the other hand, escape without
interaction for both obscured and un-obscured AGN. While this suggests a
possible correlation of neutrino observations with photons detected in the
X-ray regime for certain AGN, it also supports a correlation of neutrinos with
photons observed in the radio regime for all AGN. This is because, if an AGN
is flaring at radio wavelengths, it can signify an increase in the conditions
favorable for the neutrino production process. This means that the neutrino
signal will be directly correlated with the radio flux density measurements of
the AGN (see Jacobsen et al., 2015, for more details). Additionally, as
compared to X-ray observations, radio detection of photons from AGN is easier
and has been carried out by multiple radio observatories over time, allowing
researchers to use archival data of photon observations. Studies like Plavin
et al. (2020) report that radio measurements at higher frequencies
($>10\,$GHz) are correlated with neutrino events showing increased emission
activity as compared to lower frequencies. To test this correlation, in this
work, we use the MOJAVE XV dataset (Lister et al., 2018) which reports 15 GHz
observations of AGN sources observed over twenty years.

###  1.2 Previous AGN correlation studies

Various observational and theoretical papers exist to study photon and
neutrino correlations, however, it is unclear what are the dominant processes
for neutrino production in AGN (see review by Murase & Stecker, 2023). While
studies like Plavin et al. (2021),Hovatta et al. (2021) and Buson et al.
(2022) report a correlation of IceCube neutrinos with photons from AGN, other
studies like Zhou et al. (2021),Abbasi et al. (2022c) and Abbasi et al.
(2023b), do not find a significant correlation. One of the factors that
contribute to the discrepancy is the lack of data, which, in terms of
neutrinos can be the usage of the IceCube alert dataset instead of the full
IceCube tracks dataset (Abbasi et al., 2021). As an example, the all-sky point
source neutrino dataset made of track events (Abbasi et al., 2021) has more
muon track-like events (see Sec 2.1) as a function of energy along with a
better, more consistent coverage in time and location as compared to the
IceCube public alert sample (Abbasi et al., 2023a). For a source with an
assumed power-law spectrum with an index of $\gamma=2.5$, the public alert
selection has a factor of $\times$100 fewer astrophysical neutrinos than the
full selection. Another factor that can affect the discrepancy between the
neutrino studies can be the inclusion of additional components like energy
information in signal PDFs or signalness for neutrino events (see for example
Abbasi et al., 2022c, 2023b). Finally, apart from flaring AGN analyses,
correlation studies involving stacking are often limited to using time-
averaged measurements. This is due to the lack of time-dependent photon and
neutrino data. Including time-dependent information in a stacking study will
allow us to weight sources based on the photon data as a function of time and
increase the sensitivity of the analysis. It thus becomes important to use the
most complete IceCube neutrino dataset in combination with a large, reasonably
complete AGN source catalog preferably including time-dependent flux
information.

###  1.3 Paper Outline

This work uses the 10-year IceCube tracks dataset using a stacking analysis
(similar to Braun et al., 2008; Desai et al., 2021; Abbasi et al., 2022c) to
search for correlation between radio and neutrino data from AGN. The stacking
analysis is performed for both the time-averaged and time-dependent cases
assuming there is a 1:1 correlation between the 15 GHz radio flux density and
the high-energy neutrino flux. The article is divided as follows: Section. 2
describes in detail the neutrino and photon datasets used in this study;
Section. 3 describes the likelihood framework used; Section. 4 presents the
results derived from this work; and Section. 5 discusses the conclusion and
future implications of this work.

Figure 1: Source distribution of MOJAVE XV catalog using Equatorial (J2000)
coordinates. Note that the source distribution of the MOJAVE catalog covers
the equatorial and northern hemispheres where IceCube’s sensitivity for track-
like events is maximized (see sensitivity curve reported in Aartsen et al.,
2020).

##  2 Data Samples

###  2.1 IceCube Selection of Track-Like Events

IceCube records neutrino events following a signature of being “track-like” or
“cascade-like” depending on the type of particle interaction leading to the
detection. Charged-current muon neutrino interactions lead to elongated
“track-like” signatures produced due to long-lived muons that travel several
kilometers through the ice. On the other hand, neutral-current interactions or
charged current electron and tau neutrino interactions produce hadronic and
electromagnetic showers covering a smaller distance (Halzen & Klein, 2010)
giving “cascade-like” signatures. This work focuses on “track-like” events as
they have a better directional resolution as compared to “cascade-like”
events. The neutrino data is obtained using the all-sky point-source tracks
sample (Abbasi et al., 2021), which spans over a duration of 10 years, from
April 2008 to July 2018. The properties of this neutrino point-source events
sample along with the selection and filtering methods are described in detail
by Aartsen et al. (2020).

The ten-year dataset is further divided by the configuration of IceCube
detector strings. The first year (2008) dubbed “IC-40” uses data from the 40
string configuration, “IC-59” (2009) uses the 59 string configuration, “IC-79”
(2010) uses the 79 string configuration and “IC-86” (2011-2018) uses the 86
string configuration. For more details regarding the configurations see Abbasi
et al. (2021).

###  2.2 MOJAVE Data of AGN at 15 GHz

The AGN sources used for this study are taken from the MOJAVE XV catalog which
includes the total flux density observations of 437 sources obtained with the
Very Long Baseline Array (VLBA) at 15 GHz (Lister et al., 2018). The catalog
consists of a total of 5321 observations of these AGN made with varying
cadence and number of observations per AGN obtained between 1996 January 19
and 2016 December 26. The MOJAVE XV AGN catalog is a blazar-dominated sample
with 392 blazar sources, 27 radio galaxies, 13 unidentified AGN and 5 narrow-
line Seyfert I galaxies. The MOJAVE source list was updated over time to
include low redshift radio galaxies with spectra peaking in the GHz regime.
These sources are distributed almost uniformly over the sky at declinations
$\delta>-30\arcdeg$ ((see Fig. 1) which is better matched to the improved
sensitivity of the work presented here in the northern hemisphere (see
sensitivity curve reported in Aartsen et al., 2020). Moreover, all the
observed AGN have bright compact radio emissions with total flux densities
greater than 50 mJy. This implies that the observed sources are bright at 15
GHz and changes of the radio emission can be effectively measured. This makes
the MOJAVE catalog one of the most promising radio catalogs for correlation
studies such as this one.

As this study also uses multi-epoch observations in the form of photon flux
density light curves, the MOJAVE XV catalog is preferred over the Radio
Fundamental Catalog (RFC)222http://astrogeo.org/rfc/. The RFC catalog is more
complete but lacks time-dependent radio measurements. While many sources in
the MOJAVE XV catalog are AGN that are consistently observed with a good
cadence (see, e.g. Fig. 2), there do exist some sources that have not been
observed frequently. We remove the sources that have very few observations
(keeping sources with a minimum of three observations), reducing the sample
size of 437 sources to 397 sources.

The MOJAVE sample is considered to be complete in terms of sources observed at
15 GHz by VLBA and with flux densities with $f_{\nu}>1.5$ Jy. However, a
completeness correction is required for larger, unbiased analyses such as
this, to account for the sources not included either due to spatial coverage
or flux threshold in the catalog. The completeness correction is found to be
$44.7\pm 11.2\%$ (see Appendix A). After accounting for completeness, this
study focuses on a blazar-dominated AGN sample that follows the same
properties as the sources in the MOJAVE sample.

Figure 2: Example binned lightcurve (orange line) using MOJAVE data (blue data
points) for one source (3C279) along with different IceCube datasets, and
their time coverage is shown with the corresponding dataset name. The light
curve is binned using equal time binning within the 10-year IceCube data shown
here by the vertical lines. Left to right, the bins are shown as: green to red
(2008-2009;IC40), red to purple (2009-2010;IC59), purple to brown
(2010-2011;IC79), brown to pink (2011-2012;IC86_2011), pink to grey
(2012-2017;IC86v3_2012_2017).

##  3 Analysis Method

Using the neutrino and radio flux density described above, we search for
cross-correlations between neutrinos and photon observations in the direction
of the AGN in the MOJAVE XV catalog. This is done by using a stacking analysis
weighted according to the observed time-averaged and time-dependent radio flux
density. The basis of this work is a likelihood approach similar to the ones
described by Braun et al. (2008) and others, where the track-like neutrino
data is modeled using a background hypothesis ($B_{i}$) and signal hypothesis
($S_{i}$). The background data consists of atmospheric events from neutrinos
and muons originating in the Earth’s atmosphere, while the signal consists of
a radio source-associated point-like excess of neutrinos in the stacked data.

As described in Sec. 2.1, the point source tracks sample is further divided
into five configurations depending on the number of strings in use along with
selections, software and calibrations used (see Table I. of Abbasi et al.,
2021). If each configuration is denoted by an index $k$, let $N_{k}$ be the
number of total neutrino events in the configuration. Using the notation of
$n_{s}$ for the number of signal events from a certain direction and $N_{k}$
for the number of events in the configuration along with the signal and
background hypothesis, a likelihood function can be constructed for each event
$i$ by the following:

$\mathscr{L}(n_{s},\gamma)=\prod_{k}\prod_{i}^{N_{k}}\left(\frac{n_{s}^{k}}{N_{k}}{S}_{i}^{k}+\left(1-\frac{n_{s}^{k}}{N_{k}}\right){B}_{i}^{k}\right)$
(1)

where ${S}_{i}^{k}$ and ${B}_{i}^{k}$ represent the probability density
functions (PDFs) corresponding to the signal and background hypotheses,
respectively, for each configuration $k$. The expected number of signal events
$n_{s}^{k}$ is derived by computing the fraction of the total events $n_{s}$
in a configuration $k$ using a factor $f_{k}$ where $n_{s}^{k}=f_{k}n_{s}$.
The fractional contribution $f_{k}$ and ${S}_{i}^{k}$ are modified for a
stacking analysis to include per-source weighting information. If, for a
configuration $k$, $\omega_{j}$ is the per-source contribution for the model
being tested and ${R}_{j}^{k}$ is the detector weight at the declination of
the $j^{th}$ source, $f_{k}$ and ${S}_{i}^{k}$ can be written as:

${S}_{i}^{k}=\frac{\sum_{j}\omega_{j}{R}_{j}^{k}{S}^{k}_{ij}}{\sum_{j}\omega_{j}{R}_{j}^{k}}$
(2)

and

$f_{k}=\frac{\sum_{j}\omega_{j}{R}_{j}^{k}}{\sum_{k}\sum_{j}\omega_{j}{R}_{j}^{k}}$
(3)

The detector weights ${R}_{j}^{k}$ vary per source and account for the
detection efficiency of IceCube. The detection efficiency for signal events
depends on the source direction from which the signal neutrinos originate
along with the differential spectrum (assumed as $E^{-\gamma}$) of the
neutrinos and is calculated using Monte Carlo simulations (Huber & Coenders,
2016). The source weights $\omega_{j}$ depend on the hypothesis being tested.
In this work, these parameters are determined according to the special cases
of time-averaged stacking and time-dependent stacking described below.

###  3.1 Time-averaged Stacking

The average flux densities of each source in the MOJAVE XV catalog are used as
weights $\omega_{j}$ for the stacking in Eqs. 2 and 3 under the assumption
that there is a 1:1 correlation between the radio flux density and IceCube
neutrino flux. For an astrophysical source at direction $\vec{x_{j}}$, using a
set of neutrino data events, indexed by $i$, each with reconstructed energy
and direction given by $E_{i}$ and $\vec{x_{i}}$ respectively, we create a
power-law energy distribution $P(E_{i}|\gamma)$ where $\gamma$ indicates the
spectral index (see also Braun et al., 2008).

The signal hypothesis for a source, denoted by index j, is modeled using

${P}^{k}_{sig}(\sigma_{i},x_{i},x_{j})=\frac{1}{2\pi\sigma_{i}^{2}}\exp\left({-\frac{|\vec{x_{i}}-\vec{x_{j}}|^{2}}{2\sigma_{i}^{2}}}\right)$
(4)

where $\sigma_{i}$ is the angular reconstruction error estimate. Combining
this information with the energy-dependent signal PDF ($\epsilon_{i}^{k}$) we
get the $S_{ij}^{k}$ term, given as:

${S}_{ij}^{k}={P}^{k}_{sig}(\sigma_{i},x_{i},x_{j})\,\,\epsilon_{i}^{k}(E_{i},\delta_{i}|\gamma)$
(5)

where $\epsilon_{i}^{k}$ is computed by using a power-law energy spectrum with
index $\gamma$. This is then used in Eq. 2 to account for the per-source
weighting for the stacking. The directional uncertainty for the event
reconstruction in the configurations uses a lower limit of $0.2\arcdeg$ to
minimize the impact of any inaccuracies in ice models and to ensure the
likelihood calculation is not dominated by a single event (Abbasi et al.,
2021). Note that the radio source position uncertainty (in the order of milli-
arcseconds) is negligible compared to the above lower limit on the directional
uncertainty of the neutrino event reconstructions.

Finally, the background hypothesis is modeled using:

${B}_{i}^{k}(\vec{x_{i}},E_{i},\delta_{i})={P}_{bkg}^{k}(\vec{x_{i}})\epsilon_{B}^{k}(E_{i},\delta_{i})$
(6)

where the product of the energy-dependent background PDF $\epsilon_{B}^{k}$ is
taken with the spatial PDF ${P}_{bkg}^{k}$ at declination $\delta_{i}$. No
per-source weighting is applied to the background hypothesis.

###  3.2 Time-dependent Stacking

The time-dependent stacking analysis differs slightly from the time-averaged
case, where we test a 1:1 correlation between the time-dependent radio flux
density measurements at 15 GHz with the neutrino flux seen by IceCube. For
this case, separate light curves are created for each source using the flux
density measurements. These light curves are then binned over the 10-year
neutrino data period. The number of bins is kept fixed at 40 bins for each
source. Based on tests with multiple bin values, the 40 bin value is chosen to
ensure no temporal information is lost per AGN and the width and location of
the bin height are the same for all sources while constraining the
computational limit required to perform the analysis. Increasing the number of
bins does not give any additional increase in sensitivity but significantly
increases the computing power required while decreasing the bins leads to
reduced sensitivity due to a loss of lightcurve information in some sources.
An example of a binned light curve is shown in Fig. 2. The weighting term for
Eqs. 2 and 3 in this case is given by the flux density of the $j^{th}$ source
at time $t_{i}$. If a source has no observation during the observed period,
the time-averaged flux density measurement is used (see Fig. 2). This
correction was generally applied for either very small time periods of the
light curve or for less variable sources with fewer data points. In both
cases, this does not impact the analysis significantly.

Next, the signal and background hypothesis (described above and in Braun et
al., 2008) is modified to include the temporal information using different
time bins. For every time bin, the signal and background PDFs are computed,
changing the equations to model the hypotheses to:

${S}_{ij}^{k}(t)={P}^{k}_{sig}(\sigma_{i},x_{i},x_{j})\epsilon_{i}^{k}(E_{i},\delta_{i}|\gamma)T^{k}_{sig.j}(t_{i})$
(7)

and

${B}_{i}^{k}(\vec{x_{i}},E_{i},\delta_{i},t)={P}_{bkg}^{k}(\vec{x_{i}})\epsilon_{B}^{k}(E_{i},\delta_{i})T^{k}_{bkg}(t_{i})$
(8)

where $T^{k}_{sig.j}$ and $T^{k}_{bkg}$ gives the temporal PDF at a time
$t_{i}$ corresponding to bin $i$. Past analyses like Abbasi et al. (2022b) use
a likelihood approach to search for flares by fitting time-dependent delay and
threshold parameters which tests the possibility of a correlation with
temporal delays or signal thresholds in a non-stacking approach. This study
makes use of time-dependent radio flux densities for stacking testing a 1:1
correlation while not including the time-dependent delay and threshold
parameters and fixing them to 0.

Based on the signal and background hypothesis for the two stacking cases,
along with the fractional contribution, the likelihood is calculated using Eq.
1. The test statistic (TS), which quantifies the significance of the results,
is computed by making use of the likelihood formation in the following manner:

$TS=-2\,{\rm
sign}(\hat{n}_{s})\log\left[\frac{\mathscr{L}(\vec{x_{s}},n_{s}=0)}{\mathscr{L}(\vec{x_{s}},\hat{n}_{s},\hat{\gamma})}\right]\,.$
(9)

where the $\hat{}$ notation is used to denote a best-fit and $\vec{x_{s}}$
shows the stacked source position.

##  4 Results and Discussion

Figure 3: Likelihood maps for the $n_{s}$-gamma values for the two analyses
are shown here: time-averaged (top), time-dependent (bottom). The plot also
shows the 1, 2 and 3 sigma confidence intervals from the best fit, assuming
Wilk’s theorem with 2 degrees of freedom.

###  4.1 Stacking results:

For the time-averaged analysis, we obtain a best-fit spectral index of 3.14
and a best-fit number of signal events of 31.5. These best-fit results
correspond to a p-value of 0.49 ($0.03\sigma$ significance) and the likelihood
scan is shown in Fig. 3 (top). Similarly, for the time-dependent analysis, we
obtain a best-fit spectral index of 2.7 and a best-fit number of signal events
of 218.7, corresponding to a p-value of 0.07 ($1.51\sigma$ significance). The
likelihood contours for these best fits are shown in Fig. 3. To see how the
best-fit TS scales with respect to the background distribution, see Fig 7 in
the Appendix. We can see that the best-fit number of signal events
$\hat{n_{s}}$ derived for the time-dependent case is much higher than the
time-averaged scenario with the time-averaged scenario having a softer
spectral index value. While a 1:1 comparison for the parameters is difficult
because of the change in the best fit spectral index, we can note the increase
in the TS value of the fit for the time-dependent case. One can also see the
changes in the shape of the contours shown in Fig. 3 with the time-dependent
analysis appearing to constrain the fit in a better manner. We report the
study’s upper limits, derived using the best-fit values, as we do not obtain a
statistically significant result for both analyses.

Table 1: Best-fit results derived from the study | Time-Averaged | Time-Dependent
---|---|---
Signal Events $\hat{n_{s}}$ | 31.5 | 218.7
Spectral Index $\hat{\gamma}$ | 3.1 | 2.7
TS | 0.09 | 2.58
p-value | 0.49 | 0.07
Significance | 0.03$\sigma$ | 1.51$\sigma$
---
Figure 4: Upper Limits (UL) per neutrino ($\nu+\bar{\nu}$) flavor, for an
index of 2.0 (left) and 2.5 (right solid line), derived from the time-
dependent (blue) and time-averaged (orange solid line) analyses are shown here
along with the lower limits (grey-dashed line) reported by Plavin et al.
(2021). Note that while the samples and methodology used by the Plavin et al.
(2021), Zhou et al. (2021) (green dashed line) and this work are different,
making a 1:1 comparison difficult, they are shown here to highlight the
differences in the results. The diffuse astrophysical muon neutrino flux
measurements are taken from Abbasi et al. (2022a). The energy range of the
upper limits shown for the time-averaged analyses reflects the region where
90% of detected signal neutrinos would fall. The energy range for the time-
dependent scenario is kept similar to the time-averaged case.

Using a spectral index of 2.0 and 2.5, the energy-integrated upper limits for
the two analyses at 100 TeV are given in Fig. 4. Both of these limits are
shown after including the completeness correction described in Appendix A. We
also show the astrophysical diffuse flux reported by Abbasi et al. (2022a) and
the lower limits given by Plavin et al. (2021). The energy range for the upper
limits in the figure depicts the region where 90% of the signal neutrinos with
this spectrum will be detected. We calculate the 90% sensitivity for both
scenarios by determining the mean 90% confidence level upper limit expected in
the absence of signal (Hill & Rawlins, 2003), calculated both in terms of flux
and the number of neutrino events. The sensitivity of both the scenarios (in
terms of $E^{2}dN/dE$ flux) is comparable within uncertainties, with a value
of $\sim 2\times 10^{-12}$ TeV cm-2s-1 for a spectral index of 2.0, with the
time-dependent sensitivity being slightly higher. The discovery potential,
defined as the signal strength leading to $5\sigma$ deviation for 50% of all
cases, is found to be $\sim 1\times 10^{-11}$ TeV cm-2s-1 (time dependent) and
$\sim 8\times 10^{-12}$ TeV cm-2s-1 (time averaged), in terms of $E^{2}dN/dE$
flux for a spectral index of 2.0. However, the statistical significance of the
time-dependent analysis, used to derive the upper limit, is also higher for
the time-dependent analysis, giving it a higher upper limit (See also Fig 7 in
Appendix B which shows the median and best-fit TS value used to calculate the
sensitivity and upper limit respectively).

We also report the differential upper limits for different energy bins for
both of our analyses, which is shown in Figure. 5. The differential upper
limit highlights the energy range where the study is most sensitive. The
differential upper limits reduce the dependence on spectral assumptions and
give a per energy bin upper limit. These estimates can also be used to
highlight the energies at which neutrino production in AGN may be suppressed
due to photon intensities or relevant production mechanisms, which can be used
for AGN modeling studies.

Figure 5: Differential Upper Limits (UL), for an index of 2.0, derived from
the time-dependent (blue) and time-averaged (orange) analyses in half-decade
energy bins are shown here. The diffuse astrophysical muon neutrino flux
measurements are taken from Abbasi et al. (2022a). Note that the upper limits
shown here are differential and binned with energy, while the astrophysical
flux is an energy-integrated measurement shown here only for reference.

###  4.2 Comparison with other studies:

Previous studies have worked on similar analyses using VLBA radio data from
AGN to search for correlations and reported limits on the total neutrino flux
from these AGN (Plavin et al., 2020, 2021; Zhou et al., 2021; Abbasi et al.,
2023b).

The lower limit reported by Plavin et al. (2021), also shown in Fig. 4, lies
above the upper limits provided here, even after the inclusion of a
completeness correction. While adding temporal information for the time-
dependent analysis changes the upper limits with respect to the time-averaged
limit, it still lies below the Plavin et al. (2021) results, ruling out the
reported lower limits. Note that the radio and neutrino datasets and analysis
methodology used for the two studies are different, which makes a direct
comparison between the two studies difficult. This work uses a stacking
approach with time-dependent MOJAVE radio data while the Plavin et al. (2021)
work makes use of the direction of the radio sources given in the RFC catalog
to test for correlation with IceCube alerts. However, a more detailed study
using the complete IceCube alert dataset including additional information like
signalness, was performed recently by Abbasi et al. (2023b) which provides a
direct comparison to the Plavin et al. (2020, 2021) results. In contrast to
Plavin et al. (2020, 2021), we find in Abbasi et al. (2023b) that the signal
TS is compatible with the background and the significance goes down when a
more sophisticated description of the spatial PDF is used as opposed to a
simplified signal PDF modeled as a uniform distribution inside of the error
contour.

On the other hand, Zhou et al. (2021) use the same neutrino dataset (i.e.
Abbasi et al., 2021) as this work while using a more extensive radio sample
(more stacked sources) instead of a completeness correction. However, when
comparing the upper limits, the limits reported by Zhou et al. (2021) lie
above the limits derived by this work. The change in sensitivity of this work
as compared to Zhou et al. (2021) is mainly due to the inclusion of an energy
PDF in the likelihood framework as described above, thereby reducing the upper
limit.

##  5 Conclusion:

This work focused on using the AGN data published in the MOJAVE XV catalog to
search for radio flux density-correlated neutrino emission using time-averaged
and time-dependent analyses. We performed a stacking analysis and reported
upper limits for both analyses as no significant correlation is found. When
compared to the IceCube diffuse flux (Abbasi et al., 2022a), at 100 TeV and
for a spectral index of 2.5, the upper limits derived are $\sim 3\%$ and $\sim
9\%$ for the time-averaged and time-dependent case. Note that, as the spectral
index of the diffuse flux is different from 2.5, the percentage comparison is
done using the upper limit estimates at 100 TeV. We also compared the upper
limits presented in this work with the reported limits of Plavin et al. (2021)
and Zhou et al. (2021). While the study presented here has the limitation of
using fewer radio sources, which is made up by using a completeness
correction, it also has the advantage of using more neutrino information in a
time-dependent stacking study.

Based on the obtained results, this study shows that including time-dependent
information in the form of light curves improves the statistical power of the
stacking analysis, if the neutrino flux is directly correlated to the change
in the radio flux. While the sensitivity for both analyses is similar (see
Sec. 4), the time-dependent study includes the addition of temporal
information which increases the best-fit TS values and changes the results.
However, the time-dependent analysis depends on the light curves used to
satisfy the model assumptions for this study. The MOJAVE XV dataset used here
has per-source observations reported with a varying cadence with a few sources
being observed only a couple of times. Additionally, as compared to the number
of sources observed by VLBA, the number of MOJAVE XV catalog sources with
time-dependent observations is limited and focused on a blazar-dominated
sample with a few non-blazar AGN. This can be improved upon, in the future, by
making use of more AGN sources with observations performed with a good
cadence, preferably from a monitoring campaign.

##  6 Acknowledgements

The IceCube collaboration acknowledges the significant contributions to this
manuscript from Abhishek Desai and Justin Vandenbroucke. The authors would
like to thank M Lister, Y. Kovalev and A. Plavin for useful discussions
regarding the MOJAVE and RFC catalogs and required completeness calculation.
The authors would also like to thank M.J. Romfoe for the useful comments on
the paper draft. The authors gratefully acknowledge the support from the
following agencies and institutions: USA – U.S. National Science Foundation-
Office of Polar Programs, U.S. National Science Foundation-Physics Division,
U.S. National Science Foundation-EPSCoR, U.S. National Science Foundation-
Office of Advanced Cyberinfrastructure, Wisconsin Alumni Research Foundation,
Center for High Throughput Computing (CHTC) at the University of
Wisconsin–Madison, Open Science Grid (OSG), Partnership to Advance Throughput
Computing (PATh), Advanced Cyberinfrastructure Coordination Ecosystem:
Services & Support (ACCESS), Frontera computing project at the Texas Advanced
Computing Center, U.S. Department of Energy-National Energy Research
Scientific Computing Center, Particle astrophysics research computing center
at the University of Maryland, Institute for Cyber-Enabled Research at
Michigan State University, Astroparticle physics computational facility at
Marquette University, NVIDIA Corporation, and Google Cloud Platform; Belgium –
Funds for Scientific Research (FRS-FNRS and FWO), FWO Odysseus and Big Science
programmes, and Belgian Federal Science Policy Office (Belspo); Germany –
Bundesministerium für Bildung und Forschung (BMBF), Deutsche
Forschungsgemeinschaft (DFG), Helmholtz Alliance for Astroparticle Physics
(HAP), Initiative and Networking Fund of the Helmholtz Association, Deutsches
Elektronen Synchrotron (DESY), and High Performance Computing cluster of the
RWTH Aachen; Sweden – Swedish Research Council, Swedish Polar Research
Secretariat, Swedish National Infrastructure for Computing (SNIC), and Knut
and Alice Wallenberg Foundation; European Union – EGI Advanced Computing for
research; Australia – Australian Research Council; Canada – Natural Sciences
and Engineering Research Council of Canada, Calcul Québec, Compute Ontario,
Canada Foundation for Innovation, WestGrid, and Digital Research Alliance of
Canada; Denmark – Villum Fonden, Carlsberg Foundation, and European
Commission; New Zealand – Marsden Fund; Japan – Japan Society for Promotion of
Science (JSPS) and Institute for Global Prominent Research (IGPR) of Chiba
University; Korea – National Research Foundation of Korea (NRF); Switzerland –
Swiss National Science Foundation (SNSF).

## References

  * Aartsen et al. (2013) Aartsen, M. G., et al. 2013, Science, 342, 1242856
  * Aartsen et al. (2017) —. 2017, JINST, 12, P03012
  * Aartsen et al. (2018a) —. 2018a, Science, 361, eaat1378
  * Aartsen et al. (2018b) —. 2018b, Science, 361, 147
  * Aartsen et al. (2020) —. 2020, Phys. Rev. Lett., 124, 051103
  * Abbasi et al. (2021) Abbasi, R., et al. 2021, arXiv:2101.09836
  * Abbasi et al. (2022) Abbasi, R., Ackermann, M., Adams, J., et al. 2022, Science, 378, 538
  * Abbasi et al. (2022a) Abbasi, R., et al. 2022a, Astrophys. J., 928, 50
  * Abbasi et al. (2022b) —. 2022b, Astrophys. J. Lett., 930, L24
  * Abbasi et al. (2022c) —. 2022c, Phys. Rev. D, 106, 022005
  * Abbasi et al. (2023a) —. 2023a, Astrophys. J. Suppl., 269, 25
  * Abbasi et al. (2023b) Abbasi, R., Ackermann, M., Adams, J., et al. 2023b, The Astrophysical Journal, 954, 75
  * Ajello et al. (2012) Ajello, M., Shaw, M. S., Romani, R. W., et al. 2012, The Astrophysical Journal, 751, 108
  * Berezinsky & Ginzburg (1981) Berezinsky, V. S., & Ginzburg, V. L. 1981, Monthly Notices of the Royal Astronomical Society, 194, 3
  * Braun et al. (2008) Braun, J., Dumm, J., De Palma, F., et al. 2008, Astropart. Phys., 29, 299
  * Buson et al. (2022) Buson, S., Tramacere, A., Pfeiffer, L., et al. 2022, The Astrophysical Journal Letters, 933, L43
  * Desai et al. (2021) Desai, A., Vandenbroucke, J., & Pizzuto, A. 2021, in Proceedings of 37th International Cosmic Ray Conference — PoS(ICRC2021), Vol. 395, 949
  * Eichler (1979) Eichler, D. 1979, ApJ, 232, 106
  * Halzen & Klein (2010) Halzen, F., & Klein, S. R. 2010, Rev. Sci. Instrum., 81, 081101
  * Hill & Rawlins (2003) Hill, G. C., & Rawlins, K. 2003, Astropart. Phys., 19, 393
  * Hovatta et al. (2021) Hovatta, T., et al. 2021, Astron. Astrophys., 650, A83
  * Huber (2019) Huber, M. 2019, in Proceedings of 36th International Cosmic Ray Conference — PoS(ICRC2019), Vol. 358, 916
  * Huber & Coenders (2016) Huber, M., & Coenders, S. 2016, Verhandlungen der Deutschen Physikalischen Gesellschaft, 1
  * Jacobsen et al. (2015) Jacobsen, I. B., Wu, K., On, A. Y. L., & Saxton, C. J. 2015, Mon. Not. Roy. Astron. Soc., 451, 3649
  * Kun et al. (2022) Kun, E., Bartos, I., Tjus Becker, J., et al. 2022, Astrophys. J., 934, 180
  * Lister et al. (2018) Lister, M. L., Aller, M. F., Aller, H. D., et al. 2018, Astrophys. J. Suppl., 234, 12
  * Lister et al. (2019) Lister, M. L., Homan, D. C., Hovatta, T., et al. 2019, Astrophys. J., 874, 43
  * Murase (2022) Murase, K. 2022, Astrophys. J. Lett., 941, L17
  * Murase & Stecker (2023) Murase, K., & Stecker, F. W. 2023, in The Encyclopedia of Cosmology. Set 2: Frontiers in Cosmology. Volume 2: Neutrino Physics and Astrophysics, ed. F. W. Stecker, 483–540
  * Planck Collaboration (2016) Planck Collaboration. 2016, Astron. Astrophys., 594, A13
  * Plavin et al. (2020) Plavin, A., Kovalev, Y. Y., Kovalev, Y. A., & Troitsky, S. 2020, Astrophys. J., 894, 101
  * Plavin et al. (2021) Plavin, A. V., Kovalev, Y. Y., Kovalev, Y. A., & Troitsky, S. V. 2021, Astrophys. J., 908, 157
  * Rodrigues et al. (2021) Rodrigues, X., Garrappa, S., Gao, S., et al. 2021, The Astrophysical Journal, 912, 54
  * Sikora et al. (1987) Sikora, M., Kirk, J. G., Begelman, M. C., & Schneider, P. 1987, Astrophys. J., 320, L81
  * Stecker et al. (1991) Stecker, F. W., Done, C., Salamon, M. H., & Sommers, P. 1991, Phys. Rev. Lett., 66, 2697, [Erratum: Phys.Rev.Lett. 69, 2738 (1992)]
  * Zhou et al. (2021) Zhou, B., Kamionkowski, M., & Liang, Y.-f. 2021, Phys. Rev. D, 103, 123018

## Appendix A: Completeness Correction

While the MOJAVE sample is considered to be complete for flux densities
greater than 1.5 Jy, it is still considered to be a flux-limited and
spatially-limited sample because of its declination constraints (limited by
the region in the sky observed by VLBA telescopes; see Fig. 1) and flux
density measurement limitations of the telescopes used (Lister et al., 2018).
To account for these limitations, we compute the completeness correction for
the sample by estimating the source count distribution of the sample. The
source count distribution is the cumulative distribution of the number of
sources greater than a particular flux (see Fig. 6). This is derived using the
results of the population study performed by Lister et al. (2019) and reported
as the MOJAVE XVII study. This involves modeling the flux ($\Phi$), luminosity
($L$) and redshift ($z$) relation using luminosity evolution parameterization
similar to the ones used by Ajello et al. (2012). To model the MOJAVE sources,
we then use the parameters given by Model A of Lister et al. (2019) where
$\gamma=-3.1,k=8.0,\eta=-0.35$ and $\alpha=0$ for the equations:

$\Phi(L,z)\propto\Phi(L/e(z))$ (10) $e(z)=(1+z)^{k}e^{z/\eta}$ (11)
$\Phi(L/e(z=0))\propto L^{\gamma}$ (12)

The cosmology parameters used in the calculation is taken from Planck
Collaboration (2016) as $\Omega_{0}$=$0.308$, $\Omega_{\lambda}$=$0.692$ and
$h_{0}$=$0.678$. In addition to these parameters, we also require the Lorentz
factor distribution given by $N(\Gamma)d\Gamma\propto\Gamma^{b}$ where
$b=-1.40$ and the viewing angle distribution given by
$p(\theta)d\theta=\sin\theta$. To derive the simulated source count, the two
PDFs are used to sample different values of $\Gamma$ and $\theta$ which are
then used in the equations given above to derive the luminosity function. Once
the luminosity function is derived, it is used to calculate the flux
distribution and in turn the source count distribution. This procedure is
repeated multiple times to get the uncertainties to the source count
distribution which are shown as the shaded band in Fig. 6. Note that the mean
source count distribution curve resulting from multiple simulations follows
the $\log_{10}N=(-1.63\pm 0.02)\log_{10}S+(-19.8\pm 0.03)$ equation for higher
flux values where $N$ is the number of sources having a flux density greater
than $S$. This is close to the accepted $\log N-\log S$ relation where
$\log_{10}N=(-1.5)\log_{10}S$. We also compare this simulated source count
distribution to the sources given in the radio fundamental catalog which is
complete for flux density measurements $>=150$ mJy, and our simulation agrees
with the observations for those flux density measurements.

Figure 6: Simulated source count distribution of the blazar-dominated sample
(orange data points) as compared to the source count distribution of the
MOJAVE XV sources (blue data points). The shaded region shows the one sigma
error to the distribution due to varying the Lorentz factor and viewing angle
parameters of the jets. The green line shows the fit at flux densities higher
than $10^{-13}$erg sec-1cm-2 to the simulated sample.

Once the simulated source count distribution is derived, the area under the
curve for the distribution is compared with the area under the curve for the
observed MOJAVE source population. This gives the completeness correction for
the population which takes into account both the flux and spatial limitations
of the sample. This method adds dimmer sources to the population which might
have been missed by the MOJAVE XV catalog. As the MOJAVE XV dataset and the
population study done by Lister et al. (2019) is for a blazar-dominated source
sample, the completeness derived using this method is similarly also for a
blazar-dominated sample.

## Appendix B: Background Distribution

The TS distribution for a background-only case (${B}_{i}$) is shown here. This
is derived by setting $n_{s}$ to 0 in Eq. 1 and running multiple standalone
trials on the scrambled data which is derived by scrambling the R.A. of each
event per trial. The shaded lines show the TS required for a $2\sigma$ and
$3\sigma$ detection. The blue dashed line shows the best-fit TS values, which
are used to derive the reported upper-limit measurement.

---
Figure 7: Background distribution plots derived for the time-averaged (left)
and time-dependent (right) cases.


exist but $k_{o2}$ does, then $\arg\max_{k}h(k)$ is one of
$\\{0,1,k_{o2}+2\\}$ (see Fig. 23 for an example). If $k_{o2}$ does not exist,
then $\arg\max_{k}h(k)$ is either $0$ or $1$. The result thus follows.

Figure 23: $h(k)=0.7\times(-0.4)^{k}-0.6\times(-0.8)^{k},\
g(k)=\left(\frac{-0.4}{-0.8}\right)^{k}$

    * 1.22

$a_{2},b_{1},b_{2}<0,a_{1}<0$: obviously $h(k)$ achieves its maximum at $1$
and the result follows.

  * 2.

P has only real eigenvalues and $2$ linearly independent eigenvectors:
$h(k)=ab^{k}+ckb^{k-1}+d$.

    * 2.1

$b,c>0$: observe that

$h(k+1)>h(k)\Leftrightarrow k<\frac{ab-ab^{2}-cb}{c(b-1)}.$

Let $k_{1}\geq 0$ be the maximum integer satisfying the above inequality. If
it exists, then $h(k)$ will keep increasing until $(k_{1}+1)$ after which it
turns to be monotonically decreasing to the stationary reward $d$. Hence,
$k_{1}+1=\arg\max_{k}h(k)$ (see Fig. 24 for an example). If $k_{1}$ does not
exist, then $h(k)$ is monotonically decreasing and $\arg\max_{k}h(k)=0$. The
result thus follows.

Figure 24: $h(k)=2\times 0.7^{k}+3k\times 0.7^{k-1}$

    * 2.2

$b>0,c<0$: observe that

$h(k+1)>h(k)\Leftrightarrow k>\frac{ab-ab^{2}-cb}{c(b-1)}.$

Let $k_{1}\geq 0$ be the minimum integer satisfying the above inequality.
Clearly $h(k)$ is monotonically decreasing until $k_{1}$ after which it keeps
increasing to $d$. So $h(k)$ achieves its supremum at either $0$ or $\infty$
and the result follows.

    * 2.3

$b<0,c<0$: the proof is similar to that of (1.20) and omitted here (see Fig.
25 for an example).

Figure 25: $h(k)=-3\times(-0.8)^{k}-0.7k\times(-0.8)^{k-1}$

    * 2.4

$b<0,c>0$: the proof is similar to that of (1.21) and omitted here.

  * 3.

P has a pair of conjugate complex eigenvalues:
$h(k)=a^{\prime}A^{k}\sin(k\theta+b^{\prime})+c^{\prime}$.

    * 3.1

$d^{\prime}=\frac{r^{*}-c^{\prime}}{a^{\prime}}>0$: clearly $h(k)$ will be
smaller than $d^{\prime}$ as $k$ becomes sufficiently large and the result
follows.

    * 3.2

$d^{\prime}=\frac{r^{*}-c^{\prime}}{a^{\prime}}<0$: clearly $h(k)$ will be
larger than $d^{\prime}$ as $k$ becomes sufficiently large and the exhaustion
stops in finite time.

    * 3.3-3.8

These cases follow directly by finding the first $k\geq 0$ such that
$\sin(k\theta+b^{\prime})>0$ and we omit the details here.