# Search for Spatial Correlations of Neutrinos with Ultra-high-energy Cosmic Rays

A. Albert, S. Alves, M. André, M. Anghinolfi, M. Ardid, S. Ardid, J.-J. Aubert, J. Aublin, B. Baret, S. Basa, et al.

► To cite this version:

A. Albert, S. Alves, M. André, M. Anghinolfi, M. Ardid, et al., Search for Spatial Correlations of Neutrinos with Ultra-high-energy Cosmic Rays. Astrophys.J., 2022, 934 (2), pp.164. 10.3847/1538-4357/ac6def. hal-03623273

HAL Id: hal-03623273

https://hal.science/hal-03623273

Submitted on 18 May 2024

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Distributed under a Creative Commons Attribution 4.0 International License
# Search for Spatial Correlations of Neutrinos with Ultra-high-energy Cosmic Rays

A. Albert, S. Alves, M. André, M. Anghinolfi, M. Ardid, S. Ardid, J.-J. Aubert, J. Aublin, B. Baret, S. Basa, et al.

► To cite this version:

A. Albert, S. Alves, M. André, M. Anghinolfi, M. Ardid, et al., Search for Spatial Correlations of Neutrinos with Ultra-high-energy Cosmic Rays. Astrophys.J., 2022, 934 (2), pp.164. 10.3847/1538-4357/ac6def. hal-03623273

HAL Id: hal-03623273

https://hal.science/hal-03623273v1

Submitted on 18 May 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
OPEN ACCESS

<figure><img src="image_2.png" /></figure>

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# Search for Spatial Correlations of Neutrinos with Ultra-high-energy Cosmic Rays

The ANTARES collaboration,

A. Albert$^{1,2}$, S. Alves$^3$, M. Andre$^4$, M. Anghinolfi$^5$, M. A.$^6$, S. Ardid$^6$, J.-J. Aubert$^7$, J. Aublin$^8$, B. Baret$^8$, S. Basa$^9$, B. Belhorma$^{10}$, M. Bendahman$^{8,11}$, V. Bertin$^7$, S. Biagi$^{12}$, M. Bissinger$^{13}$, J. Boumaaza$^{11}$, M. Bouta$^{14}$, M. C. Bouwhuis$^{15}$, H. Brânzaş$^{16}$,

R. Bruijn$^{15,17}$, J. Brunner$^7$, J. Busto$^7$, B. Caiffi$^5$, D. Calvo$^3$, A. Capone$^{18,19}$, L. Caramete$^{16}$, J. Carr$^7$, V. Carretero$^3$, S. Celli$^{18,19}$, M. Chabab$^{20}$, T. N. Chau$^8$, R. Cherkaoui El Moursli$^{11}$, T. Chiarusi$^{21}$, M. Circella$^{22}$, A. Coleiro$^8$, R. Coniglione$^{12}$, P. Coyle$^7$,

A. Creusot$^8$, A. F. Díaz$^{23}$, C. Distefano$^{12}$, I. Di Palma$^{18,19}$, A. Domi$^{15,17}$, C. Donzaud$^{8,24}$, D. Dornic$^7$, D. Drouhin$^{1,2}$, T. Eberl$^{13}$, T. van Eeden$^{15}$, D. van Eijk$^{15}$, N. El Khayati$^{11}$, A. Enzenhöfer$^7$, P. Fermani$^{18,19}$, G. Ferrara$^{12}$, F. Filippini$^{21,25}$, L. Fusco$^7$, Y. Gatelet$^8$, P. Gay$^{8,26}$, H. Glotin$^{27}$, R. Gozzini$^3$, R. Gracia Ruiz$^{15}$, K. Graf$^{13}$, C. Guidi$^{5,28}$, S. Hallmann$^{13}$, H. van Haren$^{29}$, A. J. Heijboer$^{15}$, Y. Hello$^{30}$, J. J. Hernández-Rey$^3$, J. Hössl$^{13}$, J. Hofestädt$^{13}$, F. Huang$^7$, G. Illuminati$^{8,21,25}$, C. W. James$^{31}$,

B. Jisse-Jung$^{15}$, M. De Jong$^{15,32}$, P. de Jong$^{15,17}$, M. Kadler$^{33}$, O. Kalekin$^{13}$, U. Katz$^{13}$, N. R. Khan-Chowdhury$^3$, A. Kouchner$^8$, I. Kreykenbohm$^{34}$, V. Kulikovskiy$^5$, R. Lahmann$^{13}$, R. Le Breton$^8$, S. LeStum$^7$, D. Lefèvre$^{35}$, E. Leonora$^{36}$, G. Levi$^{21,25}$,

D. Lopez-Coto$^{37}$, S. Loucatos$^{8,38}$, L. Maderer$^8$, J. Manczak$^3$, M. Marcelin$^9$, A. Margiotta$^{21,25}$, A. Marinelli$^{39}$, J. A. Martínez-Mora$^6$, B. Martino$^7$, K. Melis$^{15,17}$, P. Migliozzi$^{39}$, A. Moussa$^{14}$, R. Muller$^{15}$, L. Nauta$^{15}$, S. Navas$^{37}$, E. Nezri$^9$, B. Ó. Fearraigh$^{15}$,

A. Păun$^{16}$, G. E. Păvălaş$^{16}$, C. Pellegrino$^{21,40,41}$, M. Perrin-Terrin$^7$, V. Pestel$^{15}$, P. Piattelli$^{12}$, C. Pieterse$^3$, C. Poirè$^6$, V. Popa$^{16}$, T. Pradier$^1$, N. Randazzo$^{36}$, D. Real$^3$, S. Reck$^{13}$, G. Riccobene$^{12}$, A. Romanov$^{5,28}$, A. Sánchez-Losa$^{3,22}$, F. Salesa Greus$^3$,

D. F. E. Samtleben$^{15,32}$, M. Sanguineti$^{5,28}$, P. Sapienza$^{12}$, J. Schnabel$^{13}$, J. Schumann$^{13}$, F. Schüssler$^{38}$, J. Seneca$^{15}$, M. Spurio$^{21,25}$, Th. Stolarczyk$^{38}$, M. Taiuti$^{5,28}$, Y. Tayalati$^{11}$, S. J. Tingay$^{31}$, B. Vallage$^{8,38}$, V. Van Elewyck$^{8,42}$, F. Versari$^{8,21,25}$, S. Viola$^{12}$,

D. Vivolo$^{39,43}$, J. Wilms$^{34}$, S. Zavatarelli$^5$, A. Zegarelli$^{18,19}$, J. D. Zornoza$^3$, J. Zúñiga$^3$

The IceCube collaboration,

R. Abbasi$^{44}$, M. Ackermann$^{45}$, J. Adams$^{46}$, J. A. Aguilar$^{47}$, M. Ahlers$^{48}$, M. Ahrens$^{49}$, J. M. Alameddine$^{50}$, C. Alispach$^{51}$,

A. A. Alves Jr.$^{52}$, N. M. Amin$^{53}$, K. Andeen$^{54}$, T. Anderson$^{55}$, G. Anton$^{56}$, C. Argüelles$^{57}$, Y. Ashida$^{58}$, S. Axani$^{59}$, X. Bai$^{60}$,

A. Balagopal V.$^{58}$, A. Barbano$^{51}$, S. W. Barwick$^{61}$, B. Bastian$^{45}$, V. Basu$^{58}$, S. Baur$^{47}$, R. Bay$^{62}$, J. J. Beatty$^{63,64}$, K.-H. Becker$^{65}$, J. Becker Tjus$^{66}$, C. Bellenghi$^{67}$, S. BenZvi$^{68}$, D. Berley$^{69}$, E. Bernardini$^{45,70}$, D. Z. Besson$^{71}$, G. Binder$^{62,72}$, D. Bindig$^{65}$,

E. Blaufuss$^{69}$, S. Blot$^{45}$, M. Boddenberg$^{73}$, F. Bontempo$^{52}$, J. Borowka$^{73}$, S. Böser$^{74}$, O. Botner$^{75}$, J. Böttcher$^{73}$, E. Bourbeau$^{48}$, F. Bradascio$^{45}$, J. Braun$^{58}$, B. Brinson$^{76}$, S. Bron$^{51}$, J. Brostean-Kaiser$^{45}$, S. Browne$^{77}$, A. Burgman$^{75}$, R. T. Burley$^{78}$, R. S. Busse$^{79}$,

M. A. Campana$^{80}$, E. G. Carnie-Bronca$^{78}$, C. Chen$^{76}$, Z. Chen$^{81}$, D. Chirkin$^{58}$, K. Choi$^{82}$, B. A. Clark$^{83}$, K. Clark$^{84}$, L. Classen$^{79}$, G. H. Collin$^{59}$, J. M. Conrad$^{59}$, P. Coppin$^{85}$, P. Correa$^{85}$, D. F. Cowen$^{55,86}$, R. Cross$^{68}$, C. Dappen$^{73}$, P. Dave$^{76}$, C. De Clercq$^{85}$,

J. J. DeLaunay$^{87}$, D. Delgado López$^{57}$, H. Dembinski$^{53}$, K. Deoskar$^{49}$, A. Desai$^{58}$, P. Desiati$^{58}$, K. D. de Vries$^{85}$, G. de Wasseige$^{88}$, M. de With$^{89}$, T. DeYoung$^{83}$, A. Diaz$^{59}$, J. C. Díaz-Vélez$^{58}$, M. Dittmer$^{79}$, H. Dujmovic$^{52}$, M. Dunkman$^{55}$, M. A. DuVernois$^{58}$,

E. Dvorak$^{60}$, T. Ehrhardt$^{74}$, P. Eller$^{67}$, H. Erpenbeck$^{73}$, J. Evans$^{69}$, P. A. Evenson$^{53}$, K. L. Fan$^{69}$, A. R. Fazely$^{90}$, A. Fedynitch$^{91}$, N. Feigl$^{89}$, S. Fiedlschuster$^{56}$, A. T. Fienberg$^{55}$, K. Filimonov$^{62}$, C. Finley$^{49}$, L. Fischer$^{45}$, D. Fox$^{86}$, A. Franckowiak$^{45,66}$,

E. Friedman$^{69}$, A. Fritz$^{74}$, P. Fürst$^{73}$, T. K. Gaisser$^{53}$, J. Gallagher$^{92}$, E. Ganster$^{73}$, A. Garcia$^{57}$, S. Garrappa$^{45}$, L. Gerhardt$^{72}$, A. Ghadimi$^{87}$, C. Glaser$^{75}$, T. Glauch$^{67}$, T. Glüsenkamp$^{56}$, J. G. Gonzalez$^{53}$, S. Goswami$^{87}$, D. Grant$^{83}$, T. Grégoire$^{55}$,

S. Griswold$^{68}$, C. Günther$^{73}$, P. Gutjahr$^{50}$, C. Haack$^{67}$, A. Hallgren$^{75}$, R. Halliday$^{83}$, L. Halve$^{73}$, F. Halzen$^{58}$, M. Ha Minh$^{67}$, K. Hanson$^{58}$, J. Hardin$^{58}$, A. A. Harnisch$^{83}$, A. Haungs$^{52}$, D. Hebecker$^{89}$, K. Helbing$^{65}$, F. Henningsen$^{67}$, E. C. Hettinger$^{83}$,

S. Hickford$^{65}$, J. Hignight$^{93}$, C. Hill$^{94}$, G. C. Hill$^{78}$, K. D. Hoffman$^{69}$, R. Hoffmann$^{65}$, B. Hokanson-Fasig$^{58}$, K. Hoshina$^{58,95}$, M. Huber$^{67}$, T. Huber$^{52}$, K. Hultqvist$^{49}$, M. Hünnefeld$^{50}$, R. Hussain$^{58}$, K. Hymon$^{50}$, S. In$^{82}$, N. Iovine$^{47}$, A. Ishihara$^{94}$,

M. Jansson$^{49}$, G. S. Japaridze$^{96}$, M. Jeong$^{82}$, M. Jin$^{57}$, B. J. P. Jones$^{97}$, D. Kang$^{52}$, W. Kang$^{82}$, X. Kang$^{80}$, A. Kappes$^{79}$, D. Kappesser$^{74}$, L. Kardum$^{50}$, T. Karg$^{45}$, M. Karl$^{67}$, A. Karle$^{58}$, M. Kauer$^{58}$, M. Kellermann$^{73}$, J. L. Kelley$^{58}$, A. Kheirandish$^{55}$,

K. Kin$^{94}$, T. Kintscher$^{45}$, J. Kiryluk$^{81}$, S. R. Klein$^{62,72}$, R. Koirala$^{53}$, H. Kolanoski$^{89}$, T. Kontrimas$^{67}$, L. Köpke$^{74}$, C. Kopper$^{83}$, S. Kopper$^{87}$, D. J. Koskinen$^{48}$, P. Koundal$^{52}$, M. Kovacevich$^{80}$, M. Kowalski$^{45,89}$, T. Kozynets$^{48}$, E. Kun$^{66}$, N. Kurahashi$^{80}$, N. Lad$^{45}$, C. Lagunas Gualda$^{45}$, J. L. Lanfranchi$^{55}$, M. J. Larson$^{69}$, F. Lauber$^{65}$, J. P. Lazar$^{57,58}$, J. W. Lee$^{82}$, K. Leonard$^{58}$,

A. Leszczyńska$^{77}$, Y. Li$^{55}$, M. Lincetto$^{66}$, Q. R. Liu$^{58}$, M. Liubarska$^{93}$, E. Lohfink$^{74}$, C. J. Lozano Mariscal$^{79}$, F. Lucarelli$^{51}$, A. Ludwig$^{83,98}$, W. Luszczak$^{58}$, Y. Lyu$^{62,72}$, W. Y. Ma$^{45}$, J. Madsen$^{58}$, K. B. M. Mahn$^{83}$, Y. Makino$^{58}$, S. Mancina$^{58}$,

I. Martinez-Soler$^{57}$, R. Maruyama$^{99}$, K. Mase$^{94}$, T. McElroy$^{93}$, F. McNally$^{100}$, J. V. Mead$^{48}$, K. Meagher$^{58}$, S. Mechbal$^{45}$, A. Medina$^{64}$, M. Meier$^{94}$, S. Meighen-Berger$^{67}$, J. Micallef$^{83}$, D. Mockler$^{47}$, T. Montaruli$^{51}$, R. W. Moore$^{93}$, R. Morse$^{58}$,

M. Moulai$^{59}$, R. Naab$^{45}$, R. Nagai$^{94}$, U. Naumann$^{65}$, J. Necker$^{45}$, L. V. Nguyen$^{83}$, H. Niederhausen$^{83}$, M. U. Nisa$^{83}$,

S. C. Nowicki$^{83}$, A. Obertacke Pollmann$^{65}$, M. Oehler$^{52}$, B. Oeyen$^{101}$, A. Olivas$^{69}$, E. O'Sullivan$^{75}$, H. Pandya$^{53}$, D. V. Pankova$^{55}$, N. Park$^{84}$, G. K. Parker$^{97}$, E. N. Paudel$^{53}$, L. Paul$^{54}$, C. Pérez de los Heros$^{75}$, L. Peters$^{73}$, J. Peterson$^{58}$, S. Philippen$^{73}$, S. Pieper$^{65}$,

M. Pittermann$^{77}$, A. Pizzuto$^{58}$, M. Plum$^{54}$, Y. Popovych$^{74}$, A. Porcelli$^{101}$, M. Prado Rodriguez$^{58}$, P. B. Price$^{62}$, B. Pries$^{83}$, G. T. Przybylski$^{72}$, C. Raab$^{47}$, J. Rack-Helleis$^{74}$, A. Raissi$^{46}$, M. Rameez$^{48}$, K. Rawlins$^{102}$, I. C. Rea$^{67}$, A. Rehman$^{53}$,

P. Reichherzer$^{66}$, R. Reimann$^{73}$, G. Renzi$^{47}$, E. Resconi$^{67}$, S. Reusch$^{45}$, W. Rhode$^{50}$, M. Richman$^{80}$, B. Riedel$^{58}$, E. J. Roberts$^{78}$,
OPEN ACCESS

<figure><img src="image_2.png" /></figure>

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Search for Spatial Correlations of Neutrinos with Ultra-high-energy Cosmic Rays

The ANTARES collaboration,

A. Albert<sup>1,2</sup>, S. Alves<sup>3</sup>, M. Andre<sup>4</sup>, M. Anghinolfi<sup>5</sup>, M. A.<sup>6</sup>, S. Ardid<sup>6</sup>, J.-J. Aubert<sup>7</sup>, J. Aublin<sup>8</sup>, B. Baret<sup>8</sup>, S. Basa<sup>9</sup>, B. Belhorma<sup>10</sup>, M. Bendahman<sup>8,11</sup>, V. Bertin<sup>7</sup>, S. Biagi<sup>12</sup>, M. Bissinger<sup>13</sup>, J. Boumaaza<sup>11</sup>, M. Bouta<sup>14</sup>, M. C. Bouwhuis<sup>15</sup>, H. Brânzaş<sup>16</sup>,

R. Bruijn<sup>15,17</sup>, J. Brunner<sup>7</sup>, J. Busto<sup>7</sup>, B. Caiffi<sup>5</sup>, D. Calvo<sup>3</sup>, A. Capone<sup>18,19</sup>, L. Caramete<sup>16</sup>, J. Carr<sup>7</sup>, V. Carretero<sup>3</sup>, S. Celli<sup>18,19</sup>, M. Chabab<sup>20</sup>, T. N. Chau<sup>8</sup>, R. Cherkaoui El Moursli<sup>11</sup>, T. Chiarusi<sup>21</sup>, M. Circella<sup>22</sup>, A. Coleiro<sup>8</sup>, R. Coniglione<sup>12</sup>, P. Coyle<sup>7</sup>,

A. Creusot<sup>8</sup>, A. F. Díaz<sup>23</sup>, C. Distefano<sup>12</sup>, I. Di Palma<sup>18,19</sup>, A. Domi<sup>15,17</sup>, C. Donzaud<sup>8,24</sup>, D. Dornic<sup>7</sup>, D. Drouhin<sup>1,2</sup>, T. Eberl<sup>13</sup>, T. van Eeden<sup>15</sup>, D. van Eijk<sup>15</sup>, N. El Khayati<sup>11</sup>, A. Enzenhöfer<sup>7</sup>, P. Fermani<sup>18,19</sup>, G. Ferrara<sup>12</sup>, F. Filippini<sup>21,25</sup>, L. Fusco<sup>7</sup>, Y. Gatelet<sup>8</sup>, P. Gay<sup>8,26</sup>, H. Glotin<sup>27</sup>, R. Gozzini<sup>3</sup>, R. Gracia Ruiz<sup>15</sup>, K. Graf<sup>13</sup>, C. Guidi<sup>5,28</sup>, S. Hallmann<sup>13</sup>, H. van Haren<sup>29</sup>, A. J. Heijboer<sup>15</sup>, Y. Hello<sup>30</sup>, J. J. Hernández-Rey<sup>3</sup>, J. Hössl<sup>13</sup>, J. Hofestädt<sup>13</sup>, F. Huang<sup>7</sup>, G. Illuminati<sup>8,21,25</sup>, C. W. James<sup>31</sup>,

B. Jisse-Jung<sup>15</sup>, M. De Jong<sup>15,32</sup>, P. de Jong<sup>15,17</sup>, M. Kadler<sup>33</sup>, O. Kalekin<sup>13</sup>, U. Katz<sup>13</sup>, N. R. Khan-Chowdhury<sup>3</sup>, A. Kouchner<sup>8</sup>, I. Kreykenbohm<sup>34</sup>, V. Kulikovskiy<sup>5</sup>, R. Lahmann<sup>13</sup>, R. Le Breton<sup>8</sup>, S. LeStum<sup>7</sup>, D. Lefèvre<sup>35</sup>, E. Leonora<sup>36</sup>, G. Levi<sup>21,25</sup>,

D. Lopez-Coto<sup>37</sup>, S. Loucatos<sup>8,38</sup>, L. Maderer<sup>8</sup>, J. Manczak<sup>3</sup>, M. Marcelin<sup>9</sup>, A. Margiotta<sup>21,25</sup>, A. Marinelli<sup>39</sup>, J. A. Martínez-Mora<sup>6</sup>, B. Martino<sup>7</sup>, K. Melis<sup>15,17</sup>, P. Migliozzi<sup>39</sup>, A. Moussa<sup>14</sup>, R. Muller<sup>15</sup>, L. Nauta<sup>15</sup>, S. Navas<sup>37</sup>, E. Nezri<sup>9</sup>, B. Ó. Fearraigh<sup>15</sup>,

A. Păun<sup>16</sup>, G. E. Păvălaş<sup>16</sup>, C. Pellegrino<sup>21,40,41</sup>, M. Perrin-Terrin<sup>7</sup>, V. Pestel<sup>15</sup>, P. Piattelli<sup>12</sup>, C. Pieterse<sup>3</sup>, C. Poirè<sup>6</sup>, V. Popa<sup>16</sup>, T. Pradier<sup>1</sup>, N. Randazzo<sup>36</sup>, D. Real<sup>3</sup>, S. Reck<sup>13</sup>, G. Riccobene<sup>12</sup>, A. Romanov<sup>5,28</sup>, A. Sánchez-Losa<sup>3,22</sup>, F. Salesa Greus<sup>3</sup>,

D. F. E. Samtleben<sup>15,32</sup>, M. Sanguineti<sup>5,28</sup>, P. Sapienza<sup>12</sup>, J. Schnabel<sup>13</sup>, J. Schumann<sup>13</sup>, F. Schüssler<sup>38</sup>, J. Seneca<sup>15</sup>, M. Spurio<sup>21,25</sup>, Th. Stolarczyk<sup>38</sup>,

M. Taiuti<sup>5,28</sup>, Y. Tayalati<sup>11</sup>, S. J. Tingay<sup>31</sup>, B. Vallage<sup>8,38</sup>, V. Van Elewyck<sup>8,42</sup>, F. Versari<sup>8,21,25</sup>, S. Viola<sup>12</sup>, D. Vivolo<sup>39,43</sup>, J. Wilms<sup>34</sup>, S. Zavatarelli<sup>5</sup>, A. Zegarelli<sup>18,19</sup>, J. D. Zornoza<sup>3</sup>, J. Zúñiga<sup>3</sup>

The IceCube collaboration,

R. Abbasi<sup>44</sup>, M. Ackermann<sup>45</sup>, J. Adams<sup>46</sup>, J. A. Aguilar<sup>47</sup>, M. Ahlers<sup>48</sup>, M. Ahrens<sup>49</sup>, J. M. Alameddine<sup>50</sup>, C. Alispach<sup>51</sup>,

A. A. Alves Jr.<sup>52</sup>, N. M. Amin<sup>53</sup>, K. Andeen<sup>54</sup>, T. Anderson<sup>55</sup>, G. Anton<sup>56</sup>, C. Argüelles<sup>57</sup>, Y. Ashida<sup>58</sup>, S. Axani<sup>59</sup>, X. Bai<sup>60</sup>,

A. Balagopal V.<sup>58</sup>, A. Barbano<sup>51</sup>, S. W. Barwick<sup>61</sup>, B. Bastian<sup>45</sup>, V. Basu<sup>58</sup>, S. Baur<sup>47</sup>, R. Bay<sup>62</sup>, J. J. Beatty<sup>63,64</sup>, K.-H. Becker<sup>65</sup>, J. Becker Tjus<sup>66</sup> C. Bellenghi<sup>67</sup>, S. BenZvi<sup>68</sup>, D. Berley<sup>69</sup>, E. Bernardini<sup>45,70</sup>, D. Z. Besson<sup>71</sup>, G. Binder<sup>62,72</sup>, D. Bindig<sup>65</sup>,

E. Blaufuss<sup>69</sup>, S. Blot<sup>45</sup>, M. Boddenberg<sup>73</sup>, F. Bontempo<sup>52</sup>, J. Borowka<sup>73</sup>, S. Böser<sup>74</sup>, O. Botner<sup>75</sup>, J. Böttcher<sup>73</sup>, E. Bourbeau<sup>48</sup>, F. Bradascio<sup>45</sup>, J. Braun<sup>58</sup>, B. Brinson<sup>76</sup>, S. Bron<sup>51</sup>, J. Brostean-Kaiser<sup>45</sup>, S. Browne<sup>77</sup>, A. Burgman<sup>75</sup>, R. T. Burley<sup>78</sup>, R. S. Busse<sup>79</sup>,

M. A. Campana<sup>80</sup>, E. G. Carnie-Bronca<sup>78</sup>, C. Chen<sup>76</sup>, Z. Chen<sup>81</sup>, D. Chirkin<sup>58</sup>, K. Choi<sup>82</sup>, B. A. Clark<sup>83</sup>, K. Clark<sup>84</sup>, L. Classen<sup>79</sup>, G. H. Collin<sup>59</sup>, J. M. Conrad<sup>59</sup>, P. Coppin<sup>85</sup>, P. Correa<sup>85</sup>, D. F. Cowen<sup>55,86</sup>, R. Cross<sup>68</sup>, C. Dappen<sup>73</sup>, P. Dave<sup>76</sup>, C. De Clercq<sup>85</sup>,

J. J. DeLaunay<sup>87</sup>, D. Delgado López<sup>57</sup>, H. Dembinski<sup>53</sup>, K. Deoskar<sup>49</sup>, A. Desai<sup>58</sup>, P. Desiati<sup>58</sup>, K. D. de Vries<sup>85</sup>, G. de Wasseige<sup>88</sup>, M. de With<sup>89</sup>, T. DeYoung<sup>83</sup>, A. Diaz<sup>59</sup>, J. C. Díaz-Vélez<sup>58</sup>, M. Dittmer<sup>79</sup>, H. Dujmovic<sup>52</sup>, M. Dunkman<sup>55</sup>, M. A. DuVernois<sup>58</sup>,

E. Dvorak<sup>60</sup>, T. Ehrhardt<sup>74</sup>, P. Eller<sup>67</sup>, H. Erpenbeck<sup>73</sup>, J. Evans<sup>69</sup>, P. A. Evenson<sup>53</sup>, K. L. Fan<sup>69</sup>, A. R. Fazely<sup>90</sup>, A. Fedynitch<sup>91</sup>, N. Feigl<sup>89</sup> S. Fiedlschuster<sup>56</sup> A. T. Fienberg<sup>55</sup> K. Filimonov<sup>62</sup> C. Finley<sup>49</sup> L. Fischer<sup>45</sup> D. Fox<sup>86</sup> A. Franckowiak<sup>45,66</sup>,

E. Friedman<sup>69</sup>, A. Fritz<sup>74</sup>, P. Fürst<sup>73</sup>, T. K. Gaisser<sup>53</sup>, J. Gallagher<sup>92</sup>, E. Ganster<sup>73</sup>, A. Garcia<sup>57</sup>, S. Garrappa<sup>45</sup>, L. Gerhardt<sup>72</sup>, A. Ghadimi<sup>87</sup> C. Glaser<sup>75</sup> T. Glauch<sup>67</sup> T. Glüsenkamp<sup>56</sup> J. G. Gonzalez<sup>53</sup> S. Goswami<sup>87</sup> D. Grant<sup>83</sup> T. Grégoire<sup>55</sup>,

S. Griswold<sup>68</sup> C. Günther<sup>73</sup> P. Gutjahr<sup>50</sup> C. Haack<sup>67</sup> A. Hallgren<sup>75</sup> R. Halliday<sup>83</sup> L. Halve<sup>73</sup> F. Halzen<sup>58</sup> M. Ha Minh<sup>67</sup>, K. Hanson<sup>58</sup> J. Hardin<sup>58</sup> A. A. Harnisch<sup>83</sup> A. Haungs<sup>52</sup> D. Hebecker<sup>89</sup> K. Helbing<sup>65</sup> F. Henningsen<sup>67</sup> E. C. Hettinger<sup>83</sup>,

S. Hickford<sup>65</sup> J. Hignight<sup>93</sup> C. Hill<sup>94</sup> G. C. Hill<sup>78</sup> K. D. Hoffman<sup>69</sup> R. Hoffmann<sup>65</sup> B. Hokanson-Fasig<sup>58</sup> K. Hoshina<sup>58,95</sup>, M. Huber<sup>67</sup> T. Huber<sup>52</sup> K. Hultqvist<sup>49</sup> M. Hünnefeld<sup>50</sup> R. Hussain<sup>58</sup> K. Hymon<sup>50</sup> S. In<sup>82</sup> N. Iovine<sup>47</sup> A. Ishihara<sup>94</sup>,

M. Jansson<sup>49</sup>, G. S. Japaridze<sup>96</sup>, M. Jeong<sup>82</sup> M. Jin<sup>57</sup> B. J. P. Jones<sup>97</sup> D. Kang<sup>52</sup> W. Kang<sup>82</sup> X. Kang<sup>80</sup> A. Kappes<sup>79</sup>,

D. Kappesser<sup>74</sup> L. Kardum<sup>50</sup> T. Karg<sup>45</sup> M. Karl<sup>67</sup> A. Karle<sup>58</sup> M. Kauer<sup>58</sup> M. Kellermann<sup>73</sup> J. L. Kelley<sup>58</sup> A. Kheirandish<sup>55</sup>,

K. Kin<sup>94</sup> T. Kintscher<sup>45</sup> J. Kiryluk<sup>81</sup> S. R. Klein<sup>62,72</sup> R. Koirala<sup>53</sup> H. Kolanoski<sup>89</sup> T. Kontrimas<sup>67</sup> L. Köpke<sup>74</sup> C. Kopper<sup>83</sup>, S. Kopper<sup>87</sup> D. J. Koskinen<sup>48</sup> P. Koundal<sup>52</sup> M. Kovacevich<sup>80</sup> M. Kowalski<sup>45,89</sup> T. Kozynets<sup>48</sup> E. Kun<sup>66</sup> N. Kurahashi<sup>80</sup>, N. Lad<sup>45</sup> C. Lagunas Gualda<sup>45</sup> J. L. Lanfranchi<sup>55</sup> M. J. Larson<sup>69</sup> F. Lauber<sup>65</sup> J. P. Lazar<sup>57,58</sup> J. W. Lee<sup>82</sup> K. Leonard<sup>58</sup>,

A. Leszczyńska<sup>77</sup> Y. Li<sup>55</sup> M. Lincetto<sup>66</sup> Q. R. Liu<sup>58</sup> M. Liubarska<sup>93</sup> E. Lohfink<sup>74</sup> C. J. Lozano Mariscal<sup>79</sup> F. Lucarelli<sup>51</sup>, A. Ludwig<sup>83,98</sup> W. Luszczak<sup>58</sup> Y. Lyu<sup>62,72</sup> W. Y. Ma<sup>45</sup> J. Madsen<sup>58</sup> K. B. M. Mahn<sup>83</sup> Y. Makino<sup>58</sup> S. Mancina<sup>58</sup>,

I. Martinez-Soler<sup>57</sup> R. Maruyama<sup>99</sup> K. Mase<sup>94</sup> T. McElroy<sup>93</sup> F. McNally<sup>100</sup> J. V. Mead<sup>48</sup> K. Meagher<sup>58</sup> S. Mechbal<sup>45</sup>, A. Medina<sup>64</sup> M. Meier<sup>94</sup> S. Meighen-Berger<sup>67</sup> J. Micallef<sup>83</sup> D. Mockler<sup>47</sup> T. Montaruli<sup>51</sup> R. W. Moore<sup>93</sup> R. Morse<sup>58</sup>,

M. Moulai<sup>59</sup> R. Naab<sup>45</sup> R. Nagai<sup>94</sup> U. Naumann<sup>65</sup> J. Necker<sup>45</sup> L. V. Nguyen<sup>83</sup> H. Niederhausen<sup>83</sup> M. U. Nisa<sup>83</sup>,

S. C. Nowicki<sup>83</sup> A. Obertacke Pollmann<sup>65</sup> M. Oehler<sup>52</sup> B. Oeyen<sup>101</sup> A. Olivas<sup>69</sup> E. O'Sullivan<sup>75</sup> H. Pandya<sup>53</sup> D. V. Pankova<sup>55</sup> N. Park<sup>84</sup> G. K. Parker<sup>97</sup> E. N. Paudel<sup>53</sup> L. Paul<sup>54</sup> C. Pérez de los Heros<sup>75</sup> L. Peters<sup>73</sup> J. Peterson<sup>58</sup> S. Philippen<sup>73</sup> S. Pieper<sup>65</sup>,

M. Pittermann<sup>77</sup> A. Pizzuto<sup>58</sup> M. Plum<sup>54</sup> Y. Popovych<sup>74</sup> A. Porcelli<sup>101</sup> M. Prado Rodriguez<sup>58</sup> P. B. Price<sup>62</sup> B. Pries<sup>83</sup>, G. T. Przybylski<sup>72</sup> C. Raab<sup>47</sup> J. Rack-Helleis<sup>74</sup> A. Raissi<sup>46</sup> M. Rameez<sup>48</sup> K. Rawlins<sup>102</sup> I. C. Rea<sup>67</sup> A. Rehman<sup>53</sup>,

P. Reichherzer<sup>66</sup> R. Reimann<sup>73</sup> G. Renzi<sup>47</sup> E. Resconi<sup>67</sup> S. Reusch<sup>45</sup> W. Rhode<sup>50</sup> M. Richman<sup>80</sup> B. Riedel<sup>58</sup> E. J. Roberts<sup>78</sup>,
S. Robertson$^{62,72}$, G. Roellinghoff$^{82}$, M. Rongen$^{74}$, C. Rott$^{82,103}$, T. Ruhe$^{50}$, D. Ryckbosch$^{101}$, D. Rysewyk Cantu$^{83}$, I. Safa$^{57,58}$, J. Saffer$^{77}$, S. E. Sanchez Herrera$^{83}$, A. Sandrock$^{50}$, M. Santander$^{87}$, S. Sarkar$^{104}$, S. Sarkar$^{93}$, K. Satalecka$^{45}$, M. Schaufel$^{73}$,

S. Schindler$^{56}$, T. Schmidt$^{69}$, A. Schneider$^{58}$, J. Schneider$^{56}$, F. G. Schröder$^{52,53}$, L. Schumacher$^{67}$, G. Schwefer$^{73}$, S. Sclafani$^{80}$,

D. Seckel$^{53}$, S. Seunarine$^{105}$, A. Sharma$^{75}$, S. Shefali$^{77}$, M. Silva$^{58}$, B. Skrzypek$^{57}$, B. Smithers$^{97}$, R. Snihur$^{58}$, J. Soedingrekso$^{50}$, D. Soldin$^{53}$, C. Spannfellner$^{67}$, G. M. Spiczak$^{105}$, C. Spiering$^{45}$, J. Stachurska$^{45}$, M. Stamatikos$^{64}$, T. Stanev$^{53}$, R. Stein$^{45}$, J. Stettner$^{73}$, A. Steuer$^{74}$, T. Stezelberger$^{72}$, T. Stürwald$^{65}$, T. Stuttard$^{48}$, G. W. Sullivan$^{69}$, I. Taboada$^{76}$, S. Ter-Antonyan$^{90}$,

S. Tilav$^{53}$, F. Tischbein$^{73}$, K. Tollefson$^{83}$, C. Tönnis$^{106}$, S. Toscano$^{47}$, D. Tosi$^{58}$, A. Trettin$^{45}$, M. Tselengidou$^{56}$, C. F. Tung$^{76}$,

A. Turcati$^{67}$, R. Turcotte$^{52}$, C. F. Turley$^{55}$, J. P. Twagirayezu$^{83}$, B. Ty$^{58}$, M. A. Unland Elorrieta$^{79}$, N. Valtonen-Mattila$^{75}$,

J. Vandenbroucke$^{58}$, N. van Eijndhoven$^{85}$, D. Vannerom$^{59}$, J. van Santen$^{45}$, S. Verpoest$^{101}$, C. Walck$^{49}$, T. B. Watson$^{97}$,

C. Weaver$^{83}$, P. Weigel$^{59}$, A. Weindl$^{52}$, M. J. Weiss$^{55}$, J. Weldert$^{74}$, C. Wendt$^{58}$, J. Werthebach$^{50}$, M. Weyrauch$^{77}$,

N. Whitehorn$^{83,98}$, C. H. Wiebusch$^{73}$, D. R. Williams$^{87}$, M. Wolf$^{67}$, K. Woschnagg$^{62}$, G. Wrede$^{56}$, J. Wulff$^{66}$, X. W. Xu$^{90}$,

J. P. Yanez$^{93}$, S. Yoshida$^{94}$, S. Yu$^{83}$, T. Yuan$^{58}$, Z. Zhang$^{81}$, P. Zhelnin$^{57}$

The Pierre Auger Collaboration,

P. Abreu$^{107}$, M. Aglietta$^{108,109}$, J. M. Albury$^{110}$, I. Allekotte$^{111}$, K. Almeida Cheminant$^{112}$, A. Almela$^{113,114}$, J. Alvarez-Muñiz$^{115}$, R. Alves Batista$^{116}$, G. A. Anastasi$^{108,109}$, L. Anchordoqui$^{117}$, B. Andrada$^{113}$, S. Andringa$^{107}$, C. Aramo$^{118}$,

P. R. Araújo Ferreira$^{119}$, E. Arnone$^{109,120}$, J. C. Arteaga Velázquez$^{121}$, H. Asorey$^{113}$, P. Assis$^{107}$, G. Avila$^{122}$, A. M. Badescu$^{123}$,

A. Bakalova$^{124}$, A. Balaceanu$^{125}$, F. Barbato$^{126,127}$, J. A. Bellido$^{110,128}$, C. Berat$^{129}$, M. E. Bertaina$^{109,120}$, X. Bertou$^{111}$,

G. Bhatta$^{112}$, P. L. Biermann$^{130}$, V. Binet$^{131}$, K. Bismark$^{113,132}$, T. Bister$^{119}$, J. Biteau$^{133}$, J. Blazek$^{124}$, C. Bleve$^{129}$, J. Blümer$^{134}$,

M. Boháčová$^{124}$, D. Boncioli$^{127,135}$, C. Bonifazi$^{136,137}$, L. Bonneau Arbeletche$^{138}$, N. Borodai$^{112}$, A. M. Botti$^{113}$, J. Brack$^{139}$,

T. Bretz$^{119}$, P. G. Brichetto Orchera$^{113}$, F. L. Briechle$^{119}$, P. Buchholz$^{140}$, A. Bueno$^{141}$, S. Buitink$^{142}$, M. Buscemi$^{143}$,

M. Büsken$^{113,132}$, K. S. Caballero-Mora$^{144}$, L. Caccianiga$^{145,146}$, F. Canfora$^{116,147}$, I. Caracas$^{148}$, R. Caruso$^{143,149}$,

A. Castellina$^{108,109}$, F. Catalani$^{150}$, G. Cataldi$^{151}$, L. Cazon$^{115}$, M. Cerda$^{152}$, J. A. Chinellato$^{138}$, J. Chudoba$^{124}$, L. Chytka$^{153}$,

R. W. Clay$^{110}$, A. C. Cobos Cerutti$^{154}$, R. Colalillo$^{118,155}$, A. Coleman$^{156}$, M. R. Coluccia$^{151}$, R. Conceição$^{107}$, A. Condorelli$^{126,127}$,

G. Consolati$^{146,157}$, F. Contreras$^{122}$, F. Convenga$^{134}$, D. Correia dos Santos$^{158}$, C. E. Covault$^{159}$, S. Dasso$^{160,161}$, K. Daumiller$^{134}$,

B. R. Dawson$^{110}$, J. A. Day$^{110}$, R. M. de Almeida$^{158}$, J. de Jesús$^{113,134}$, S. J. de Jong$^{116,147}$, J. R. T. de Mello Neto$^{137,162}$,

I. De Mitri$^{126,127}$, J. de Oliveira$^{163}$, D. de Oliveira Franco$^{138}$, F. de Palma$^{151,164}$, V. de Souza$^{165}$, E. De Vito$^{151,164}$,

A. Del Popolo$^{143,149}$, M. del Río$^{122}$, O. Deligny$^{166}$, L. Deval$^{113,134}$, A. di Matteo$^{109}$, M. Dobre$^{125}$, C. Dobrigkeit$^{138}$,

J. C. D'Olivo$^{167}$, L. M. Domingues Mendes$^{107}$, R. C. dos Anjos$^{168}$, M. T. Dova$^{169}$, J. Ebr$^{124}$, R. Engel$^{132,134}$, I. Epicoco$^{151,164}$,

M. Erdmann$^{119}$, C. O. Escobar$^{170}$, A. Etchegoyen$^{113,114}$, H. Falcke$^{116,147,171}$, J. Farmer$^{172}$, G. Farrar$^{173}$, A. C. Fauth$^{138}$,

N. Fazzini$^{170}$, F. Feldbusch$^{174}$, F. Fenu$^{109,120}$, B. Fick$^{175}$, J. M. Figueira$^{113}$, A. Filipčič$^{176,177}$, T. Fitoussi$^{134}$, T. Fodran$^{116}$,

T. Fujii$^{172,178}$, A. Fuster$^{113,114}$, C. Galea$^{116}$, C. Galelli$^{145,146}$, B. García$^{154}$, A. L. Garcia Vegas$^{119}$, H. Gemmeke$^{174}$,

F. Gesualdi$^{113,134}$, A. Gherghel-Lascu$^{125}$, P. L. Ghia$^{166}$, U. Giaccari$^{116}$, M. Giammarchi$^{146}$, J. Glombitza$^{119}$, F. Gobbi$^{152}$,

F. Gollan$^{113}$, G. Golup$^{111}$, M. Gómez Berisso$^{111}$, P. F. Gómez Vitale$^{122}$, J. P. Gongora$^{122}$, J. M. González$^{111}$, N. González$^{179}$,

I. Goos$^{111,134}$, D. Góra$^{112}$, A. Gorgi$^{108,109}$, M. Gottowik$^{148}$, T. D. Grubb$^{110}$, F. Guarino$^{118,155}$, G. P. Guedes$^{180}$, E. Guido$^{109,120}$,

S. Hahn$^{113,134}$, P. Hamal$^{124}$, M. R. Hampel$^{113}$, P. Hansen$^{169}$, D. Harari$^{111}$, V. M. Harvey$^{110}$, T. Hebbeker$^{119}$, D. Heck$^{134}$,

C. Hojvat$^{170}$, J. R. Hörandel$^{116,147}$, P. Horvath$^{153}$, M. Hrabovsky$^{153}$, T. Huege$^{134,142}$, A. Insolia$^{143,149}$, P. G. Isar$^{181}$, P. Janecek$^{124}$,

J. A. Johnsen$^{182}$, J. Jurysek$^{124}$, A. Kääpä$^{148}$, K. H. Kampert$^{148}$, N. Karastathis$^{134}$, B. Keilhauer$^{134}$, A. Khakurdikar$^{116}$,

V. V. Kizakke Covilakam$^{113,134}$, H. O. Klages$^{134}$, M. Kleifges$^{174}$, J. Kleinfeller$^{152}$, F. Knapp$^{132}$, N. Kunka$^{174}$, B. L. Lago$^{183}$,

R. G. Lang$^{165}$, N. Langner$^{119}$, M. A. Leigui de Oliveira$^{184}$, V. Lenok$^{134}$, A. Letessier-Selvon$^{185}$, I. Lhenry-Yvon$^{166}$,

D. Lo Presti$^{143,149}$, L. Lopes$^{107}$, R. López$^{186}$, L. Lu$^{187}$, Q. Luce$^{132}$, J. P. Lundquist$^{177}$, A. Machado Payeras$^{138}$,

G. Mancarella$^{151,164}$, D. Mandat$^{124}$, B. C. Manning$^{110}$, J. Manshanden$^{188}$, P. Mantsch$^{170}$, S. Marafico$^{166}$, F. M. Mariani$^{145,146}$,

A. G. Mariazzi$^{169}$, I. C. Mariš$^{179}$, G. Marsella$^{143,189}$, D. Martello$^{151,164}$, S. Martinelli$^{113,134}$, O. Martínez Bravo$^{186}$,

M. Mastrodicasa$^{127,135}$, H. J. Mathes$^{134}$, J. Matthews$^{190}$, G. Matthiae$^{191,192}$, E. Mayotte$^{148,182}$, S. Mayotte$^{182}$, P. O. Mazur$^{170}$,

G. Medina-Tanco$^{167}$, D. Melo$^{113}$, A. Menshikov$^{174}$, S. Michal$^{153}$, M. I. Micheletti$^{131}$, L. Miramonti$^{145,146}$, S. Mollerach$^{111}$,

F. Montanet$^{129}$, L. Morejon$^{148}$, C. Morello$^{108,109}$, M. Mostafá$^{193}$, A. L. Müller$^{124}$, M. A. Muller$^{138}$, K. Mulrey$^{116,147}$, R. Mussa$^{109}$,

M. Muzio$^{173}$, W. M. Namasaka$^{148}$, A. Nasr-Esfahani$^{148}$, L. Nellen$^{167}$, G. Nicora$^{194}$, M. Niculescu-Oglinzanu$^{125}$, M. Niechciol$^{140}$,

D. Nitz$^{175}$, D. Nosek$^{195}$, V. Novotny$^{195}$, L. Nožka$^{153}$, A. Nucita$^{151,164}$, L. A. Núñez$^{196}$, C. Oliveira$^{165}$, M. Palatka$^{124}$, J. Pallotta$^{194}$,

P. Papenbreer$^{148}$, G. Parente$^{115}$, A. Parra$^{186}$, J. Pawlowsky$^{148}$, M. Pech$^{124}$, J. Pěkala$^{112}$, R. Pelayo$^{197}$, J. Peña-Rodriguez$^{196}$,

E. E. Pereira Martins$^{113,132}$, J. Perez Armand$^{198}$, C. Pérez Bertolli$^{113,134}$, M. Perlin$^{113,134}$, L. Perrone$^{151,164}$, S. Petrera$^{126,127}$,

C. Petrucci$^{127,135}$, T. Pierog$^{134}$, M. Pimenta$^{107}$, V. Pirronello$^{143,149}$, M. Platino$^{113}$, B. Pont$^{116}$, M. Pothast$^{116,147}$, P. Privitera$^{172}$,

M. Prouza$^{124}$, A. Puyleart$^{175}$, S. Querchfeld$^{148}$, J. Rautenberg$^{148}$, D. Ravignani$^{113}$, M. Reininghaus$^{113,134}$, J. Ridky$^{124}$, F. Riehn$^{107}$,

M. Risse$^{140}$, V. Rizi$^{127,135}$, W. Rodrigues de Carvalho$^{116}$, J. Rodriguez Rojo$^{122}$, M. J. Roncoroni$^{113}$, S. Rossoni$^{188}$, M. Roth$^{134}$,

E. Roulet$^{111}$, A. C. Rovero$^{160}$, P. Ruehl$^{140}$, A. Saftoiu$^{125}$, M. Saharan$^{116}$, F. Salamida$^{127,135}$, H. Salazar$^{186}$, G. Salina$^{192}$,

J. D. Sanabria Gomez$^{196}$, F. Sánchez$^{113}$, E. M. Santos$^{198}$, E. Santos$^{124}$, F. Sarazin$^{182}$, R. Sarmento$^{107}$, C. Sarmiento-Cano$^{113}$,

R. Sato$^{122}$, P. Savina$^{187}$, C. M. Schäfer$^{134}$, V. Scherini$^{151,164}$, H. Schieler$^{134}$, M. Schimassek$^{113,132}$, M. Schimp$^{148}$,

F. Schlüter$^{113,134}$, D. Schmidt$^{132}$, O. Scholten$^{142}$, H. Schoorlemmer$^{116,147}$, P. Schovánek$^{124}$, J. Schulte$^{119}$, T. Schulz$^{134}$,

S. J. Sciutto$^{169}$, M. Scornavacche$^{113,134}$, A. Segreto$^{143,199}$, S. Sehgal$^{148}$, R. C. Shellard$^{200}$, G. Sigl$^{188}$, G. Silli$^{113,134}$, O. Sima$^{125,201}$,
S. Robertson$^{62,72}$, G. Roellinghoff$^{82}$, M. Rongen$^{74}$, C. Rott$^{82,103}$, T. Ruhe$^{50}$, D. Ryckbosch$^{101}$, D. Rysewyk Cantu$^{83}$, I. Safa$^{57,58}$, J. Saffer$^{77}$, S. E. Sanchez Herrera$^{83}$, A. Sandrock$^{50}$, M. Santander$^{87}$, S. Sarkar$^{104}$, S. Sarkar$^{93}$, K. Satalecka$^{45}$, M. Schaufel$^{73}$, S. Schindler$^{56}$, T. Schmidt$^{69}$, A. Schneider$^{58}$, J. Schneider$^{56}$, F. G. Schröder$^{52,53}$, L. Schumacher$^{67}$, G. Schwefer$^{73}$, S. Sclafani$^{80}$, D. Seckel$^{53}$, S. Seunarine$^{105}$, A. Sharma$^{75}$, S. Shefali$^{77}$, M. Silva$^{58}$, B. Skrzypek$^{57}$, B. Smithers$^{97}$, R. Snihur$^{58}$, J. Soedingrekso$^{50}$, D. Soldin$^{53}$, C. Spannfellner$^{67}$, G. M. Spiczak$^{105}$, C. Spiering$^{45}$, J. Stachurska$^{45}$, M. Stamatikos$^{64}$, T. Stanev$^{53}$, R. Stein$^{45}$, J. Stettner$^{73}$, A. Steuer$^{74}$, T. Stezelberger$^{72}$, T. Stürwald$^{65}$, T. Stuttard$^{48}$, G. W. Sullivan$^{69}$, I. Taboada$^{76}$, S. Ter-Antonyan$^{90}$, S. Tilav$^{53}$, F. Tischbein$^{73}$, K. Tollefson$^{83}$, C. Tönnis$^{106}$, S. Toscano$^{47}$, D. Tosi$^{58}$, A. Trettin$^{45}$, M. Tselengidou$^{56}$, C. F. Tung$^{76}$, A. Turcati$^{67}$, R. Turcotte$^{52}$, C. F. Turley$^{55}$, J. P. Twagirayezu$^{83}$, B. Ty$^{58}$, M. A. Unland Elorrieta$^{79}$, N. Valtonen-Mattila$^{75}$, J. Vandenbroucke$^{58}$, N. van Eijndhoven$^{85}$, D. Vannerom$^{59}$, J. van Santen$^{45}$, S. Verpoest$^{101}$, C. Walck$^{49}$, T. B. Watson$^{97}$, C. Weaver$^{83}$, P. Weigel$^{59}$, A. Weindl$^{52}$, M. J. Weiss$^{55}$, J. Weldert$^{74}$, C. Wendt$^{58}$, J. Werthebach$^{50}$, M. Weyrauch$^{77}$, N. Whitehorn$^{83,98}$, C. H. Wiebusch$^{73}$, D. R. Williams$^{87}$, M. Wolf$^{67}$, K. Woschnagg$^{62}$, G. Wrede$^{56}$, J. Wulff$^{66}$, X. W. Xu$^{90}$, J. P. Yanez$^{93}$, S. Yoshida$^{94}$, S. Yu$^{83}$, T. Yuan$^{58}$, Z. Zhang$^{81}$, P. Zhelnin$^{57}$

The Pierre Auger Collaboration,

P. Abreu$^{107}$, M. Aglietta$^{108,109}$, J. M. Albury$^{110}$, I. Allekotte$^{111}$, K. Almeida Cheminant$^{112}$, A. Almela$^{113,114}$, J. Alvarez-Muñiz$^{115}$, R. Alves Batista$^{116}$, G. A. Anastasi$^{108,109}$, L. Anchordoqui$^{117}$, B. Andrada$^{113}$, S. Andringa$^{107}$, C. Aramo$^{118}$,

P. R. Araújo Ferreira$^{119}$, E. Arnone$^{109,120}$, J. C. Arteaga Velázquez$^{121}$, H. Asorey$^{113}$, P. Assis$^{107}$, G. Avila$^{122}$, A. M. Badescu$^{123}$, A. Bakalova$^{124}$, A. Balaceanu$^{125}$, F. Barbato$^{126,127}$, J. A. Bellido$^{110,128}$, C. Berat$^{129}$, M. E. Bertaina$^{109,120}$, X. Bertou$^{111}$,

G. Bhatta$^{112}$, P. L. Biermann$^{130}$, V. Binet$^{131}$, K. Bismark$^{113,132}$, T. Bister$^{119}$, J. Biteau$^{133}$, J. Blazek$^{124}$, C. Bleve$^{129}$, J. Blümer$^{134}$, M. Boháčová$^{124}$, D. Boncioli$^{127,135}$, C. Bonifazi$^{136,137}$, L. Bonneau Arbeletche$^{138}$, N. Borodai$^{112}$, A. M. Botti$^{113}$, J. Brack$^{139}$, T. Bretz$^{119}$, P. G. Brichetto Orchera$^{113}$, F. L. Briechle$^{119}$, P. Buchholz$^{140}$, A. Bueno$^{141}$, S. Buitink$^{142}$, M. Buscemi$^{143}$, M. Büsken$^{113,132}$, K. S. Caballero-Mora$^{144}$, L. Caccianiga$^{145,146}$, F. Canfora$^{116,147}$, I. Caracas$^{148}$, R. Caruso$^{143,149}$,

A. Castellina$^{108,109}$, F. Catalani$^{150}$, G. Cataldi$^{151}$, L. Cazon$^{115}$, M. Cerda$^{152}$, J. A. Chinellato$^{138}$, J. Chudoba$^{124}$, L. Chytka$^{153}$,

R. W. Clay$^{110}$, A. C. Cobos Cerutti$^{154}$, R. Colalillo$^{118,155}$, A. Coleman$^{156}$, M. R. Coluccia$^{151}$, R. Conceição$^{107}$, A. Condorelli$^{126,127}$,

G. Consolati$^{146,157}$, F. Contreras$^{122}$, F. Convenga$^{134}$, D. Correia dos Santos$^{158}$, C. E. Covault$^{159}$, S. Dasso$^{160,161}$, K. Daumiller$^{134}$, B. R. Dawson$^{110}$, J. A. Day$^{110}$, R. M. de Almeida$^{158}$, J. de Jesús$^{113,134}$, S. J. de Jong$^{116,147}$, J. R. T. de Mello Neto$^{137,162}$,

I. De Mitri$^{126,127}$, J. de Oliveira$^{163}$, D. de Oliveira Franco$^{138}$, F. de Palma$^{151,164}$, V. de Souza$^{165}$, E. De Vito$^{151,164}$,

A. Del Popolo$^{143,149}$, M. del Río$^{122}$, O. Deligny$^{166}$, L. Deval$^{113,134}$, A. di Matteo$^{109}$, M. Dobre$^{125}$, C. Dobrigkeit$^{138}$,

J. C. D'Olivo$^{167}$, L. M. Domingues Mendes$^{107}$, R. C. dos Anjos$^{168}$, M. T. Dova$^{169}$, J. Ebr$^{124}$, R. Engel$^{132,134}$, I. Epicoco$^{151,164}$, M. Erdmann$^{119}$, C. O. Escobar$^{170}$, A. Etchegoyen$^{113,114}$, H. Falcke$^{116,147,171}$, J. Farmer$^{172}$, G. Farrar$^{173}$, A. C. Fauth$^{138}$,

N. Fazzini$^{170}$, F. Feldbusch$^{174}$, F. Fenu$^{109,120}$, B. Fick$^{175}$, J. M. Figueira$^{113}$, A. Filipčič$^{176,177}$, T. Fitoussi$^{134}$, T. Fodran$^{116}$,

T. Fujii$^{172,178}$, A. Fuster$^{113,114}$, C. Galea$^{116}$, C. Galelli$^{145,146}$, B. García$^{154}$, A. L. Garcia Vegas$^{119}$, H. Gemmeke$^{174}$,

F. Gesualdi$^{113,134}$, A. Gherghel-Lascu$^{125}$, P. L. Ghia$^{166}$, U. Giaccari$^{116}$, M. Giammarchi$^{146}$, J. Glombitza$^{119}$, F. Gobbi$^{152}$,

F. Gollan$^{113}$, G. Golup$^{111}$, M. Gómez Berisso$^{111}$, P. F. Gómez Vitale$^{122}$, J. P. Gongora$^{122}$, J. M. González$^{111}$, N. González$^{179}$,

I. Goos$^{111,134}$, D. Góra$^{112}$, A. Gorgi$^{108,109}$, M. Gottowik$^{148}$, T. D. Grubb$^{110}$, F. Guarino$^{118,155}$, G. P. Guedes$^{180}$, E. Guido$^{109,120}$, S. Hahn$^{113,134}$, P. Hamal$^{124}$, M. R. Hampel$^{113}$, P. Hansen$^{169}$, D. Harari$^{111}$, V. M. Harvey$^{110}$, T. Hebbeker$^{119}$, D. Heck$^{134}$,

C. Hojvat$^{170}$, J. R. Hörandel$^{116,147}$, P. Horvath$^{153}$, M. Hrabovsky$^{153}$, T. Huege$^{134,142}$, A. Insolia$^{143,149}$, P. G. Isar$^{181}$, P. Janecek$^{124}$, J. A. Johnsen$^{182}$, J. Jurysek$^{124}$, A. Kääpä$^{148}$, K. H. Kampert$^{148}$, N. Karastathis$^{134}$, B. Keilhauer$^{134}$, A. Khakurdikar$^{116}$,

V. V. Kizakke Covilakam$^{113,134}$, H. O. Klages$^{134}$, M. Kleifges$^{174}$, J. Kleinfeller$^{152}$, F. Knapp$^{132}$, N. Kunka$^{174}$, B. L. Lago$^{183}$,

R. G. Lang$^{165}$, N. Langner$^{119}$, M. A. Leigui de Oliveira$^{184}$, V. Lenok$^{134}$, A. Letessier-Selvon$^{185}$, I. Lhenry-Yvon$^{166}$,

D. Lo Presti$^{143,149}$, L. Lopes$^{107}$, R. López$^{186}$, L. Lu$^{187}$, Q. Luce$^{132}$, J. P. Lundquist$^{177}$, A. Machado Payeras$^{138}$,

G. Mancarella$^{151,164}$, D. Mandat$^{124}$, B. C. Manning$^{110}$, J. Manshanden$^{188}$, P. Mantsch$^{170}$, S. Marafico$^{166}$, F. M. Mariani$^{145,146}$, A. G. Mariazzi$^{169}$, I. C. Mariš$^{179}$, G. Marsella$^{143,189}$, D. Martello$^{151,164}$, S. Martinelli$^{113,134}$, O. Martínez Bravo$^{186}$,

M. Mastrodicasa$^{127,135}$, H. J. Mathes$^{134}$, J. Matthews$^{190}$, G. Matthiae$^{191,192}$, E. Mayotte$^{148,182}$, S. Mayotte$^{182}$, P. O. Mazur$^{170}$,

G. Medina-Tanco$^{167}$, D. Melo$^{113}$, A. Menshikov$^{174}$, S. Michal$^{153}$, M. I. Micheletti$^{131}$, L. Miramonti$^{145,146}$, S. Mollerach$^{111}$,

F. Montanet$^{129}$, L. Morejon$^{148}$, C. Morello$^{108,109}$, M. Mostafá$^{193}$, A. L. Müller$^{124}$, M. A. Muller$^{138}$, K. Mulrey$^{116,147}$, R. Mussa$^{109}$,

M. Muzio$^{173}$, W. M. Namasaka$^{148}$, A. Nasr-Esfahani$^{148}$, L. Nellen$^{167}$, G. Nicora$^{194}$, M. Niculescu-Oglinzanu$^{125}$, M. Niechciol$^{140}$,

D. Nitz$^{175}$, D. Nosek$^{195}$, V. Novotny$^{195}$, L. Nožka$^{153}$, A. Nucita$^{151,164}$, L. A. Núñez$^{196}$, C. Oliveira$^{165}$, M. Palatka$^{124}$, J. Pallotta$^{194}$, P. Papenbreer$^{148}$, G. Parente$^{115}$, A. Parra$^{186}$, J. Pawlowsky$^{148}$, M. Pech$^{124}$, J. Pěkala$^{112}$, R. Pelayo$^{197}$, J. Peña-Rodriguez$^{196}$, E. E. Pereira Martins$^{113,132}$, J. Perez Armand$^{198}$, C. Pérez Bertolli$^{113,134}$, M. Perlin$^{113,134}$, L. Perrone$^{151,164}$, S. Petrera$^{126,127}$,

C. Petrucci$^{127,135}$, T. Pierog$^{134}$, M. Pimenta$^{107}$, V. Pirronello$^{143,149}$, M. Platino$^{113}$, B. Pont$^{116}$, M. Pothast$^{116,147}$, P. Privitera$^{172}$,

M. Prouza$^{124}$, A. Puyleart$^{175}$, S. Querchfeld$^{148}$, J. Rautenberg$^{148}$, D. Ravignani$^{113}$, M. Reininghaus$^{113,134}$, J. Ridky$^{124}$, F. Riehn$^{107}$,

M. Risse$^{140}$, V. Rizi$^{127,135}$, W. Rodrigues de Carvalho$^{116}$, J. Rodriguez Rojo$^{122}$, M. J. Roncoroni$^{113}$, S. Rossoni$^{188}$, M. Roth$^{134}$, E. Roulet$^{111}$, A. C. Rovero$^{160}$, P. Ruehl$^{140}$, A. Saftoiu$^{125}$, M. Saharan$^{116}$, F. Salamida$^{127,135}$, H. Salazar$^{186}$, G. Salina$^{192}$,

J. D. Sanabria Gomez$^{196}$, F. Sánchez$^{113}$, E. M. Santos$^{198}$, E. Santos$^{124}$, F. Sarazin$^{182}$, R. Sarmento$^{107}$, C. Sarmiento-Cano$^{113}$,

R. Sato$^{122}$, P. Savina$^{187}$, C. M. Schäfer$^{134}$, V. Scherini$^{151,164}$, H. Schieler$^{134}$, M. Schimassek$^{113,132}$, M. Schimp$^{148}$,

F. Schlüter$^{113,134}$, D. Schmidt$^{132}$, O. Scholten$^{142}$, H. Schoorlemmer$^{116,147}$, P. Schovánek$^{124}$, J. Schulte$^{119}$, T. Schulz$^{134}$,

S. J. Sciutto$^{169}$, M. Scornavacche$^{113,134}$, A. Segreto$^{143,199}$, S. Sehgal$^{148}$, R. C. Shellard$^{200}$, G. Sigl$^{188}$, G. Silli$^{113,134}$, O. Sima$^{125,201}$,
R. Smau<sup>125</sup>, R. Šmída<sup>172</sup>, P. Sommers<sup>193</sup>, J. F. Soriano<sup>117</sup>, R. Squartini<sup>152</sup>, M. Stadelmaier<sup>113,134</sup>, D. Stanca<sup>125</sup>, S. Stanič<sup>177</sup>, J. Stasielak<sup>112</sup>, P. Stassi<sup>129</sup>, A. Streich<sup>113,132</sup>, M. Suárez-Durán<sup>179</sup>, T. Sudholz<sup>110</sup>, T. Suomijärvi<sup>133</sup>, A. D. Supanitsky<sup>113</sup>, Z. Szadkowski<sup>202</sup>, A. Tapia<sup>203</sup>, C. Taricco<sup>109,120</sup>, C. Timmermans<sup>116,147</sup>, O. Tkachenko<sup>134</sup>, P. Tobiska<sup>124</sup>, C. J. Todero Peixoto<sup>150</sup>, B. Tome<sup>107</sup>, Z. Torrès<sup>129</sup>, A. Travaini<sup>152</sup>, P. Travnicek<sup>124</sup>, C. Trimarelli<sup>127,135</sup>, M. Tueros<sup>169</sup>, R. Ulrich<sup>134</sup>, M. Unger<sup>134</sup>, L. Vaclavek<sup>153</sup>, M. Vacula<sup>153</sup>, J. F. Valdés Galicia<sup>167</sup>, L. Valore<sup>118,155</sup>, E. Varela<sup>186</sup>, A. Vásquez-Ramírez<sup>196</sup>, D. Veberič<sup>134</sup>, C. Ventura<sup>162</sup>, I. D. Vergara Quispe<sup>169</sup>, V. Verzi<sup>192</sup>, J. Vicha<sup>124</sup>, J. Vink<sup>204</sup>, S. Vorobiov<sup>177</sup>, H. Wahlberg<sup>169</sup>, C. Watanabe<sup>137</sup>, A. A. Watson<sup>205</sup>, L. Wiencke<sup>182</sup>, H. Wilczyński<sup>112</sup>, D. Wittkowski<sup>148</sup>, B. Wundheiler<sup>113</sup>, A. Yushkov<sup>124</sup>, O. Zapparrata<sup>179</sup>, E. Zas<sup>115</sup>, D. Zavrtanik<sup>176,177</sup>, M. Zavrtanik<sup>176,177</sup>, L. Zehrer<sup>177</sup>

The Telescope Array collaboration,

T. Abu-Zayyad<sup>206,207</sup>, M. Allen<sup>207</sup>, Y. Arai<sup>208</sup>, R. Arimura<sup>208</sup>, E. Barcikowski<sup>207</sup>, J. W. Belz<sup>207</sup>, D. R. Bergman<sup>207</sup>, S. A. Blake<sup>207</sup>, I. Buckland<sup>207</sup>, R. Cady<sup>207</sup>, B. G. Cheon<sup>209</sup>, J. Chiba<sup>210</sup>, M. Chikawa<sup>211</sup>, K. Fujisue<sup>211</sup>, K. Fujita<sup>208</sup>, R. Fujiwara<sup>208</sup>, M. Fukushima<sup>211</sup>, R. Fukushima<sup>208</sup>, G. Furlich<sup>207</sup>, N. Globus<sup>212,213,214</sup>, R. Gonzalez<sup>207</sup>, W. Hanlon<sup>207</sup>, M. Hayashi<sup>215</sup>, N. Hayashida<sup>216</sup>, H. He<sup>214</sup>, K. Hibino<sup>216</sup>, R. Higuchi<sup>211</sup>, K. Honda<sup>217</sup>, D. Ikeda<sup>216</sup>, T. Inadomi<sup>215</sup>, N. Inoue<sup>218</sup>, T. Ishii<sup>217</sup>, H. Ito<sup>214</sup>, D. Ivanov<sup>207</sup>, H. Iwakura<sup>215</sup>, A. Iwasaki<sup>208</sup>, H. M. Jeong<sup>219</sup>, S. Jeong<sup>219</sup>, C. C. H. Jui<sup>207</sup>, K. Kadota<sup>220</sup>, F. Kakimoto<sup>216</sup>, O. Kalashev<sup>221</sup>, K. Kasahara<sup>222</sup>, S. Kasami<sup>223</sup>, S. Kawakami<sup>208</sup>, S. Kawana<sup>218</sup>, K. Kawata<sup>211</sup>, I. Kharuk<sup>221</sup>, E. Kido<sup>214</sup>, H. B. Kim<sup>209</sup>, J. H. Kim<sup>207</sup>, J. H. Kim<sup>207</sup>, S. W. Kim<sup>219</sup>, Y. Kimura<sup>208</sup>, S. Kishigami<sup>208</sup>, Y. Kubota<sup>215</sup>, S. Kurisu<sup>215</sup>, V. Kuzmin<sup>221,239</sup>, M. Kuznetsov<sup>221,224</sup>, Y. J. Kwon<sup>225</sup>, K. H. Lee<sup>219</sup>, B. Lubsandorzhiev<sup>221</sup>, K. Machida<sup>217</sup>, H. Matsumiya<sup>208</sup>, T. Matsuyama<sup>208</sup>, J. N. Matthews<sup>207</sup>, R. Mayta<sup>208</sup>, M. Minamino<sup>208</sup>, K. Mukai<sup>217</sup>, I. Myers<sup>207</sup>, S. Nagataki<sup>214</sup>, K. Nakai<sup>208</sup>, R. Nakamura<sup>215</sup>, T. Nakamura<sup>226</sup>, T. Nakamura<sup>215</sup>, Y. Nakamura<sup>215</sup>, A. Nakazawa<sup>215</sup>, E. Nishio<sup>223</sup>, T. Nonaka<sup>211</sup>, H. Oda<sup>208</sup>, S. Ogio<sup>208,227</sup>, M. Ohnishi<sup>211</sup>, H. Ohoka<sup>211</sup>, Y. Oku<sup>223</sup>, T. Okuda<sup>228</sup>, Y. Omura<sup>208</sup>, M. Ono<sup>214</sup>, R. Onogi<sup>208</sup>, A. Oshima<sup>229</sup>, S. Ozawa<sup>230</sup>, I. H. Park<sup>219</sup>, M. Potts<sup>207</sup>, M. S. Pshirkov<sup>221,231</sup>, J. Remington<sup>207</sup>, D. C. Rodriguez<sup>207</sup>, G. I. Rubtsov<sup>221</sup>, D. Ryu<sup>232</sup>, H. Sagawa<sup>211</sup>, R. Sahara<sup>208</sup>, Y. Saito<sup>215</sup>, N. Sakaki<sup>211</sup>, T. Sako<sup>211</sup>, N. Sakurai<sup>208</sup>, K. Sano<sup>215</sup>, K. Sato<sup>208</sup>, T. Seki<sup>215</sup>, K. Sekino<sup>211</sup>, P. D. Shah<sup>207</sup>, Y. Shibasaki<sup>215</sup>, F. Shibata<sup>217</sup>, N. Shibata<sup>223</sup>, T. Shibata<sup>211</sup>, H. Shimodaira<sup>211</sup>, B. K. Shin<sup>232</sup>, H. S. Shin<sup>211</sup>, D. Shinto<sup>223</sup>, J. D. Smith<sup>207</sup>, P. Sokolsky<sup>207</sup>, N. Sone<sup>215</sup>, B. T. Stokes<sup>207</sup>, T. A. Stroman<sup>207</sup>, Y. Takagi<sup>208</sup>, Y. Takahashi<sup>208</sup>, M. Takamura<sup>210</sup>, M. Takeda<sup>211</sup>, R. Takeishi<sup>211</sup>, A. Taketa<sup>233</sup>, M. Takita<sup>211</sup>, Y. Tameda<sup>223</sup>, H. Tanaka<sup>208</sup>, K. Tanaka<sup>234</sup>, M. Tanaka<sup>235</sup>, Y. Tanoue<sup>208</sup>, S. B. Thomas<sup>207</sup>, G. B. Thomson<sup>207</sup>, P. Tinyakov<sup>221,224</sup>, I. Tkachev<sup>221</sup>, H. Tokuno<sup>236</sup>, T. Tomida<sup>215</sup>, S. Troitsky<sup>221</sup>, R. Tsuda<sup>208</sup>, Y. Tsunesada<sup>208,227</sup>, Y. Uchihori<sup>237</sup>, S. Udo<sup>216</sup>, T. Uehama<sup>215</sup>, F. Urban<sup>238</sup>, D. Warren<sup>214</sup>, T. Wong<sup>207</sup>, M. Yamamoto<sup>215</sup>, K. Yamazaki<sup>229</sup>, K. Yashiro<sup>210</sup>, F. Yoshida<sup>223</sup>, Y. Yoshioka<sup>215</sup>, Y. Zhezher<sup>211,221</sup>, and Z. Zundel<sup>207</sup>

<footnote n="1">¹ Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France</footnote>

<footnote n="2">² Université de Haute Alsace, F-68100 Mulhouse, France</footnote>

<footnote n="3">³ IFIC—Instituto de Física Corpuscular (CSIC—Universitat de València) c/ Catedrático José Beltrán, 2 E-46980 Paterna, Valencia, Spain</footnote>

<footnote n="4">⁴ Technical University of Catalonia, Laboratory of Applied Bioacoustics, Rambla Exposició, E-08800 Vilanova i la Geltrú, Barcelona, Spain</footnote>

<footnote n="5">⁵ INFN—Sezione di Genova, Via Dodecaneso 33, I-16146 Genova, Italy</footnote>

<footnote n="6">⁶ Institut d'Investigació per a la Gestió Integrada de les Zones Costaneres (IGIC) - Universitat Politècnica de València. C/ Paranimf 1, E-46730 Gandia, Spain</footnote>

<footnote>Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France</footnote>

<footnote n="7">⁷ Université de Paris, CNRS, Astroparticule et Cosmologie, F-75013 Paris, France</footnote>

<footnote n="8">⁸ Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France</footnote>

<footnote n="10">¹⁰ National Center for Energy Sciences and Nuclear Techniques, B.P.1382, R.P.10001 Rabat, Morocco</footnote>

<footnote n="11">¹¹ University Mohammed V in Rabat, Faculty of Sciences, 4 av. Ibn Battouta, B.P. 1014, R.P. 10000, Rabat, Morocco</footnote>

<footnote n="12">¹² INFN—Laboratori Nazionali del Sud (LNS), Via S. Sofia 62, I-95123 Catania, Italy</footnote>

<footnote n="13">¹³ Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen Centre for Astroparticle Physics, Erwin-Rommel-Str. 1, D-91058 Erlangen, Germany</footnote>

<footnote n="14">¹⁴ University Mohammed I, Laboratory of Physics of Matter and Radiations, B.P.717, Oujda 6000, Morocco</footnote>

<footnote n="15">¹⁵ Nikhef, Science Park, Amsterdam, The Netherlands</footnote>

<footnote n="16">¹⁶ Institute of Space Science, RO-077125 Bucharest, Măgurele, Romania</footnote>

<footnote n="17">¹⁷ Universiteit van Amsterdam, Instituut voor Hoge-Energie Fysica, Science Park 105, 1098 XG Amsterdam, The Netherlands</footnote>

<footnote n="18">¹⁸ INFN—Sezione di Roma, P.le Aldo Moro 2, I-00185 Roma, Italy</footnote>

<footnote n="19">¹⁹ Dipartimento di Fisica dell'Università La Sapienza, P.le Aldo Moro 2, I-00185 Roma, Italy</footnote>

<footnote n="20">²⁰ LPHEA, Faculty of Science—Semlalı, Cadi Ayyad University, P.O.B. 2390, Marrakech, Morocco</footnote>

<footnote n="21">²¹ INFN—Sezione di Bologna, Viale Berti-Pichat 6/2, I-40127 Bologna, Italy</footnote>

<footnote n="22">²² INFN—Sezione di Bari, Via E. Orabona 4, I-70126 Bari, Italy</footnote>

<footnote n="23">²³ Department of Computer Architecture and Technology/CITIC, University of Granada, E-18071 Granada, Spain</footnote>

<footnote n="24">²⁴ Université Paris-Sud, F-91405 Orsay Cedex, France</footnote>

<footnote n="25">²⁵ Dipartimento di Fisica e Astronomia dell'Università, Viale Berti Pichat 6/2, I-40127 Bologna, Italy</footnote>

<footnote n="26">²⁶ Laboratoire de Physique Corpusculaire, Clermont Université, Université Blaise Pascal, CNRS/IN2P3, BP 10448, F-63000 Clermont-Ferrand, France</footnote>

<footnote n="27">²⁷ LIS, UMR Université de Toulon, Aix Marseille Université, CNRS, F-83041 Toulon, France</footnote>

<footnote n="28">²⁸ Dipartimento di Fisica dell'Università, Via Dodecaneso 33, I-16146 Genova, Italy</footnote>

<footnote n="29">²⁹ Royal Netherlands Institute for Sea Research (NIOZ), Landsdiep 4, 1797 SZ 't Horntje (Texel), The Netherlands</footnote>

<footnote n="30">³⁰ Géoazur, UCA, CNRS, IRD, Observatoire de la Côte d'Azur, Sophia Antipolis, France</footnote>

<footnote n="31">³¹ International Centre for Radio Astronomy Research—Curtin University, Bentley, WA 6102, Australia</footnote>

<footnote n="32">³² Huygens-Kamerlingh Onnes Laboratorium, Universiteit Leiden, The Netherlands</footnote>

<footnote n="33">³³ Institut für Theoretische Physik und Astrophysik, Universität Würzburg, Emil-Fischer Str. 31, D-97074 Würzburg, Germany</footnote>

<footnote n="34">³⁴ Dr. Remeis-Sternwarte und ECAP, Friedrich-Alexander-Universität Erlangen-Nürnberg, Sternwartstr. 7, D-96049 Bamberg, Germany</footnote>

<footnote n="35">³⁵ Mediterranean Institute of Oceanography (MIO), Aix-Marseille University, 13288, Marseille, Cedex 9, France; Université du Sud Toulon-Var, CNRS-INSU/IRD UM 110, F-83957, La Garde Cedex, France</footnote>

<footnote n="36">³⁶ INFN—Sezione di Catania, Via S. Sofia 64, I-95123 Catania, Italy</footnote>

<footnote n="37">³⁷ Dpto. de Física Teórica y del Cosmos & C.A.F.P.E., University of Granada, E-18071 Granada, Spain</footnote>

<footnote n="38">³⁸ IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France</footnote>
R. Smau<sup>125</sup>, R. Šmída<sup>172</sup>, P. Sommers<sup>193</sup>, J. F. Soriano<sup>117</sup>, R. Squartini<sup>152</sup>, M. Stadelmaier<sup>113,134</sup>, D. Stanca<sup>125</sup>, S. Stanič<sup>177</sup>, J. Stasielak<sup>112</sup>, P. Stassi<sup>129</sup>, A. Streich<sup>113,132</sup>, M. Suárez-Durán<sup>179</sup>, T. Sudholz<sup>110</sup>, T. Suomijärvi<sup>133</sup>, A. D. Supanitsky<sup>113</sup>, Z. Szadkowski<sup>202</sup>, A. Tapia<sup>203</sup>, C. Taricco<sup>109,120</sup>, C. Timmermans<sup>116,147</sup>, O. Tkachenko<sup>134</sup>, P. Tobiska<sup>124</sup>, C. J. Todero Peixoto<sup>150</sup>, B. Tome<sup>107</sup>, Z. Torrès<sup>129</sup>, A. Travaini<sup>152</sup>, P. Travnicek<sup>124</sup>, C. Trimarelli<sup>127,135</sup>, M. Tueros<sup>169</sup>, R. Ulrich<sup>134</sup>, M. Unger<sup>134</sup>, L. Vaclavek<sup>153</sup>, M. Vacula<sup>153</sup>, J. F. Valdés Galicia<sup>167</sup>, L. Valore<sup>118,155</sup>, E. Varela<sup>186</sup>, A. Vásquez-Ramírez<sup>196</sup>, D. Veberič<sup>134</sup>, C. Ventura<sup>162</sup>, I. D. Vergara Quispe<sup>169</sup>, V. Verzi<sup>192</sup>, J. Vicha<sup>124</sup>, J. Vink<sup>204</sup>, S. Vorobiov<sup>177</sup>, H. Wahlberg<sup>169</sup>, C. Watanabe<sup>137</sup>, A. A. Watson<sup>205</sup>, L. Wiencke<sup>182</sup>, H. Wilczyński<sup>112</sup>, D. Wittkowski<sup>148</sup>, B. Wundheiler<sup>113</sup>, A. Yushkov<sup>124</sup>, O. Zapparrata<sup>179</sup>, E. Zas<sup>115</sup>, D. Zavrtanik<sup>176,177</sup>, M. Zavrtanik<sup>176,177</sup>, L. Zehrer<sup>177</sup>

The Telescope Array collaboration,

T. Abu-Zayyad<sup>206,207</sup>, M. Allen<sup>207</sup>, Y. Arai<sup>208</sup>, R. Arimura<sup>208</sup>, E. Barcikowski<sup>207</sup>, J. W. Belz<sup>207</sup>, D. R. Bergman<sup>207</sup>, S. A. Blake<sup>207</sup>, I. Buckland<sup>207</sup>, R. Cady<sup>207</sup>, B. G. Cheon<sup>209</sup>, J. Chiba<sup>210</sup>, M. Chikawa<sup>211</sup>, K. Fujisue<sup>211</sup>, K. Fujita<sup>208</sup>, R. Fujiwara<sup>208</sup>, M. Fukushima<sup>211</sup>, R. Fukushima<sup>208</sup>, G. Furlich<sup>207</sup>, N. Globus<sup>212,213,214</sup>, R. Gonzalez<sup>207</sup>, W. Hanlon<sup>207</sup>, M. Hayashi<sup>215</sup>, N. Hayashida<sup>216</sup>, H. He<sup>214</sup>, K. Hibino<sup>216</sup>, R. Higuchi<sup>211</sup>, K. Honda<sup>217</sup>, D. Ikeda<sup>216</sup>, T. Inadomi<sup>215</sup>, N. Inoue<sup>218</sup>, T. Ishii<sup>217</sup>, H. Ito<sup>214</sup>, D. Ivanov<sup>207</sup>, H. Iwakura<sup>215</sup>, A. Iwasaki<sup>208</sup>, H. M. Jeong<sup>219</sup>, S. Jeong<sup>219</sup>, C. C. H. Jui<sup>207</sup>, K. Kadota<sup>220</sup>, F. Kakimoto<sup>216</sup>, O. Kalashev<sup>221</sup>, K. Kasahara<sup>222</sup>, S. Kasami<sup>223</sup>, S. Kawakami<sup>208</sup>, S. Kawana<sup>218</sup>, K. Kawata<sup>211</sup>, I. Kharuk<sup>221</sup>, E. Kido<sup>214</sup>, H. B. Kim<sup>209</sup>, J. H. Kim<sup>207</sup>, J. H. Kim<sup>207</sup>, S. W. Kim<sup>219</sup>, Y. Kimura<sup>208</sup>, S. Kishigami<sup>208</sup>, Y. Kubota<sup>215</sup>, S. Kurisu<sup>215</sup>, V. Kuzmin<sup>221,239</sup>, M. Kuznetsov<sup>221,224</sup>, Y. J. Kwon<sup>225</sup>, K. H. Lee<sup>219</sup>, B. Lubsandorzhiev<sup>221</sup>, K. Machida<sup>217</sup>, H. Matsumiya<sup>208</sup>, T. Matsuyama<sup>208</sup>, J. N. Matthews<sup>207</sup>, R. Mayta<sup>208</sup>, M. Minamino<sup>208</sup>, K. Mukai<sup>217</sup>, I. Myers<sup>207</sup>, S. Nagataki<sup>214</sup>, K. Nakai<sup>208</sup>, R. Nakamura<sup>215</sup>, T. Nakamura<sup>226</sup>, T. Nakamura<sup>215</sup>, Y. Nakamura<sup>215</sup>, A. Nakazawa<sup>215</sup>, E. Nishio<sup>223</sup>, T. Nonaka<sup>211</sup>, H. Oda<sup>208</sup>, S. Ogio<sup>208,227</sup>, M. Ohnishi<sup>211</sup>, H. Ohoka<sup>211</sup>, Y. Oku<sup>223</sup>, T. Okuda<sup>228</sup>, Y. Omura<sup>208</sup>, M. Ono<sup>214</sup>, R. Onogi<sup>208</sup>, A. Oshima<sup>229</sup>, S. Ozawa<sup>230</sup>, I. H. Park<sup>219</sup>, M. Potts<sup>207</sup>, M. S. Pshirkov<sup>221,231</sup>, J. Remington<sup>207</sup>, D. C. Rodriguez<sup>207</sup>, G. I. Rubtsov<sup>221</sup>, D. Ryu<sup>232</sup>, H. Sagawa<sup>211</sup>, R. Sahara<sup>208</sup>, Y. Saito<sup>215</sup>, N. Sakaki<sup>211</sup>, T. Sako<sup>211</sup>, N. Sakurai<sup>208</sup>, K. Sano<sup>215</sup>, K. Sato<sup>208</sup>, T. Seki<sup>215</sup>, K. Sekino<sup>211</sup>, P. D. Shah<sup>207</sup>, Y. Shibasaki<sup>215</sup>, F. Shibata<sup>217</sup>, N. Shibata<sup>223</sup>, T. Shibata<sup>211</sup>, H. Shimodaira<sup>211</sup>, B. K. Shin<sup>232</sup>, H. S. Shin<sup>211</sup>, D. Shinto<sup>223</sup>, J. D. Smith<sup>207</sup>, P. Sokolsky<sup>207</sup>, N. Sone<sup>215</sup>, B. T. Stokes<sup>207</sup>, T. A. Stroman<sup>207</sup>, Y. Takagi<sup>208</sup>, Y. Takahashi<sup>208</sup>, M. Takamura<sup>210</sup>, M. Takeda<sup>211</sup>, R. Takeishi<sup>211</sup>, A. Taketa<sup>233</sup>, M. Takita<sup>211</sup>, Y. Tameda<sup>223</sup>, H. Tanaka<sup>208</sup>, K. Tanaka<sup>234</sup>, M. Tanaka<sup>235</sup>, Y. Tanoue<sup>208</sup>, S. B. Thomas<sup>207</sup>, G. B. Thomson<sup>207</sup>, P. Tinyakov<sup>221,224</sup>, I. Tkachev<sup>221</sup>, H. Tokuno<sup>236</sup>, T. Tomida<sup>215</sup>, S. Troitsky<sup>221</sup>, R. Tsuda<sup>208</sup>, Y. Tsunesada<sup>208,227</sup>, Y. Uchihori<sup>237</sup>, S. Udo<sup>216</sup>, T. Uehama<sup>215</sup>, F. Urban<sup>238</sup>, D. Warren<sup>214</sup>, T. Wong<sup>207</sup>, M. Yamamoto<sup>215</sup>, K. Yamazaki<sup>229</sup>, K. Yashiro<sup>210</sup>, F. Yoshida<sup>223</sup>, Y. Yoshioka<sup>215</sup>, Y. Zhezher<sup>211,221</sup>, and Z. Zundel<sup>207</sup>

<footnote n="1">¹ Université de Strasbourg, CNRS, IPHC UMR 7178, F-67000 Strasbourg, France</footnote>

<footnote n="2">² Université de Haute Alsace, F-68100 Mulhouse, France</footnote>

<footnote n="3">³ IFIC—Instituto de Física Corpuscular (CSIC—Universitat de València) c/ Catedrático José Beltrán, 2 E-46980 Paterna, Valencia, Spain</footnote>

<footnote n="4">⁴ Technical University of Catalonia, Laboratory of Applied Bioacoustics, Rambla Exposició, E-08800 Vilanova i la Geltrú, Barcelona, Spain</footnote>

<footnote n="5">⁵ INFN—Sezione di Genova, Via Dodecaneso 33, I-16146 Genova, Italy</footnote>

<footnote n="6">⁶ Institut d'Investigació per a la Gestió Integrada de les Zones Costaneres (IGIC) - Universitat Politècnica de València. C/ Paranimf 1, E-46730 Gandia, Spain</footnote>

<footnote>Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France</footnote>

<footnote n="7">⁷ Université de Paris, CNRS, Astroparticule et Cosmologie, F-75013 Paris, France</footnote>

<footnote n="8">⁸ Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France</footnote>

<footnote n="10">¹⁰ National Center for Energy Sciences and Nuclear Techniques, B.P.1382, R.P.10001 Rabat, Morocco</footnote>

<footnote n="11">¹¹ University Mohammed V in Rabat, Faculty of Sciences, 4 av. Ibn Battouta, B.P. 1014, R.P. 10000, Rabat, Morocco</footnote>

<footnote n="12">¹² INFN—Laboratori Nazionali del Sud (LNS), Via S. Sofia 62, I-95123 Catania, Italy</footnote>

<footnote n="13">¹³ Friedrich-Alexander-Universität Erlangen-Nürnberg, Erlangen Centre for Astroparticle Physics, Erwin-Rommel-Str. 1, D-91058 Erlangen, Germany</footnote>

<footnote n="14">¹⁴ University Mohammed I, Laboratory of Physics of Matter and Radiations, B.P.717, Oujda 6000, Morocco</footnote>

<footnote n="15">¹⁵ Nikhef, Science Park, Amsterdam, The Netherlands</footnote>

<footnote n="16">¹⁶ Institute of Space Science, RO-077125 Bucharest, Măgurele, Romania</footnote>

<footnote n="17">¹⁷ Universiteit van Amsterdam, Instituut voor Hoge-Energie Fysica, Science Park 105, 1098 XG Amsterdam, The Netherlands</footnote>

<footnote n="18">¹⁸ INFN—Sezione di Roma, P.le Aldo Moro 2, I-00185 Roma, Italy</footnote>

<footnote n="19">¹⁹ Dipartimento di Fisica dell'Università La Sapienza, P.le Aldo Moro 2, I-00185 Roma, Italy</footnote>

<footnote n="20">²⁰ LPHEA, Faculty of Science—Semlalı, Cadi Ayyad University, P.O.B. 2390, Marrakech, Morocco</footnote>

<footnote n="21">²¹ INFN—Sezione di Bologna, Viale Berti-Pichat 6/2, I-40127 Bologna, Italy</footnote>

<footnote n="22">²² INFN—Sezione di Bari, Via E. Orabona 4, I-70126 Bari, Italy</footnote>

<footnote n="23">²³ Department of Computer Architecture and Technology/CITIC, University of Granada, E-18071 Granada, Spain</footnote>

<footnote n="24">²⁴ Université Paris-Sud, F-91405 Orsay Cedex, France</footnote>

<footnote n="25">²⁵ Dipartimento di Fisica e Astronomia dell'Università, Viale Berti Pichat 6/2, I-40127 Bologna, Italy</footnote>

<footnote n="26">²⁶ Laboratoire de Physique Corpusculaire, Clermont Université, Université Blaise Pascal, CNRS/IN2P3, BP 10448, F-63000 Clermont-Ferrand, France</footnote>

<footnote n="27">²⁷ LIS, UMR Université de Toulon, Aix Marseille Université, CNRS, F-83041 Toulon, France</footnote>

<footnote n="28">²⁸ Dipartimento di Fisica dell'Università, Via Dodecaneso 33, I-16146 Genova, Italy</footnote>

<footnote n="29">²⁹ Royal Netherlands Institute for Sea Research (NIOZ), Landsdiep 4, 1797 SZ 't Horntje (Texel), The Netherlands</footnote>

<footnote n="30">³⁰ Géoazur, UCA, CNRS, IRD, Observatoire de la Côte d'Azur, Sophia Antipolis, France</footnote>

<footnote n="31">³¹ International Centre for Radio Astronomy Research—Curtin University, Bentley, WA 6102, Australia</footnote>

<footnote n="32">³² Huygens-Kamerlingh Onnes Laboratorium, Universiteit Leiden, The Netherlands</footnote>

<footnote n="33">³³ Institut für Theoretische Physik und Astrophysik, Universität Würzburg, Emil-Fischer Str. 31, D-97074 Würzburg, Germany</footnote>

<footnote n="34">³⁴ Dr. Remeis-Sternwarte und ECAP, Friedrich-Alexander-Universität Erlangen-Nürnberg, Sternwartstr. 7, D-96049 Bamberg, Germany</footnote>

<footnote n="35">³⁵ Mediterranean Institute of Oceanography (MIO), Aix-Marseille University, 13288, Marseille, Cedex 9, France; Université du Sud Toulon-Var, CNRS-INSU/IRD UM 110, F-83957, La Garde Cedex, France</footnote>

<footnote n="36">³⁶ INFN—Sezione di Catania, Via S. Sofia 64, I-95123 Catania, Italy</footnote>

<footnote n="37">³⁷ Dpto. de Física Teórica y del Cosmos & C.A.F.P.E., University of Granada, E-18071 Granada, Spain</footnote>

<footnote n="38">³⁸ IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France</footnote>
³⁹ INFN—Sezione di Napoli, Via Cintia I-80126 Napoli, Italy

⁴⁰ Museo Storico della Fisica e Centro Studi e Ricerche Enrico Fermi, Piazza del Viminale 1, I-00184, Roma, Italy

⁴¹ INFN—CNAF, Viale C. Berti Pichat 6/2, 40127, Bologna

⁴² Institut Universitaire de France, F-75005 Paris, France

⁴³ Dipartimento di Fisica dell'Università Federico II di Napoli, Via Cintia I-80126, Napoli, Italy

⁴⁴ Department of Physics, Loyola University Chicago, Chicago, IL 60660, USA

⁴⁵ DESY, D-15738 Zeuthen, Germany

⁴⁶ Dept. of Physics and Astronomy, University of Canterbury, Private Bag 4800, Christchurch, New Zealand

⁴⁷ Université Libre de Bruxelles, Science Faculty CP230, B-1050 Brussels, Belgium

⁴⁸ Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark

⁴⁹ Oskar Klein Centre and Dept. of Physics, Stockholm University, SE-10691 Stockholm, Sweden

⁵⁰ Dept. of Physics, TU Dortmund University, D-44221 Dortmund, Germany

⁵¹ Département de physique nucléaire et corpusculaire, Université de Genève, CH-1211 Genève, Switzerland

⁵² Karlsruhe Institute of Technology, Institute for Astroparticle Physics, D-76021 Karlsruhe, Germany

⁵³ Bartol Research Institute and Dept. of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA

⁵⁴ Department of Physics, Marquette University, Milwaukee, WI 53201, USA

⁵⁵ Dept. of Physics, Pennsylvania State University, University Park, PA 16802, USA

⁵⁶ Erlangen Centre for Astroparticle Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg, D-91058 Erlangen, Germany

⁵⁷ Department of Physics and Laboratory for Particle Physics and Cosmology, Harvard University, Cambridge, MA 02138, USA

⁵⁸ Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center, University of Wisconsin–Madison, Madison, WI 53706, USA; analysis@icecube.wisc.edu

⁵⁹ Dept. of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

⁶⁰ Physics Department, South Dakota School of Mines and Technology, Rapid City, SD 57701, USA

⁶¹ Dept. of Physics and Astronomy, University of California, Irvine, CA 92697, USA

⁶² Dept. of Physics, University of California, Berkeley, CA 94720, USA

⁶³ Dept. of Astronomy, Ohio State University, Columbus, OH 43210, USA

⁶⁴ Dept. of Physics and Center for Cosmology and Astro-Particle Physics, Ohio State University, Columbus, OH 43210, USA

⁶⁵ Dept. of Physics, University of Wuppertal, D-42119 Wuppertal, Germany

⁶⁶ Fakultät für Physik & Astronomie, Ruhr-Universität Bochum, D-44780 Bochum, Germany

⁶⁷ Physik-department, Technische Universität München, D-85748 Garching, Germany

⁶⁸ Dept. of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA

⁶⁹ Dept. of Physics, University of Maryland, College Park, MD 20742, USA

⁷⁰ Also at Università di Padova, I-35131 Padova, Italy

⁷¹ Dept. of Physics and Astronomy, University of Kansas, Lawrence, KS 66045, USA

⁷² Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

⁷³ III. Physikalisches Institut, RWTH Aachen University, D-52056 Aachen, Germany

⁷⁴ Institute of Physics, University of Mainz, Staudinger Weg 7, D-55099 Mainz, Germany

⁷⁵ Dept. of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden

⁷⁶ School of Physics and Center for Relativistic Astrophysics, Georgia Institute of Technology, Atlanta, GA 30332, USA

⁷⁷ Karlsruhe Institute of Technology, Institute of Experimental Particle Physics, D-76021 Karlsruhe, Germany

⁷⁸ Department of Physics, University of Adelaide, Adelaide, 5005, Australia

⁷⁹ Institut für Kernphysik, Westfälische Wilhelms-Universität Münster, D-48149 Münster, Germany

⁸⁰ Dept. of Physics, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, USA

⁸¹ Dept. of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794-3800, USA

⁸² Dept. of Physics, Sungkyunkwan University, Suwon 16419, Republic of Korea

⁸³ Dept. of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA

⁸⁴ Dept. of Physics, Engineering Physics, and Astronomy, Queen's University, Kingston, ON K7L 3N6, Canada

⁸⁵ Vrije Universiteit Brussel (VUB), Dienst ELEM, B-1050 Brussels, Belgium

⁸⁶ Dept. of Astronomy and Astrophysics, Pennsylvania State University, University Park, PA 16802, USA

⁸⁷ Dept. of Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487, USA

⁸⁸ Centre for Cosmology, Particle Physics and Phenomenology—CP3, Université catholique de Louvain, Louvain-la-Neuve, Belgium

⁸⁹ Institut für Physik, Humboldt-Universität zu Berlin, D-12489 Berlin, Germany

⁹⁰ Dept. of Physics, Southern University, Baton Rouge, LA 70813, USA

⁹¹ Institute of Physics, Academia Sinica, Taipei, 11529, Taiwan

⁹² Dept. of Astronomy, University of Wisconsin–Madison, Madison, WI 53706, USA

⁹³ Dept. of Physics, University of Alberta, Edmonton, AB T6G 2E1, Canada

⁹⁴ Dept. of Physics and Institute for Global Prominent Research, Chiba University, Chiba 263-8522, Japan

⁹⁵ Also at Earthquake Research Institute, University of Tokyo, Bunkyo, Tokyo 113-0032, Japan

⁹⁶ CTSPS, Clark-Atlanta University, Atlanta, GA 30314, USA

⁹⁷ Dept. of Physics, University of Texas at Arlington, 502 Yates St., Science Hall Rm 108, Box 19059, Arlington, TX 76019, USA

⁹⁸ Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095, USA

⁹⁹ Dept. of Physics, Yale University, New Haven, CT 06520, USA

¹⁰⁰ Department of Physics, Mercer University, Macon, GA 31207-0001, USA

¹⁰¹ Dept. of Physics and Astronomy, University of Gent, B-9000 Gent, Belgium

¹⁰² Dept. of Physics and Astronomy, University of Alaska Anchorage, 3211 Providence Dr., Anchorage, AK 99508, USA

¹⁰³ Department of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112, USA

¹⁰⁴ Dept. of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, UK

¹⁰⁵ Dept. of Physics, University of Wisconsin, River Falls, WI 54022, USA

¹⁰⁶ Institute of Basic Science, Sungkyunkwan University, Suwon 16419, Republic of Korea

¹⁰⁷ Laboratório de Instrumentação e Física Experimental de Partículas—LIP and Instituto Superior Técnico—IST, Universidade de Lisboa—UL, Lisboa, Portugal

¹⁰⁸ Osservatorio Astrofisico di Torino (INAF), Torino, Italy

¹⁰⁹ INFN, Sezione di Torino, Torino, Italy

¹¹⁰ University of Adelaide, Adelaide, S.A., Australia

¹¹¹ Centro Atómico Bariloche and Instituto Balseiro (CNEA-UNCuyo-CONICET), San Carlos de Bariloche, Argentina

¹¹² Institute of Nuclear Physics PAN, Krakow, Poland

¹¹³ Instituto de Tecnologías en Deteción y Astropartículas (CNEA, CONICET, UNSAM), Buenos Aires, Argentina

¹¹⁴ Universidad Tecnológica Nacional—Facultad Regional Buenos Aires, Buenos Aires, Argentina
³⁹ INFN—Sezione di Napoli, Via Cintia I-80126 Napoli, Italy

⁴⁰ Museo Storico della Fisica e Centro Studi e Ricerche Enrico Fermi, Piazza del Viminale 1, I-00184, Roma, Italy

⁴¹ INFN—CNAF, Viale C. Berti Pichat 6/2, 40127, Bologna

⁴² Institut Universitaire de France, F-75005 Paris, France

⁴³ Dipartimento di Fisica dell'Università Federico II di Napoli, Via Cintia I-80126, Napoli, Italy

⁴⁴ Department of Physics, Loyola University Chicago, Chicago, IL 60660, USA

⁴⁵ DESY, D-15738 Zeuthen, Germany

⁴⁶ Dept. of Physics and Astronomy, University of Canterbury, Private Bag 4800, Christchurch, New Zealand

⁴⁷ Université Libre de Bruxelles, Science Faculty CP230, B-1050 Brussels, Belgium

⁴⁸ Niels Bohr Institute, University of Copenhagen, DK-2100 Copenhagen, Denmark

⁴⁹ Oskar Klein Centre and Dept. of Physics, Stockholm University, SE-10691 Stockholm, Sweden

⁵⁰ Dept. of Physics, TU Dortmund University, D-44221 Dortmund, Germany

⁵¹ Département de physique nucléaire et corpusculaire, Université de Genève, CH-1211 Genève, Switzerland

⁵² Karlsruhe Institute of Technology, Institute for Astroparticle Physics, D-76021 Karlsruhe, Germany

⁵³ Bartol Research Institute and Dept. of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA

⁵⁴ Department of Physics, Marquette University, Milwaukee, WI 53201, USA

⁵⁵ Dept. of Physics, Pennsylvania State University, University Park, PA 16802, USA

⁵⁶ Erlangen Centre for Astroparticle Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg, D-91058 Erlangen, Germany

⁵⁷ Department of Physics and Laboratory for Particle Physics and Cosmology, Harvard University, Cambridge, MA 02138, USA

⁵⁸ Dept. of Physics and Wisconsin IceCube Particle Astrophysics Center, University of Wisconsin–Madison, Madison, WI 53706, USA; analysis@icecube.wisc.edu

⁵⁹ Dept. of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

⁶⁰ Physics Department, South Dakota School of Mines and Technology, Rapid City, SD 57701, USA

⁶¹ Dept. of Physics and Astronomy, University of California, Irvine, CA 92697, USA

⁶² Dept. of Physics, University of California, Berkeley, CA 94720, USA

⁶³ Dept. of Astronomy, Ohio State University, Columbus, OH 43210, USA

⁶⁴ Dept. of Physics and Center for Cosmology and Astro-Particle Physics, Ohio State University, Columbus, OH 43210, USA

⁶⁵ Dept. of Physics, University of Wuppertal, D-42119 Wuppertal, Germany

⁶⁶ Fakultät für Physik & Astronomie, Ruhr-Universität Bochum, D-44780 Bochum, Germany

⁶⁷ Physik-department, Technische Universität München, D-85748 Garching, Germany

⁶⁸ Dept. of Physics and Astronomy, University of Rochester, Rochester, NY 14627, USA

⁶⁹ Dept. of Physics, University of Maryland, College Park, MD 20742, USA

⁷⁰ Also at Università di Padova, I-35131 Padova, Italy

⁷¹ Dept. of Physics and Astronomy, University of Kansas, Lawrence, KS 66045, USA

⁷² Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

⁷³ III. Physikalisches Institut, RWTH Aachen University, D-52056 Aachen, Germany

⁷⁴ Institute of Physics, University of Mainz, Staudinger Weg 7, D-55099 Mainz, Germany

⁷⁵ Dept. of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden

⁷⁶ School of Physics and Center for Relativistic Astrophysics, Georgia Institute of Technology, Atlanta, GA 30332, USA

⁷⁷ Karlsruhe Institute of Technology, Institute of Experimental Particle Physics, D-76021 Karlsruhe, Germany

⁷⁸ Department of Physics, University of Adelaide, Adelaide, 5005, Australia

⁷⁹ Institut für Kernphysik, Westfälische Wilhelms-Universität Münster, D-48149 Münster, Germany

⁸⁰ Dept. of Physics, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, USA

⁸¹ Dept. of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794-3800, USA

⁸² Dept. of Physics, Sungkyunkwan University, Suwon 16419, Republic of Korea

⁸³ Dept. of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA

⁸⁴ Dept. of Physics, Engineering Physics, and Astronomy, Queen's University, Kingston, ON K7L 3N6, Canada

⁸⁵ Vrije Universiteit Brussel (VUB), Dienst ELEM, B-1050 Brussels, Belgium

⁸⁶ Dept. of Astronomy and Astrophysics, Pennsylvania State University, University Park, PA 16802, USA

⁸⁷ Dept. of Physics and Astronomy, University of Alabama, Tuscaloosa, AL 35487, USA

⁸⁸ Centre for Cosmology, Particle Physics and Phenomenology—CP3, Université catholique de Louvain, Louvain-la-Neuve, Belgium

⁸⁹ Institut für Physik, Humboldt-Universität zu Berlin, D-12489 Berlin, Germany

⁹⁰ Dept. of Physics, Southern University, Baton Rouge, LA 70813, USA

⁹¹ Institute of Physics, Academia Sinica, Taipei, 11529, Taiwan

⁹² Dept. of Astronomy, University of Wisconsin–Madison, Madison, WI 53706, USA

⁹³ Dept. of Physics, University of Alberta, Edmonton, AB T6G 2E1, Canada

⁹⁴ Dept. of Physics and Institute for Global Prominent Research, Chiba University, Chiba 263-8522, Japan

⁹⁵ Also at Earthquake Research Institute, University of Tokyo, Bunkyo, Tokyo 113-0032, Japan

⁹⁶ CTSPS, Clark-Atlanta University, Atlanta, GA 30314, USA

⁹⁷ Dept. of Physics, University of Texas at Arlington, 502 Yates St., Science Hall Rm 108, Box 19059, Arlington, TX 76019, USA

⁹⁸ Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095, USA

⁹⁹ Dept. of Physics, Yale University, New Haven, CT 06520, USA

¹⁰⁰ Department of Physics, Mercer University, Macon, GA 31207-0001, USA

¹⁰¹ Dept. of Physics and Astronomy, University of Gent, B-9000 Gent, Belgium

¹⁰² Dept. of Physics and Astronomy, University of Alaska Anchorage, 3211 Providence Dr., Anchorage, AK 99508, USA

¹⁰³ Department of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112, USA

¹⁰⁴ Dept. of Physics, University of Oxford, Parks Road, Oxford OX1 3PU, UK

¹⁰⁵ Dept. of Physics, University of Wisconsin, River Falls, WI 54022, USA

¹⁰⁶ Institute of Basic Science, Sungkyunkwan University, Suwon 16419, Republic of Korea

¹⁰⁷ Laboratório de Instrumentação e Física Experimental de Partículas—LIP and Instituto Superior Técnico—IST, Universidade de Lisboa—UL, Lisboa, Portugal

¹⁰⁸ Osservatorio Astrofisico di Torino (INAF), Torino, Italy

¹⁰⁹ INFN, Sezione di Torino, Torino, Italy

¹¹⁰ University of Adelaide, Adelaide, S.A., Australia

¹¹¹ Centro Atómico Bariloche and Instituto Balseiro (CNEA-UNCuyo-CONICET), San Carlos de Bariloche, Argentina

¹¹² Institute of Nuclear Physics PAN, Krakow, Poland

¹¹³ Instituto de Tecnologías en Deteción y Astropartículas (CNEA, CONICET, UNSAM), Buenos Aires, Argentina

¹¹⁴ Universidad Tecnológica Nacional—Facultad Regional Buenos Aires, Buenos Aires, Argentina
¹¹⁵ Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de Santiago de Compostela, Santiago de Compostela, Spain

¹¹⁶ IMAPP, Radboud University Nijmegen, Nijmegen, The Netherlands

¹¹⁷ Department of Physics and Astronomy, Lehman College, City University of New York, Bronx, NY 10468, USA

¹¹⁸ INFN, Sezione di Napoli, Napoli, Italy

¹¹⁹ RWTH Aachen University, III. Physikalisches Institut A, Aachen, Germany

¹²⁰ Università Torino, Dipartimento di Fisica, Torino, Italy

¹²¹ Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Michoacán, México

¹²² Observatorio Pierre Auger and Comisión Nacional de Energía Atómica, Malargüe, Argentina

¹²³ University Politehnică of Bucharest, Bucharest, Romania

¹²⁴ Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic

¹²⁵ "Horia Hulubei" National Institute for Physics and Nuclear Engineering, Bucharest-Magurele, Romania

¹²⁶ Gran Sasso Science Institute, L'Aquila, Italy

¹²⁷ INFN Laboratori Nazionali del Gran Sasso, Assergi (L'Aquila), Italy

¹²⁸ Universidad Nacional de San Agustín de Arequipa, Facultad de Ciencias Naturales y Formales, Arequipa, Peru

¹²⁹ Univ. Grenoble Alpes, CNRS, Grenoble Institute of Engineering Univ. Grenoble Alpes, LPSC-IN2P3, F-38000 Grenoble, France

¹³⁰ Max-Planck-Institut für Radioastronomie, Bonn, Germany

¹³¹ Instituto de Física de Rosario (IFIR)—CONICET/U.N.R. and Facultad de Ciencias Bioquímicas y Farmacéuticas U.N.R., Rosario, Argentina

¹³² Karlsruhe Institute of Technology (KIT), Institute for Experimental Particle Physics, Karlsruhe, Germany

¹³³ Université Paris-Saclay, CNRS/IN2P3, IJCLab, Orsay, France

¹³⁴ Karlsruhe Institute of Technology (KIT), Institute for Astroparticle Physics, Karlsruhe, Germany

¹³⁵ Università dell'Aquila, Dipartimento di Scienze Fisiche e Chimiche, L'Aquila, Italy

¹³⁶ International Center of Advanced Studies and Instituto de Ciencias Físicas, ECYT-UNSAM and CONICET, Campus Miguelete—San Martín, Buenos Aires, Argentina

¹³⁷ Universidade Federal do Rio de Janeiro, Instituto de Física, Rio de Janeiro, RJ, Brazil

¹³⁸ Universidade Estadual de Campinas, IFGW, Campinas, SP, Brazil

¹³⁹ Colorado State University, Fort Collins, CO 80521, USA

¹⁴⁰ Université Siegen, Department Physik—Experimentelle Teilchenphysik, Siegen, Germany

¹⁴¹ Universidad de Granada and C.A.F.P.E., Granada, Spain

¹⁴² Vrije Universiteit Brussels, Brussels, Belgium

¹⁴³ INFN, Sezione di Catania, Catania, Italy

¹⁴⁴ Universidad Autónoma de Chiapas, Tuxtla Gutiérrez, Chiapas, México

¹⁴⁵ Università di Milano, Dipartimento di Fisica, Milano, Italy

¹⁴⁶ INFN, Sezione di Milano, Milano, Italy

¹⁴⁷ Nationaal Instituut voor Kernfysica en Hoge Energie Fysica (NIKHEF), Science Park, Amsterdam, The Netherlands

¹⁴⁸ Bergische Universität Wuppertal, Department of Physics, Wuppertal, Germany

¹⁴⁹ Università di Catania, Dipartimento di Fisica e Astronomia "Ettore Majorana", Catania, Italy

¹⁵⁰ Universidade de São Paulo, Escola de Engenharia de Lorena, Lorena, SP, Brazil

¹⁵¹ INFN, Sezione di Lecce, Lecce, Italy

¹⁵² Observatorio Pierre Auger, Malargüe, Argentina

¹⁵³ Palacky University, RCPTM, Olomouc, Czech Republic

¹⁵⁴ Instituto de Tecnologías en Detección y Astropartículas (CNEA, CONICET, UNSAM), and Universidad Tecnológica Nacional—Facultad Regional Mendoza (CONICET/CNEA), Mendoza, Argentina

¹⁵⁵ Università di Napoli "Federico II" Dipartimento di Fisica "Ettore Pancini", Napoli, Italy

¹⁵⁶ University of Delaware, Department of Physics and Astronomy, Bartol Research Institute, Newark, DE 19716, USA

¹⁵⁷ Politecnico di Milano, Dipartimento di Scienze e Tecnologie Aerospaziali, Milano, Italy

¹⁵⁸ Universidade Federal Fluminense, EEIMVR, Volta Redonda, RJ, Brazil

¹⁵⁹ Case Western Reserve University, Cleveland, OH 44106, USA

¹⁶⁰ Instituto de Astronomía y Física del Espacio (IAFE, CONICET-UBA), Buenos Aires, Argentina

¹⁶¹ Departamento de Física and Departamento de Ciencias de la Atmósfera y los Océanos, FCEyN, Universidad de Buenos Aires and CONICET, Buenos Aires, Argentina

¹⁶² Universidade Federal do Rio de Janeiro (UFRJ), Observatório do Valongo, Rio de Janeiro, RJ, Brazil

¹⁶³ Instituto Federal de Educação, Ciência e Tecnologia do Rio de Janeiro (IFRJ), Brazil

¹⁶⁴ Università del Salento, Dipartimento di Matematica e Fisica "E. De Giorgi", Lecce, Italy

¹⁶⁵ Universidade de São Paulo, Instituto de Física de São Carlos, São Carlos, SP, Brazil

¹⁶⁶ CNRS/IN2P3, IJCLab, Université Paris-Saclay, Orsay, France

¹⁶⁷ Universidad Nacional Autónoma de México, México, D.F., México

¹⁶⁸ Universidade Federal do Paraná, Setor Palotina, Palotina, Brazil

¹⁶⁹ IFLP, Universidad Nacional de La Plata and CONICET, La Plata, Argentina

¹⁷⁰ Fermi National Accelerator Laboratory, Fermilab, Batavia, IL 60510, USA

¹⁷¹ Stichting Astronomisch Onderzoek in Nederland (ASTRON), Dwingeloo, The Netherlands

¹⁷² University of Chicago, Enrico Fermi Institute, Chicago, IL 60637, USA

¹⁷³ New York University, New York, NY 10012, USA

¹⁷⁴ Karlsruhe Institute of Technology (KIT), Institut für Prozessdatenverarbeitung und Elektronik, Karlsruhe, Germany

¹⁷⁵ Michigan Technological University, Houghton, MI 49931, USA

¹⁷⁶ Experimental Particle Physics Department, J. Stefan Institute, Ljubljana, Slovenia

¹⁷⁷ Center for Astrophysics and Cosmology (CAC), University of Nova Gorica, Nova Gorica, Slovenia

¹⁷⁸ Now at Hakubi Center for Advanced Research and Graduate School of Science, Kyoto University, Kyoto, Japan

¹⁷⁹ Université Libre de Bruxelles (ULB), Brussels, Belgium

¹⁸⁰ Universidade Estadual de Feira de Santana, Feira de Santana, Brazil

¹⁸¹ Institute of Space Science, Bucharest-Magurele, Romania

¹⁸² Colorado School of Mines, Golden, CO 80401, USA

¹⁸³ Centro Federal de Educação Tecnológica Celso Suckow da Fonseca, Nova Friburgo, Brazil

¹⁸⁴ Universidade Federal do ABC, Santo André, SP, Brazil

¹⁸⁵ Laboratoire de Physique Nucléaire et de Hautes Energies (LPNHE), Sorbonne Université, Université de Paris, CNRS-IN2P3, Paris, France

¹⁸⁶ Benemérita Universidad Autónoma de Puebla, Puebla, México

¹⁸⁷ University of Wisconsin-Madison, Department of Physics and WIPAC, Madison, WI 53706, USA

¹⁸⁸ Universität Hamburg, II. Institut für Theoretische Physik, Hamburg, Germany

¹⁸⁹ Università di Palermo, Dipartimento di Fisica e Chimica "E. Segrè", Palermo, Italy
¹¹⁵ Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de Santiago de Compostela, Santiago de Compostela, Spain

¹¹⁶ IMAPP, Radboud University Nijmegen, Nijmegen, The Netherlands

¹¹⁷ Department of Physics and Astronomy, Lehman College, City University of New York, Bronx, NY 10468, USA

¹¹⁸ INFN, Sezione di Napoli, Napoli, Italy

¹¹⁹ RWTH Aachen University, III. Physikalisches Institut A, Aachen, Germany

¹²⁰ Università Torino, Dipartimento di Fisica, Torino, Italy

¹²¹ Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Michoacán, México

¹²² Observatorio Pierre Auger and Comisión Nacional de Energía Atómica, Malargüe, Argentina

¹²³ University Politehnică of Bucharest, Bucharest, Romania

¹²⁴ Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic

¹²⁵ "Horia Hulubei" National Institute for Physics and Nuclear Engineering, Bucharest-Magurele, Romania

¹²⁶ Gran Sasso Science Institute, L'Aquila, Italy

¹²⁷ INFN Laboratori Nazionali del Gran Sasso, Assergi (L'Aquila), Italy

¹²⁸ Universidad Nacional de San Agustín de Arequipa, Facultad de Ciencias Naturales y Formales, Arequipa, Peru

¹²⁹ Univ. Grenoble Alpes, CNRS, Grenoble Institute of Engineering Univ. Grenoble Alpes, LPSC-IN2P3, F-38000 Grenoble, France

¹³⁰ Max-Planck-Institut für Radioastronomie, Bonn, Germany

¹³¹ Instituto de Física de Rosario (IFIR)—CONICET/U.N.R. and Facultad de Ciencias Bioquímicas y Farmacéuticas U.N.R., Rosario, Argentina

¹³² Karlsruhe Institute of Technology (KIT), Institute for Experimental Particle Physics, Karlsruhe, Germany

¹³³ Université Paris-Saclay, CNRS/IN2P3, IJCLab, Orsay, France

¹³⁴ Karlsruhe Institute of Technology (KIT), Institute for Astroparticle Physics, Karlsruhe, Germany

¹³⁵ Università dell'Aquila, Dipartimento di Scienze Fisiche e Chimiche, L'Aquila, Italy

¹³⁶ International Center of Advanced Studies and Instituto de Ciencias Físicas, ECYT-UNSAM and CONICET, Campus Miguelete—San Martín, Buenos Aires, Argentina

¹³⁷ Universidade Federal do Rio de Janeiro, Instituto de Física, Rio de Janeiro, RJ, Brazil

¹³⁸ Universidade Estadual de Campinas, IFGW, Campinas, SP, Brazil

¹³⁹ Colorado State University, Fort Collins, CO 80521, USA

¹⁴⁰ Université Siegen, Department Physik—Experimentelle Teilchenphysik, Siegen, Germany

¹⁴¹ Universidad de Granada and C.A.F.P.E., Granada, Spain

¹⁴² Vrije Universiteit Brussels, Brussels, Belgium

¹⁴³ INFN, Sezione di Catania, Catania, Italy

¹⁴⁴ Universidad Autónoma de Chiapas, Tuxtla Gutiérrez, Chiapas, México

¹⁴⁵ Università di Milano, Dipartimento di Fisica, Milano, Italy

¹⁴⁶ INFN, Sezione di Milano, Milano, Italy

¹⁴⁷ Nationaal Instituut voor Kernfysica en Hoge Energie Fysica (NIKHEF), Science Park, Amsterdam, The Netherlands

¹⁴⁸ Bergische Universität Wuppertal, Department of Physics, Wuppertal, Germany

¹⁴⁹ Università di Catania, Dipartimento di Fisica e Astronomia "Ettore Majorana", Catania, Italy

¹⁵⁰ Universidade de São Paulo, Escola de Engenharia de Lorena, Lorena, SP, Brazil

¹⁵¹ INFN, Sezione di Lecce, Lecce, Italy

¹⁵² Observatorio Pierre Auger, Malargüe, Argentina

¹⁵³ Palacky University, RCPTM, Olomouc, Czech Republic

¹⁵⁴ Instituto de Tecnologías en Detección y Astropartículas (CNEA, CONICET, UNSAM), and Universidad Tecnológica Nacional—Facultad Regional Mendoza (CONICET/CNEA), Mendoza, Argentina

¹⁵⁵ Università di Napoli "Federico II" Dipartimento di Fisica "Ettore Pancini", Napoli, Italy

¹⁵⁶ University of Delaware, Department of Physics and Astronomy, Bartol Research Institute, Newark, DE 19716, USA

¹⁵⁷ Politecnico di Milano, Dipartimento di Scienze e Tecnologie Aerospaziali, Milano, Italy

¹⁵⁸ Universidade Federal Fluminense, EEIMVR, Volta Redonda, RJ, Brazil

¹⁵⁹ Case Western Reserve University, Cleveland, OH 44106, USA

¹⁶⁰ Instituto de Astronomía y Física del Espacio (IAFE, CONICET-UBA), Buenos Aires, Argentina

¹⁶¹ Departamento de Física and Departamento de Ciencias de la Atmósfera y los Océanos, FCEyN, Universidad de Buenos Aires and CONICET, Buenos Aires, Argentina

¹⁶² Universidade Federal do Rio de Janeiro (UFRJ), Observatório do Valongo, Rio de Janeiro, RJ, Brazil

¹⁶³ Instituto Federal de Educação, Ciência e Tecnologia do Rio de Janeiro (IFRJ), Brazil

¹⁶⁴ Università del Salento, Dipartimento di Matematica e Fisica "E. De Giorgi", Lecce, Italy

¹⁶⁵ Universidade de São Paulo, Instituto de Física de São Carlos, São Carlos, SP, Brazil

¹⁶⁶ CNRS/IN2P3, IJCLab, Université Paris-Saclay, Orsay, France

¹⁶⁷ Universidad Nacional Autónoma de México, México, D.F., México

¹⁶⁸ Universidade Federal do Paraná, Setor Palotina, Palotina, Brazil

¹⁶⁹ IFLP, Universidad Nacional de La Plata and CONICET, La Plata, Argentina

¹⁷⁰ Fermi National Accelerator Laboratory, Fermilab, Batavia, IL 60510, USA

¹⁷¹ Stichting Astronomisch Onderzoek in Nederland (ASTRON), Dwingeloo, The Netherlands

¹⁷² University of Chicago, Enrico Fermi Institute, Chicago, IL 60637, USA

¹⁷³ New York University, New York, NY 10012, USA

¹⁷⁴ Karlsruhe Institute of Technology (KIT), Institut für Prozessdatenverarbeitung und Elektronik, Karlsruhe, Germany

¹⁷⁵ Michigan Technological University, Houghton, MI 49931, USA

¹⁷⁶ Experimental Particle Physics Department, J. Stefan Institute, Ljubljana, Slovenia

¹⁷⁷ Center for Astrophysics and Cosmology (CAC), University of Nova Gorica, Nova Gorica, Slovenia

¹⁷⁸ Now at Hakubi Center for Advanced Research and Graduate School of Science, Kyoto University, Kyoto, Japan

¹⁷⁹ Université Libre de Bruxelles (ULB), Brussels, Belgium

¹⁸⁰ Universidade Estadual de Feira de Santana, Feira de Santana, Brazil

¹⁸¹ Institute of Space Science, Bucharest-Magurele, Romania

¹⁸² Colorado School of Mines, Golden, CO 80401, USA

¹⁸³ Centro Federal de Educação Tecnológica Celso Suckow da Fonseca, Nova Friburgo, Brazil

¹⁸⁴ Universidade Federal do ABC, Santo André, SP, Brazil

¹⁸⁵ Laboratoire de Physique Nucléaire et de Hautes Energies (LPNHE), Sorbonne Université, Université de Paris, CNRS-IN2P3, Paris, France

¹⁸⁶ Benemérita Universidad Autónoma de Puebla, Puebla, México

¹⁸⁷ University of Wisconsin-Madison, Department of Physics and WIPAC, Madison, WI 53706, USA

¹⁸⁸ Universität Hamburg, II. Institut für Theoretische Physik, Hamburg, Germany

¹⁸⁹ Università di Palermo, Dipartimento di Fisica e Chimica "E. Segrè", Palermo, Italy
190 Louisiana State University, Baton Rouge, LA 70803, USA

191 Università di Roma “Tor Vergata” Dipartimento di Fisica, Roma, Italy

192 INFN, Sezione di Roma “Tor Vergata”, Roma, Italy

193 Pennsylvania State University, University Park, PA 16802, USA

194 Centro de Investigaciones en Láseres y Aplicaciones, CITEDEF and CONICET, Villa Martelli, Argentina

195 Charles University, Faculty of Mathematics and Physics, Institute of Particle and Nuclear Physics, Prague, Czech Republic

196 Universidad Industrial de Santander, Bucaramanga, Colombia

197 Unidad Profesional Interdisciplinaria en Ingeniería y Tecnologías Avanzadas del Instituto Politécnico Nacional (UPIITA-IPN), México, D.F., México

198 Universidade de São Paulo, Instituto de Física, São Paulo, SP, Brazil

199 Instituto di Astrofisica Spaziale e Fisica Cosmica di Palermo (INAF), Palermo, Italy

200 Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro, RJ, Brazil

201 Also at University of Bucharest, Physics Department, Bucharest, Romania

202 University of Łódź, Faculty of High-Energy Astrophysics, Łódź, Poland

203 Universidad de Medellín, Medellín, Colombia

204 Universiteit van Amsterdam, Faculty of Science, Amsterdam, The Netherlands

205 School of Physics and Astronomy, University of Leeds, Leeds, UK

206 Department of Physics, Loyola University Chicago, Chicago, IL 60611, USA

207 High Energy Astrophysics Institute and Department of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112, USA

208 Graduate School of Science, Osaka City University, Osaka, Osaka, Japan

209 Department of Physics and The Research Institute of Natural Science, Hanyang University, Seongdong-gu, Seoul, Republic of Korea

210 Department of Physics, Tokyo University of Science, Noda, Chiba, Japan

211 Institute for Cosmic Ray Research, University of Tokyo, Kashiwa, Chiba, Japan

212 Currently at University of California—Santa Cruz, CA 95064, USA

213 Currently at Flatiron Institute, Simons Foundation, NY 10010, USA

214 Astrophysical Big Bang Laboratory, RIKEN, Wako, Saitama, Japan

215 Academic Assembly School of Science and Technology Institute of Engineering, Shinshu University, Nagano, Nagano, Japan

216 Faculty of Engineering, Kanagawa University, Yokohama, Kanagawa, Japan

217 Interdisciplinary Graduate School of Medicine and Engineering, University of Yamanashi, Kofu, Yamanashi, Japan

218 The Graduate School of Science and Engineering, Saitama University, Saitama, Saitama, Japan

219 Department of Physics, SungKyunKwan University, Jang-an-gu, Suwon, Republic of Korea

220 Department of Physics, Tokyo City University, Setagaya-ku, Tokyo, Japan

221 Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia

222 Faculty of Systems Engineering and Science, Shibaura Institute of Technology, Minato-ku, Tokyo, Japan

223 Department of Engineering Science, Faculty of Engineering, Osaka Electro-Communication University, Neyagawa-shi, Osaka, Japan

224 Service de Physique Théorique, Université Libre de Bruxelles, Brussels, Belgium

225 Department of Physics, Yonsei University, Seodaemun-gu, Seoul, Republic of Korea

226 Faculty of Science, Kochi University, Kochi, Kochi, Japan

227 Nambu Yoichiro Institute of Theoretical and Experimental Physics, Osaka City University, Osaka, Osaka, Japan

228 Department of Physical Sciences, Ritsumeikan University, Kusatsu, Shiga, Japan

229 College of Engineering, Chubu University, Kasugai, Aichi, Japan

230 Quantum ICT Advanced Development Center, National Institute for Information and Communications Technology, Koganei, Tokyo, Japan

231 Sternberg Astronomical Institute, Moscow M.V. Lomonosov State University, Moscow, Russia

232 Department of Physics, School of Natural Sciences, Ulsan National Institute of Science and Technology, UNIST-gil, Ulsan, Republic of Korea

233 Earthquake Research Institute, University of Tokyo, Bunkyo-ku, Tokyo, Japan

234 Graduate School of Information Sciences, Hiroshima City University, Hiroshima, Hiroshima, Japan

235 Institute of Particle and Nuclear Studies, KEK, Tsukuba, Ibaraki, Japan

236 Graduate School of Science and Engineering, Tokyo Institute of Technology, Meguro, Tokyo, Japan

237 Department of Research Planning and Promotion, Quantum Medical Science Directorate, National Institutes for Quantum and Radiological Science and Technology, Chiba, Chiba, Japan

238 CEICO, Institute of Physics, Czech Academy of Sciences, Prague, Czech Republic

Received 2022 January 31; revised 2022 April 12; accepted 2022 May 5; published 2022 August 3

Abstract

For several decades, the origin of ultra-high-energy cosmic rays (UHECRs) has been an unsolved question of high-energy astrophysics. One approach for solving this puzzle is to correlate UHECRs with high-energy neutrinos, since neutrinos are a direct probe of hadronic interactions of cosmic rays and are not deflected by magnetic fields. In this paper, we present three different approaches for correlating the arrival directions of neutrinos with the arrival directions of UHECRs. The neutrino data are provided by the IceCube Neutrino Observatory and ANTARES, while the UHECR data with energies above ~50 EeV are provided by the Pierre Auger Observatory and the Telescope Array. All experiments provide increased statistics and improved reconstructions with respect to our previous results reported in 2015. The first analysis uses a high-statistics neutrino sample optimized for point-source searches to search for excesses of neutrino clustering in the vicinity of UHECR directions. The second analysis searches for an excess of UHECRs in the direction of the highest-energy neutrinos. The third analysis searches for an excess of pairs of UHECRs and highest-energy neutrinos

<figure><img src="image_3.png" /></figure>

<footnote n="239">239 Deceased.</footnote>
190 Louisiana State University, Baton Rouge, LA 70803, USA

191 Università di Roma “Tor Vergata” Dipartimento di Fisica, Roma, Italy

192 INFN, Sezione di Roma “Tor Vergata”, Roma, Italy

193 Pennsylvania State University, University Park, PA 16802, USA

194 Centro de Investigaciones en Láseres y Aplicaciones, CITEDEF and CONICET, Villa Martelli, Argentina

195 Charles University, Faculty of Mathematics and Physics, Institute of Particle and Nuclear Physics, Prague, Czech Republic

196 Universidad Industrial de Santander, Bucaramanga, Colombia

197 Unidad Profesional Interdisciplinaria en Ingeniería y Tecnologías Avanzadas del Instituto Politécnico Nacional (UPIITA-IPN), México, D.F., México

198 Universidade de São Paulo, Instituto de Física, São Paulo, SP, Brazil

199 Instituto di Astrofisica Spaziale e Fisica Cosmica di Palermo (INAF), Palermo, Italy

200 Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro, RJ, Brazil

201 Also at University of Bucharest, Physics Department, Bucharest, Romania

202 University of Łódź, Faculty of High-Energy Astrophysics, Łódź, Poland

203 Universidad de Medellín, Medellín, Colombia

204 Universiteit van Amsterdam, Faculty of Science, Amsterdam, The Netherlands

205 School of Physics and Astronomy, University of Leeds, Leeds, UK

206 Department of Physics, Loyola University Chicago, Chicago, IL 60611, USA

207 High Energy Astrophysics Institute and Department of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112, USA

208 Graduate School of Science, Osaka City University, Osaka, Osaka, Japan

209 Department of Physics and The Research Institute of Natural Science, Hanyang University, Seongdong-gu, Seoul, Republic of Korea

210 Department of Physics, Tokyo University of Science, Noda, Chiba, Japan

211 Institute for Cosmic Ray Research, University of Tokyo, Kashiwa, Chiba, Japan

212 Currently at University of California—Santa Cruz, CA 95064, USA

213 Currently at Flatiron Institute, Simons Foundation, NY 10010, USA

214 Astrophysical Big Bang Laboratory, RIKEN, Wako, Saitama, Japan

215 Academic Assembly School of Science and Technology Institute of Engineering, Shinshu University, Nagano, Nagano, Japan

216 Faculty of Engineering, Kanagawa University, Yokohama, Kanagawa, Japan

217 Interdisciplinary Graduate School of Medicine and Engineering, University of Yamanashi, Kofu, Yamanashi, Japan

218 The Graduate School of Science and Engineering, Saitama University, Saitama, Saitama, Japan

219 Department of Physics, SungKyunKwan University, Jang-an-gu, Suwon, Republic of Korea

220 Department of Physics, Tokyo City University, Setagaya-ku, Tokyo, Japan

221 Institute for Nuclear Research of the Russian Academy of Sciences, Moscow, Russia

222 Faculty of Systems Engineering and Science, Shibaura Institute of Technology, Minato-ku, Tokyo, Japan

223 Department of Engineering Science, Faculty of Engineering, Osaka Electro-Communication University, Neyagawa-shi, Osaka, Japan

224 Service de Physique Théorique, Université Libre de Bruxelles, Brussels, Belgium

225 Department of Physics, Yonsei University, Seodaemun-gu, Seoul, Republic of Korea

226 Faculty of Science, Kochi University, Kochi, Kochi, Japan

227 Nambu Yoichiro Institute of Theoretical and Experimental Physics, Osaka City University, Osaka, Osaka, Japan

228 Department of Physical Sciences, Ritsumeikan University, Kusatsu, Shiga, Japan

229 College of Engineering, Chubu University, Kasugai, Aichi, Japan

230 Quantum ICT Advanced Development Center, National Institute for Information and Communications Technology, Koganei, Tokyo, Japan

231 Sternberg Astronomical Institute, Moscow M.V. Lomonosov State University, Moscow, Russia

232 Department of Physics, School of Natural Sciences, Ulsan National Institute of Science and Technology, UNIST-gil, Ulsan, Republic of Korea

233 Earthquake Research Institute, University of Tokyo, Bunkyo-ku, Tokyo, Japan

234 Graduate School of Information Sciences, Hiroshima City University, Hiroshima, Hiroshima, Japan

235 Institute of Particle and Nuclear Studies, KEK, Tsukuba, Ibaraki, Japan

236 Graduate School of Science and Engineering, Tokyo Institute of Technology, Meguro, Tokyo, Japan

237 Department of Research Planning and Promotion, Quantum Medical Science Directorate, National Institutes for Quantum and Radiological Science and Technology, Chiba, Chiba, Japan

238 CEICO, Institute of Physics, Czech Academy of Sciences, Prague, Czech Republic

Received 2022 January 31; revised 2022 April 12; accepted 2022 May 5; published 2022 August 3

Abstract

For several decades, the origin of ultra-high-energy cosmic rays (UHECRs) has been an unsolved question of high-energy astrophysics. One approach for solving this puzzle is to correlate UHECRs with high-energy neutrinos, since neutrinos are a direct probe of hadronic interactions of cosmic rays and are not deflected by magnetic fields. In this paper, we present three different approaches for correlating the arrival directions of neutrinos with the arrival directions of UHECRs. The neutrino data are provided by the IceCube Neutrino Observatory and ANTARES, while the UHECR data with energies above ~50 EeV are provided by the Pierre Auger Observatory and the Telescope Array. All experiments provide increased statistics and improved reconstructions with respect to our previous results reported in 2015. The first analysis uses a high-statistics neutrino sample optimized for point-source searches to search for excesses of neutrino clustering in the vicinity of UHECR directions. The second analysis searches for an excess of UHECRs in the direction of the highest-energy neutrinos. The third analysis searches for an excess of pairs of UHECRs and highest-energy neutrinos

<figure><img src="image_3.png" /></figure>

<footnote n="239">239 Deceased.</footnote>
on different angular scales. None of the analyses have found a significant excess, and previously reported overfluctuations
are reduced in significance. Based on these results, we further constrain the neutrino flux spatially correlated with
UHECRs.

*Key words:* Neutrino astronomy – High energy astrophysics – Ultra-high-energy cosmic radiation

# 1. Introduction

Earth is continuously bombarded by high-energy cosmic rays, most of which are charged atomic nuclei (Particle Data Group et al. 2020). It is generally believed that cosmic rays with energies above 1 eV (10¹⁸ eV), known as ultra-high-energy cosmic rays (UHECRs), mostly originate from extragalactic sources in the nearby universe. Based on the estimated magnitude of galactic magnetic fields (Nagano & Watson 2000), cosmic rays below this energy are believed to diffuse within their host galaxy, whereas cosmic rays above this energy escape from the galaxy. These assumptions are confirmed by the observation of large-scale anisotropies in the arrival directions of UHECRs above 8 eV with the excess flux directed from outside of our Galaxy (Aab et al. 2017). The two largest observatories for UHECRs are the Pierre Auger Observatory (Auger; The Pierre Auger Collaboration 2015) in Argentina in the Southern Hemisphere and the Telescope Array (TA; Abu-Zayyad et al. 2012) in the United States in the Northern Hemisphere. They have performed detailed analyses of the arrival directions of detected UHECRs and have revealed interesting features, such as possible correlations with the directions of known starburst and active galaxies in the nearby universe observed by Auger (Aab et al. 2018; Biteau et al. 2022), and intermediate-scale directional clustering observed by TA (Abbasi et al. 2014; Tkachev et al. 2021). However, due to the small number of events detected at the highest energies, these indications of anisotropic arrival directions have not yet been confirmed on the 5σ level. Thus, neither Auger nor TA reports an unambiguous identification of the UHECR sources to date. One of the main reasons is the deflection of cosmic rays by magnetic fields during propagation from their sources to Earth, which alters their directional information with respect to the source positions. The deflection of UHECRs increases with increasing charge, which is not well determined at the highest energies above 50 eV considered in this work. This uncertainty additionally complicates the identification of UHECR sources. Nevertheless, UHECRs at the highest energies are deflected the least owing to their extremely high magnetic rigidity (Alves Batista et al. 2019), which makes them suitable for directional correlation searches.

We search for the sources of the highest-energy cosmic rays with a multimessenger approach using high-energy neutrinos and UHECRs with energies above ~50 EeV. High-energy neutrinos are direct tracers of hadronic interactions of cosmic rays and are thus expected to be produced at the acceleration site or during propagation (see, e.g., Murase 2015; Mészáros 2018; Anchordoqui et al. 2008). Furthermore, the electrically neutral neutrinos carry the directional information of their origin, since they are not deflected by magnetic fields. A combined analysis of UHECRs and high-energy neutrinos is further motivated by the observation of a diffuse flux of high-energy neutrinos of astrophysical origin above the background of atmospheric neutrinos. The flux follows a hard power law that extends to energies of multiple PeV and possibly beyond (The IceCube Collaboration 2013; Aartsen et al. 2016; Stettner 2019; Aartsen et al. 2020d; Abbasi et al. 2021). The measured flux is closely below the Waxman–Bahcall bound (Waxman & Bahcall 1999; Bahcall & Waxman 2001), which is

a theoretical upper bound on the astrophysical neutrino flux derived from the observed UHECR flux, suggesting a connection between UHECRs and high-energy neutrinos (Murase et al. 2013; Ahlers & Halzen 2018). We note that the Seyfert and starburst galaxy NGC 1068 appears both as a neutrino source candidate in Aartsen et al. (2020c) and as a UHECR source candidate in Aab et al. (2018). Recently, the potential of high-energy neutrinos in a multimessenger approach has been demonstrated together with observations of high-energy photons (Aartsen et al. 2018b, 2018c), finding compelling evidence for neutrino emission from the blazar TXS 0506+056. At the same time, it can be concluded that blazars seem insufficient to saturate the total observed neutrino flux (Murase & Waxman 2016; Aartsen et al. 2017a) based on the nonobservation of steady neutrino fluxes from known blazars in the universe. This result motivates different approaches to identify hadronic sources in the local universe; a promising method is to correlate the sources traced with UHECRs with high-energy neutrinos, and vice versa.

In this paper, we report results from three conceptually different approaches. All are based on correlating the arrival directions of high-energy neutrinos detected by the IceCube Neutrino Observatory (IC; Aartsen et al. 2017d) and the ANTARES experiment (ANT; Ageron et al. 2011) with the arrival directions of UHECRs with energies above ~50 eV measured by Auger and TA:

1. Analysis 1 (described in Section 4.1) uses the measured UHECR directions, as well as magnetic deflection estimates, for identifying regions in which we search for point-like neutrino sources.

2. Analysis 2 (described in Section 4.2) uses the arrival directions of neutrinos with a high probability of astrophysical origin to search for a correlated clustering of UHECR arrival directions, while also accounting for the magnetic deflection.

3. Analysis 3 (described in Section 4.3) is based on a largely model-independent two-point correlation analysis of the arrival directions of UHECRs and neutrinos with a high probability of astrophysical origin.

With these three approaches, we follow up on previous searches performed by the Pierre Auger, Telescope Array, and IceCube Collaborations (The IceCube Collaboration et al. 2016), which showed an interesting correlation at a significance level close to three standard deviations.

Since 2015, the analyses have been improved with substantially enlarged data sets, extending the Auger data from 10 to 13 yr (see Section 2.3) and TA data from 6 to 9 yr (see Section 2.4). The different data sets from IceCube are enlarged from 4 to 10.5 yr, from 4 to 7.5 yr, and from 2 to 8 yr, depending on the specific detection channel as described in Section 2.1. Newly included are two data sets of 9 and 11 yr from ANTARES, as further specified in Section 2.2. Due to the updated data sets by Auger and TA, their respective shift of the energy scale is updated from a sole shift of TA energies by -13% to a shift of +14% for Auger and -14% TA energies
on different angular scales. None of the analyses have found a significant excess, and previously reported overfluctuations
are reduced in significance. Based on these results, we further constrain the neutrino flux spatially correlated with
UHECRs.

Key words: Neutrino astronomy – High energy astrophysics – Ultra-high-energy cosmic radiation

**1. Introduction**

Earth is continuously bombarded by high-energy cosmic rays, most of which are charged atomic nuclei (Particle Data Group et al. 2020). It is generally believed that cosmic rays with energies above 1 eV (10¹⁸ eV), known as ultra-high-energy cosmic rays (UHECRs), mostly originate from extragalactic sources in the nearby universe. Based on the estimated magnitude of galactic magnetic fields (Nagano & Watson 2000), cosmic rays below this energy are believed to diffuse within their host galaxy, whereas cosmic rays above this energy escape from the galaxy. These assumptions are confirmed by the observation of large-scale anisotropies in the arrival directions of UHECRs above 8 eV with the excess flux directed from outside of our Galaxy (Aab et al. 2017). The two largest observatories for UHECRs are the Pierre Auger Observatory (Auger; The Pierre Auger Collaboration 2015) in Argentina in the Southern Hemisphere and the Telescope Array (TA; Abu-Zayyad et al. 2012) in the United States in the Northern Hemisphere. They have performed detailed analyses of the arrival directions of detected UHECRs and have revealed interesting features, such as possible correlations with the directions of known starburst and active galaxies in the nearby universe observed by Auger (Aab et al. 2018; Biteau et al. 2022), and intermediate-scale directional clustering observed by TA (Abbasi et al. 2014; Tkachev et al. 2021). However, due to the small number of events detected at the highest energies, these indications of anisotropic arrival directions have not yet been confirmed on the 5σ level. Thus, neither Auger nor TA reports an unambiguous identification of the UHECR sources to date. One of the main reasons is the deflection of cosmic rays by magnetic fields during propagation from their sources to Earth, which alters their directional information with respect to the source positions. The deflection of UHECRs increases with increasing charge, which is not well determined at the highest energies above 50 eV considered in this work. This uncertainty additionally complicates the identification of UHECR sources. Nevertheless, UHECRs at the highest energies are deflected the least owing to their extremely high magnetic rigidity (Alves Batista et al. 2019), which makes them suitable for directional correlation searches.

We search for the sources of the highest-energy cosmic rays with a multimessenger approach using high-energy neutrinos and UHECRs with energies above ~50 EeV. High-energy neutrinos are direct tracers of hadronic interactions of cosmic rays and are thus expected to be produced at the acceleration site or during propagation (see, e.g., Murase 2015; Mészáros 2018; Anchordoqui et al. 2008). Furthermore, the electrically neutral neutrinos carry the directional information of their origin, since they are not deflected by magnetic fields. A combined analysis of UHECRs and high-energy neutrinos is further motivated by the observation of a diffuse flux of high-energy neutrinos of astrophysical origin above the background of atmospheric neutrinos. The flux follows a hard power law that extends to energies of multiple PeV and possibly beyond (The IceCube Collaboration 2013; Aartsen et al. 2016; Stettner 2019; Aartsen et al. 2020d; Abbasi et al. 2021). The measured flux is closely below the Waxman–Bahcall bound (Waxman & Bahcall 1999; Bahcall & Waxman 2001), which is

a theoretical upper bound on the astrophysical neutrino flux derived from the observed UHECR flux, suggesting a connection between UHECRs and high-energy neutrinos (Murase et al. 2013; Ahlers & Halzen 2018). We note that the Seyfert and starburst galaxy NGC 1068 appears both as a neutrino source candidate in Aartsen et al. (2020c) and as a UHECR source candidate in Aab et al. (2018). Recently, the potential of high-energy neutrinos in a multimessenger approach has been demonstrated together with observations of high-energy photons (Aartsen et al. 2018b, 2018c), finding compelling evidence for neutrino emission from the blazar TXS 0506+056. At the same time, it can be concluded that blazars seem insufficient to saturate the total observed neutrino flux (Murase & Waxman 2016; Aartsen et al. 2017a) based on the nonobservation of steady neutrino fluxes from known blazars in the universe. This result motivates different approaches to identify hadronic sources in the local universe; a promising method is to correlate the sources traced with UHECRs with high-energy neutrinos, and vice versa.

In this paper, we report results from three conceptually different approaches. All are based on correlating the arrival directions of high-energy neutrinos detected by the IceCube Neutrino Observatory (IC; Aartsen et al. 2017d) and the ANTARES experiment (ANT; Ageron et al. 2011) with the arrival directions of UHECRs with energies above ~50 eV measured by Auger and TA:

1. Analysis 1 (described in Section 4.1) uses the measured UHECR directions, as well as magnetic deflection estimates, for identifying regions in which we search for point-like neutrino sources.

2. Analysis 2 (described in Section 4.2) uses the arrival directions of neutrinos with a high probability of astrophysical origin to search for a correlated clustering of UHECR arrival directions, while also accounting for the magnetic deflection.

3. Analysis 3 (described in Section 4.3) is based on a largely model-independent two-point correlation analysis of the arrival directions of UHECRs and neutrinos with a high probability of astrophysical origin.

With these three approaches, we follow up on previous searches performed by the Pierre Auger, Telescope Array, and IceCube Collaborations (The IceCube Collaboration et al. 2016), which showed an interesting correlation at a significance level close to three standard deviations.

Since 2015, the analyses have been improved with substantially enlarged data sets, extending the Auger data from 10 to 13 yr (see Section 2.3) and TA data from 6 to 9 yr (see Section 2.4). The different data sets from IceCube are enlarged from 4 to 10.5 yr, from 4 to 7.5 yr, and from 2 to 8 yr, depending on the specific detection channel as described in Section 2.1. Newly included are two data sets of 9 and 11 yr from ANTARES, as further specified in Section 2.2. Due to the updated data sets by Auger and TA, their respective shift of the energy scale is updated from a sole shift of TA energies by -13% to a shift of +14% for Auger and -14% TA energies
<figure><img src="image_4.png" /><figcaption>**Figure 1.** Left: sky map of the arrival directions of UHECR events and high-energy neutrinos. The high-energy neutrino track-like events from IceCube consist of the HESE, HENU, and EHE data sets, while the cascade-like events are only of the IceCube-HESE data sets. From ANTARES, only high-energy tracks are selected for the analyses. Right: histogram of the decl. of UHECR events, separated into Auger and TA contributions.</figcaption></figure>

above the energy threshold of ~50 EeV (Biteau et al. 2019). Furthermore, we include an improved magnetic deflection model that distinguishes between the Northern and Southern Hemispheres for analysis 2. We report the results from the three improved correlation searches, which update the preliminary reported results in Schumacher (2019), Aublin et al. (2019), and Barbano (2019). In addition, we report upper limits on the correlated fluxes of UHECRs and neutrinos based on benchmark models for the magnetic deflections.

## 2. Observatories and Data Sets

All data sets used in this paper are used in previous work by the four respective collaborations. This section focuses on the main aspects relevant for our analyses.

### 2.1. The IceCube Neutrino Observatory

The IceCube Neutrino Observatory (Aartsen et al. 2017d) is an ice-Cherenkov detector sensitive to neutrinos with energies $\ge 5$ GeV. It is located at the geographic South Pole, about 1.45 –2.45 km deep in the ice. Its main component consists of a volume of about $1 \text{ km}^3$ glacial ice instrumented with 5160 photomultipliers that are connected to the surface by 86 cable strings.

Two classes of neutrino-induced events can be phenomenologically distinguished: elongated, track-like events that are produced by muons that originate mostly from charged-current $\nu_\mu$ interactions; and the spherical, cascade-like events that originate from charged-current $\nu_e$ and $\nu_\tau$ interactions with hadronic and electromagnetic decays, as well as neutral–current interactions of all flavors. Typically, track-like events enable a better angular resolution than cascade-like events owing to their different topologies, but they provide a poorer energy resolution (Aartsen et al. 2014a; Wandkowsky 2018). One method of suppressing the dominant background of down-going muons produced by cosmic-ray interactions in the atmosphere is by selecting events with the interaction vertex within the detector (Aartsen et al. 2014c; Kopper 2017; Wandkowsky 2018). Alternatively, through-going tracks with either horizontal or up-going directions are selected, such that

the atmospheric muons are blocked out by Earth (Aartsen et al. 2016; Haack & Wiebusch 2017). In the case of down-going tracks, a high-energy threshold and elaborate selection procedures are necessary to filter out atmospheric muons (Aartsen et al. 2017b, 2017c, 2018c). In all cases, the remaining event rate is dominated by atmospheric neutrinos. The selection of astrophysical neutrinos can be achieved on a statistical basis by selecting very energetic events or, in the case of the very down-going region, by vetoing events where an atmospheric shower is observed in IceTop, IceCube’s surface detector for cosmic rays (Abbasi et al. 2013).

For the three analyses, data from multiple detection channels are used, which are (i) a data set of through-going tracks from the full sky optimized for point-source searches (PS), (ii) a data set of high-energy starting events (HESE) of both topologies from the full sky, (iii) a data set of high-energy neutrinos (HENU) selected from through-going tracks with horizontal and up-going directions, and (iv) a data set of tracks from a selection of extremely high energy events (EHE). The PS data set is used for analysis 1, while the HESE, HENU, and EHE data sets are used for analyses 2 and 3. For analyses 2 and 3, track-like events from the HESE, HENU, and EHE data sets are combined, while multiple instances of identical events are removed. This results in a data set of 81 track-like events. In analyses 2 and 3, the 76 cascade-like events from the HESE data set are analyzed separately owing to their larger directional uncertainty. The sky distribution of selected events is shown in Figure 1, and an overview of the nomenclature is presented in Table 1.

The PS data set consists of a combination of data collected from 7 yr of operation between 2008 and 2015 that were used for point-source searches (Aartsen et al. 2017b) and data from 3.5 yr of operation between 2015 and 2018 that were selected for the real-time gamma-ray follow-up (GFU) program of IceCube (Aartsen et al. 2017c, 2018c). The combined data set consists of about 1.4 million track-like events above ~100 GeV. It is dominated by atmospheric neutrinos in the Northern Hemisphere and by atmospheric muons in the Southern Hemisphere. The median of the angular resolution ($\Psi$) is better than 0.5° above energies of a few TeV. Figure 2 shows the
<figure><img src="image_4.png" /><figcaption>**Figure 1.** Left: sky map of the arrival directions of UHECR events and high-energy neutrinos. The high-energy neutrino track-like events from IceCube consist of the HESE, HENU, and EHE data sets, while the cascade-like events are only of the IceCube-HESE data sets. From ANTARES, only high-energy tracks are selected for the analyses. Right: histogram of the decl. of UHECR events, separated into Auger and TA contributions.</figcaption></figure>

above the energy threshold of ~50 EeV (Biteau et al. 2019). Furthermore, we include an improved magnetic deflection model that distinguishes between the Northern and Southern Hemispheres for analysis 2. We report the results from the three improved correlation searches, which update the preliminary reported results in Schumacher (2019), Aublin et al. (2019), and Barbano (2019). In addition, we report upper limits on the correlated fluxes of UHECRs and neutrinos based on benchmark models for the magnetic deflections.

## 2. Observatories and Data Sets

All data sets used in this paper are used in previous work by the four respective collaborations. This section focuses on the main aspects relevant for our analyses.

### 2.1. The IceCube Neutrino Observatory

The IceCube Neutrino Observatory (Aartsen et al. 2017d) is an ice-Cherenkov detector sensitive to neutrinos with energies $\ge 5$ GeV. It is located at the geographic South Pole, about 1.45 –2.45 km deep in the ice. Its main component consists of a volume of about $1 \text{ km}^3$ glacial ice instrumented with 5160 photomultipliers that are connected to the surface by 86 cable strings.

Two classes of neutrino-induced events can be phenomenologically distinguished: elongated, track-like events that are produced by muons that originate mostly from charged-current $\nu_\mu$ interactions; and the spherical, cascade-like events that originate from charged-current $\nu_e$ and $\nu_\tau$ interactions with hadronic and electromagnetic decays, as well as neutral–current interactions of all flavors. Typically, track-like events enable a better angular resolution than cascade-like events owing to their different topologies, but they provide a poorer energy resolution (Aartsen et al. 2014a; Wandkowsky 2018). One method of suppressing the dominant background of down-going muons produced by cosmic-ray interactions in the atmosphere is by selecting events with the interaction vertex within the detector (Aartsen et al. 2014c; Kopper 2017; Wandkowsky 2018). Alternatively, through-going tracks with either horizontal or up-going directions are selected, such that

the atmospheric muons are blocked out by Earth (Aartsen et al. 2016; Haack & Wiebusch 2017). In the case of down-going tracks, a high-energy threshold and elaborate selection procedures are necessary to filter out atmospheric muons (Aartsen et al. 2017b, 2017c, 2018c). In all cases, the remaining event rate is dominated by atmospheric neutrinos. The selection of astrophysical neutrinos can be achieved on a statistical basis by selecting very energetic events or, in the case of the very down-going region, by vetoing events where an atmospheric shower is observed in IceTop, IceCube’s surface detector for cosmic rays (Abbasi et al. 2013).

For the three analyses, data from multiple detection channels are used, which are (i) a data set of through-going tracks from the full sky optimized for point-source searches (PS), (ii) a data set of high-energy starting events (HESE) of both topologies from the full sky, (iii) a data set of high-energy neutrinos (HENU) selected from through-going tracks with horizontal and up-going directions, and (iv) a data set of tracks from a selection of extremely high energy events (EHE). The PS data set is used for analysis 1, while the HESE, HENU, and EHE data sets are used for analyses 2 and 3. For analyses 2 and 3, track-like events from the HESE, HENU, and EHE data sets are combined, while multiple instances of identical events are removed. This results in a data set of 81 track-like events. In analyses 2 and 3, the 76 cascade-like events from the HESE data set are analyzed separately owing to their larger directional uncertainty. The sky distribution of selected events is shown in Figure 1, and an overview of the nomenclature is presented in Table 1.

The PS data set consists of a combination of data collected from 7 yr of operation between 2008 and 2015 that were used for point-source searches (Aartsen et al. 2017b) and data from 3.5 yr of operation between 2015 and 2018 that were selected for the real-time gamma-ray follow-up (GFU) program of IceCube (Aartsen et al. 2017c, 2018c). The combined data set consists of about 1.4 million track-like events above ~100 GeV. It is dominated by atmospheric neutrinos in the Northern Hemisphere and by atmospheric muons in the Southern Hemisphere. The median of the angular resolution ($\Psi$) is better than 0.5° above energies of a few TeV. Figure 2 shows the
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Median of the angular resolution, $\Psi$, which is the angle between the true neutrino direction and reconstructed muon direction for IceCube and ANTARES point-source data as a function of the true neutrino energy. This is calculated based on simulation data sets. The data from partial detector configurations of IceCube are denoted by the number of operating strings (IC40, IC59, IC79; Aartsen et al. 2014b) and are shown in blue. The data of the full detector configuration with 86 strings, including the GFU data set (Aartsen et al. 2017b, 2017c, 2018c), are identified by the years of data taking and are shown in orange. The angular resolution of ANTARES data for the 11 yr data set (Albert et al. 2018; Illuminati et al. 2019) is shown in black.</figcaption></figure>

<table><caption>**Table 1**

Overview of Different Neutrino Data Sets Used in the Different Analyses</caption><thead><tr><th>Detector</th><th>Analysis 1</th><th>Analysis 2</th><th>Analysis 3</th></tr></thead><tbody><tr><td>ANTARES</td><td>PS</td><td>HENU</td><td>HENU</td></tr><tr><td>IceCube</td><td>PS</td><td>HESE + HENU<br>+ EHE</td><td>HESE + HENU<br>+ EHE</td></tr><tr><th>Data set</th><th>Description</th><th></th><th>Topology</th></tr><tr><td>PS</td><td>Optimized for point-source searches, v<sub>&mu;</sub> candidates</td><td></td><td>Tracks</td></tr><tr><td>HESE</td><td>High-energy starting events, all flavors</td><td></td><td>Tracks and<br>cascades</td></tr><tr><td>HENU</td><td>High-energy selection of v<sub>&mu;</sub> candidates</td><td></td><td>Tracks</td></tr><tr><td>EHE</td><td>Extremely high energy v<sub>&mu;</sub> candidates</td><td></td><td>Tracks</td></tr></tbody></table>

median of the angular resolution for the different detector configurations of IceCube and for data from the ANTARES detector (see Section 2.2). Note that analysis 1 is based not on the median of the angular resolution but on the estimator for the angular resolution on an event-by-event basis ($\sigma$).

The HESE data set contains 76 cascade-like events and 26 track-like events that have been collected between 2010 and 2017, as presented in Wandkowsky (2018). It consists of neutrinos of all flavors that interacted inside the detection volume, called starting events, with deposited energies ranging between about 20 TeV and 2 PeV. Integrated above 60 TeV, which corresponds to 60 events in total, the percentage of events of astrophysical origin, i.e., the astrophysical purity, is larger than 75% (Wandkowsky 2018), while the percentage is lower below 60 TeV. The angular resolution is about 1° for track-like events and 15° for cascade-like events above 100 TeV. The resolution of the deposited energy for tracks and cascades is around 10% (Aartsen et al. 2014a) without accounting for systematic uncertainties, but the cascades have a

better correlation with the primary neutrino energy since they deposit most of their energy inside the detector, while tracks do not.

The HENU data set consists of the 35 highest-energy track-like events with a reconstructed decl. $\ge -5°$, which have been collected between 2009 and 2016 (Haack & Wiebusch 2017) for the measurement of the diffuse muon-neutrino flux (Aartsen et al. 2016). From the original data set starting at ~100 GeV, only events of high probability of nonatmospheric origin have been selected by applying an energy threshold of $\ge 200$ TeV on the reconstructed muon energy. This corresponds to an astrophysical purity of more than 50%. The astrophysical purity here is defined as the flux ratio of the astrophysical to the sum of atmospheric and astrophysical differential fluxes and thus depends on the assumed astrophysical flux. For the estimation of the astrophysical purity, the best-fit astrophysical flux from Aartsen et al. (2016), $d\phi/dE = 1.01 \times 10^{18} \text{ GeV}^{-1} \text{ cm}^{-2} \text{ s}^{-1} \text{ sr}^{-1} (E/100 \text{ TeV})^{-2.19}$, is used, as well as the best-fit atmospheric flux from the same reference.
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Median of the angular resolution, $\Psi$, which is the angle between the true neutrino direction and reconstructed muon direction for IceCube and ANTARES point-source data as a function of the true neutrino energy. This is calculated based on simulation data sets. The data from partial detector configurations of IceCube are denoted by the number of operating strings (IC40, IC59, IC79; Aartsen et al. 2014b) and are shown in blue. The data of the full detector configuration with 86 strings, including the GFU data set (Aartsen et al. 2017b, 2017c, 2018c), are identified by the years of data taking and are shown in orange. The angular resolution of ANTARES data for the 11 yr data set (Albert et al. 2018; Illuminati et al. 2019) is shown in black.</figcaption></figure>

<table><caption>**Table 1**

Overview of Different Neutrino Data Sets Used in the Different Analyses</caption><thead><tr><th>Detector</th><th>Analysis 1</th><th>Analysis 2</th><th>Analysis 3</th></tr></thead><tbody><tr><td>ANTARES</td><td>PS</td><td>HENU</td><td>HENU</td></tr><tr><td>IceCube</td><td>PS</td><td>HESE + HENU<br>+ EHE</td><td>HESE + HENU<br>+ EHE</td></tr><tr><th>Data set</th><th colspan="2">Description</th><th>Topology</th></tr><tr><td>PS</td><td colspan="2">Optimized for point-source searches, v<sub>&mu;</sub> candidates</td><td>Tracks</td></tr><tr><td>HESE</td><td colspan="2">High-energy starting events, all flavors</td><td>Tracks and<br>cascades</td></tr><tr><td>HENU</td><td colspan="2">High-energy selection of v<sub>&mu;</sub> candidates</td><td>Tracks</td></tr><tr><td>EHE</td><td colspan="2">Extremely high energy v<sub>&mu;</sub> candidates</td><td>Tracks</td></tr></tbody></table>

median of the angular resolution for the different detector configurations of IceCube and for data from the ANTARES detector (see Section 2.2). Note that analysis 1 is based not on the median of the angular resolution but on the estimator for the angular resolution on an event-by-event basis ($\sigma$).

The HESE data set contains 76 cascade-like events and 26 track-like events that have been collected between 2010 and 2017, as presented in Wandkowsky (2018). It consists of neutrinos of all flavors that interacted inside the detection volume, called starting events, with deposited energies ranging between about 20 TeV and 2 PeV. Integrated above 60 TeV, which corresponds to 60 events in total, the percentage of events of astrophysical origin, i.e., the astrophysical purity, is larger than 75% (Wandkowsky 2018), while the percentage is lower below 60 TeV. The angular resolution is about 1° for track-like events and 15° for cascade-like events above 100 TeV. The resolution of the deposited energy for tracks and cascades is around 10% (Aartsen et al. 2014a) without accounting for systematic uncertainties, but the cascades have a

better correlation with the primary neutrino energy since they deposit most of their energy inside the detector, while tracks do not.

The HENU data set consists of the 35 highest-energy track-like events with a reconstructed decl. $\ge -5^\circ$, which have been collected between 2009 and 2016 (Haack & Wiebusch 2017) for the measurement of the diffuse muon-neutrino flux (Aartsen et al. 2016). From the original data set starting at ~100 GeV, only events of high probability of nonatmospheric origin have been selected by applying an energy threshold of $\ge 200$ TeV on the reconstructed muon energy. This corresponds to an astrophysical purity of more than 50%. The astrophysical purity here is defined as the flux ratio of the astrophysical to the sum of atmospheric and astrophysical differential fluxes and thus depends on the assumed astrophysical flux. For the estimation of the astrophysical purity, the best-fit astrophysical flux from Aartsen et al. (2016), $d\phi/dE = 1.01 \times 10^{18} \text{ GeV}^{-1} \text{ cm}^{-2} \text{ s}^{-1} \text{ sr}^{-1} (E/100 \text{ TeV})^{-2.19}$, is used, as well as the best-fit atmospheric flux from the same reference.
The EHE data set consists of 20 events that have been
collected between 2008 and 2017 (Aartsen et al. 2017c, 2018a).
The selection is targeting high-energy track-like events of good
angular resolution ≪1°. The selection has been optimized to be
sensitive to events in the energy range of 0.5–10.0 PeV. The
integrated astrophysical purity depends on the assumption on
the spectrum of astrophysical events at the highest energies but
can be estimated as approximately 60% purity (see Table 2 in
Aartsen et al. 2017c).

## 2.2. The ANTARES Neutrino Telescope

The ANTARES telescope (Ageron et al. 2011) is a deep-sea
Cherenkov neutrino detector located 40 km offshore from
Toulon, France, in the Mediterranean Sea. The detector is
composed of 12 vertical strings anchored at the sea floor at a
depth of 2475 m. The strings are spaced at distances of about
70 m from each other, instrumenting a total volume of
~0.01 km³. Each string is equipped with 25 storeys of three
optical modules (ANTARES Collaboration et al. 2002),
vertically spaced by 14.5 m, for a total of 885 optical modules.
Each optical module houses a 10-inch photomultiplier tube
facing 45° downward to optimize the detection of light from
upward-going charged particles. The detector was completed
in 2008.

The ANTARES and IceCube detection principles are very
similar (see Section 2.1). Particles above the Cherenkov
threshold induce coherent radiation emitted in a cone with a
characteristic angle of 42° in water. The position, time, and
collected charge of the signals induced in the photomultiplier
tubes by detected photons are used to reconstruct the direction
and energy of events induced by neutrino interactions and
atmospheric muons. Trigger conditions based on combinations
of local coincidences are applied to identify signals due to
physics events over the environmental light background due to
⁴⁰K decays and bioluminescence (Aguilar et al. 2007).
ANTARES is thus also capable of detecting charged-current
and neutral-current neutrino interactions of all flavors.

For analysis 1, we use all track-like events from the 11 yr
data set used for point-source searches (PS) recorded between
2007 and 2017 (Albert et al. 2018; Illuminati et al. 2019). The
high-energy events (HENU) for analyses 2 and 3 are selected
from an earlier data set of track-like and cascade-like events
collected between 2007 and 2015 (Albert et al. 2017). In order
to ensure a high probability of astrophysical origin for analyses
2 and 3, we require an astrophysical purity ≥40% based on the
same definition as used for the HENU data set of IceCube. This
selection results in a total of three track-like events and no
cascade-like events, of which the arrival directions are shown
in Figure 1. An overview of the nomenclature is presented in
Table 1. The track-like events are combined with the respective
IceCube data sets. All events have a good angular resolution
that is below 0.4 above 10 TeV (Albert et al. 2017). Overall,
the angular resolution of ANTARES tracks is better than for
IceCube tracks for energies below 100 TeV and comparable
around 100 TeV and above. The median angular resolution, Ψ,
of the 11 yr data set compared to the IceCube PS data set is
shown in Figure 2.

Despite the smaller detection volume, the inclusion of
ANTARES data significantly improves the all-sky coverage of
the neutrino data set used in analysis 1. This is shown in
Section 4.1 and Figure 5 through the relative contribution to the
expected number of signal events for the individual PS data

sets of ANTARES and IceCube. Particularly in the Southern
Hemisphere, where the background from atmospheric muons
results in a higher energy threshold in IceCube, the ANTARES
data contribute substantially to the signal acceptance for soft
source spectra.

## 2.3. The Pierre Auger Observatory

The Pierre Auger Observatory is located in Argentina (32° S, 69° W) at a mean altitude of 1.4 km above sea level (The Pierre Auger Collaboration 2015). The observatory has a hybrid design combining an array of particle detectors at ground (surface detector, SD) and an atmospheric fluorescence detector (FD) for detecting the air showers caused by UHECRs interacting with the atmosphere. The SD array is composed of 1660 water-Cherenkov detectors spread over an area of 3000 km². The SD area is overlooked by the FD, which consists of 27 wide-angle optical telescopes located at four sites. The reconstruction of SD events is described in detail in Aab et al. (2020b). The energy estimate is based on the signal at 1000 m from the reconstructed intersection of the shower axis with the ground, which is extracted with a fit of the lateral distribution of signals in the individual detectors. This value is then corrected to take into account the different absorption suffered by showers coming at different angles. Due to the hybrid design of the observatory, this energy estimator is calibrated via the correlation with the near-calorimetric energy measured by the FD with the events observed by both SD and FD. Through this calibration, the energy estimate for SD is done without relying on Monte Carlo. At the energies considered in this work, the systematic uncertainty on the energy scale is 14% (Dawson 2019), the statistical uncertainty on the energy due to the number of triggered SD stations and uncertainties of their response is smaller than 12% (Abreu et al. 2011; Aab et al. 2020b), and the angular uncertainty is less than 0.9 (Bonifazi & The Pierre Auger Collaboration 2009).

The UHECR data set used here consists of 324 events
observed with the SD between 2004 January and 2017 April
(Aab et al. 2018) with reconstructed energies ≥52 EeV and
zenith angles ≪80°. The data statistics is enlarged with respect
to the previously used data set (Aab et al. 2015) by 93 events
and includes an updated energy calibration based on a larger
number of hybrid events and an improved absorption
correction. The angular acceptance translates into a field of
view ranging from −90° to 45° in decl. All energies are scaled
up by a constant factor of 14% to match both Auger and TA
energies on a common energy scale (see Section 2.4),
following the recommendation of the Auger–TA joint working
group (Biteau et al. 2019). This scaling factor was deteminied
by cross-calibrating the measured UHECR fluxes in the
declination band around the celestial equator covered by both
observatories. It is chosen such that the fluxes in the common
declination band match at 40 EeV of the Auger data set and at
53 EeV of the TA data set, respectively. The celestial
distribution of the UHECR arrival directions is shown in
Figure 1. The directional exposure as a function of decl. can be
found in Figure 3, which amounts to an integrated geometric
exposure of 91,300 km² yr sr.

## 2.4. The Telescope Array

The TA is located in Millard County, Utah, USA (39.3°N,
112.9°W) at an altitude of about 1.4 km (Kawai et al. 2008). It
The EHE data set consists of 20 events that have been
collected between 2008 and 2017 (Aartsen et al. 2017c, 2018a).
The selection is targeting high-energy track-like events of good
angular resolution $\le 1°$. The selection has been optimized to be
sensitive to events in the energy range of 0.5–10.0 PeV. The
integrated astrophysical purity depends on the assumption on
the spectrum of astrophysical events at the highest energies but
can be estimated as approximately 60% purity (see Table 2 in
Aartsen et al. 2017c).

## 2.2. The ANTARES Neutrino Telescope

The ANTARES telescope (Ageron et al. 2011) is a deep-sea
Cherenkov neutrino detector located 40 km offshore from
Toulon, France, in the Mediterranean Sea. The detector is
composed of 12 vertical strings anchored at the sea floor at a
depth of 2475 m. The strings are spaced at distances of about
70 m from each other, instrumenting a total volume of
~0.01 km³. Each string is equipped with 25 storeys of three
optical modules (ANTARES Collaboration et al. 2002),
vertically spaced by 14.5 m, for a total of 885 optical modules.
Each optical module houses a 10-inch photomultiplier tube
facing 45° downward to optimize the detection of light from
upward-going charged particles. The detector was completed
in 2008.

The ANTARES and IceCube detection principles are very
similar (see Section 2.1). Particles above the Cherenkov
threshold induce coherent radiation emitted in a cone with a
characteristic angle of 42° in water. The position, time, and
collected charge of the signals induced in the photomultiplier
tubes by detected photons are used to reconstruct the direction
and energy of events induced by neutrino interactions and
atmospheric muons. Trigger conditions based on combinations
of local coincidences are applied to identify signals due to
physics events over the environmental light background due to
⁴⁰K decays and bioluminescence (Aguilar et al. 2007).
ANTARES is thus also capable of detecting charged-current
and neutral-current neutrino interactions of all flavors.

For analysis 1, we use all track-like events from the 11 yr
data set used for point-source searches (PS) recorded between
2007 and 2017 (Albert et al. 2018; Illuminati et al. 2019). The
high-energy events (HENU) for analyses 2 and 3 are selected
from an earlier data set of track-like and cascade-like events
collected between 2007 and 2015 (Albert et al. 2017). In order
to ensure a high probability of astrophysical origin for analyses
2 and 3, we require an astrophysical purity $\ge 40\%$ based on the
same definition as used for the HENU data set of IceCube. This
selection results in a total of three track-like events and no
cascade-like events, of which the arrival directions are shown
in Figure 1. An overview of the nomenclature is presented in
Table 1. The track-like events are combined with the respective
IceCube data sets. All events have a good angular resolution
that is below 0.4 above 10 TeV (Albert et al. 2017). Overall,
the angular resolution of ANTARES tracks is better than for
IceCube tracks for energies below 100 TeV and comparable
around 100 TeV and above. The median angular resolution, $\Psi$,
of the 11 yr data set compared to the IceCube PS data set is
shown in Figure 2.

Despite the smaller detection volume, the inclusion of
ANTARES data significantly improves the all-sky coverage of
the neutrino data set used in analysis 1. This is shown in
Section 4.1 and Figure 5 through the relative contribution to the
expected number of signal events for the individual PS data

sets of ANTARES and IceCube. Particularly in the Southern
Hemisphere, where the background from atmospheric muons
results in a higher energy threshold in IceCube, the ANTARES
data contribute substantially to the signal acceptance for soft
source spectra.

## 2.3. The Pierre Auger Observatory

The Pierre Auger Observatory is located in Argentina (32° S,
69° W) at a mean altitude of 1.4 km above sea level (The Pierre
Auger Collaboration 2015). The observatory has a hybrid
design combining an array of particle detectors at ground
(surface detector, SD) and an atmospheric fluorescence detector
(FD) for detecting the air showers caused by UHECRs
interacting with the atmosphere. The SD array is composed
of 1660 water-Cherenkov detectors spread over an area of
3000 km². The SD area is overlooked by the FD, which
consists of 27 wide-angle optical telescopes located at four
sites. The reconstruction of SD events is described in detail in
Aab et al. (2020b). The energy estimate is based on the signal
at 1000 m from the reconstructed intersection of the shower
axis with the ground, which is extracted with a fit of the lateral
distribution of signals in the individual detectors. This value is
then corrected to take into account the different absorption
suffered by showers coming at different angles. Due to the
hybrid design of the observatory, this energy estimator is
calibrated via the correlation with the near-calorimetric energy
measured by the FD with the events observed by both SD and
FD. Through this calibration, the energy estimate for SD is
done without relying on Monte Carlo. At the energies
considered in this work, the systematic uncertainty on the
energy scale is 14% (Dawson 2019), the statistical uncertainty
on the energy due to the number of triggered SD stations and
uncertainties of their response is smaller than 12% (Abreu et al.
2011; Aab et al. 2020b), and the angular uncertainty is less than
0.9 (Bonifazi & The Pierre Auger Collaboration 2009).

The UHECR data set used here consists of 324 events
observed with the SD between 2004 January and 2017 April
(Aab et al. 2018) with reconstructed energies $\ge 52$ EeV and
zenith angles $\le 80°$. The data statistics is enlarged with respect
to the previously used data set (Aab et al. 2015) by 93 events
and includes an updated energy calibration based on a larger
number of hybrid events and an improved absorption
correction. The angular acceptance translates into a field of
view ranging from -90° to 45° in decl. All energies are scaled
up by a constant factor of 14% to match both Auger and TA
energies on a common energy scale (see Section 2.4),
following the recommendation of the Auger-TA joint working
group (Biteau et al. 2019). This scaling factor was detemined
by cross-calibrating the measured UHECR fluxes in the
declination band around the celestial equator covered by both
observatories. It is chosen such that the fluxes in the common
declination band match at 40 EeV of the Auger data set and at
53 EeV of the TA data set, respectively. The celestial
distribution of the UHECR arrival directions is shown in
Figure 1. The directional exposure as a function of decl. can be
found in Figure 3, which amounts to an integrated geometric
exposure of 91,300 km² yr sr.

## 2.4. The Telescope Array

The TA is located in Millard County, Utah, USA (39.3°N,
112.9°W) at an altitude of about 1.4 km (Kawai et al. 2008). It
<figure><img src="image_6.png" /><figcaption>**Figure 3.** Directional exposures of TA, Auger, and their sum. The underlying geometric exposure functions are derived from Sommers (2001). See also Figure 1 of Biteau et al. (2019).</figcaption></figure>

consists of a surface detector (SD) array, composed of 507 plastic scintillation detectors of $3\,m^2$ each. The SD stations are located on a square grid with 1.2 km separation, which extends over an area of $700\,km^2$ (Abu-Zayyad et al. 2012). The TA SD array observes UHECRs with a duty cycle near 100%. With its wide field of view, the SD array covers a range from -15° to 90° in decl. In addition to the SD, there are three fluorescence telescope stations, instrumented with 12–14 telescopes each (Tokuno et al. 2012). The telescope stations observe the sky above the SD array and measure the longitudinal development of the air showers as they traverse the atmosphere.

The data set for this analysis follows the selection in Abbasi et al. (2014), which has been updated to 9 yr of data in Abbasi et al. (2018). The data set is identical to the data set used for anisotropy analyses presented in Troitsky et al. (2017). It consists of 143 events observed with the SD between 2008 May and 2017 May with reconstructed energies ≥57 EeV and zenith angles ≤55°. At these energies, the angular uncertainty is about 1°.5. The statistical uncertainty on the reconstructed energy is about 15%–20%, with an additional systematic uncertainty on the energy scale of 21% (Abbasi et al. 2018). All energies are scaled down by -14% to match both Auger and TA energies on a common energy scale (see Section 2.3), following the recommendation of the Auger–TA joint working group (Biteau et al. 2019).

The celestial distribution of selected events is shown in Figure 1. The right-hand side of the figure is a histogram of the sine of decl. of all UHECR events, showing that TA data substantially contributes to the full-sky exposure of the combined UHECR data set. Figure 3 shows the directional exposure as a function of decl., with an integrated total exposure of 11,600 km$^2$ yr sr (Biteau et al. 2019).

### 3. Magnetic Deflections

Despite their extremely high rigidity, UHECRs are deflected in galactic and in extragalactic magnetic fields (GMF/EGMF) by a nonnegligible amount. Neither the strength and correlation length of the extragalactic magnetic fields (Durrer & Neronov 2013; Kronberg 1994) nor the distances of the UHECR sources are known well. Measurements of the Faraday rotation of extragalactic sources indicate that the extragalactic magnetic fields are weaker than 1 nG (Pshirkov et al. 2016). Assuming a correlation length of ~1 Mpc, this results in a deflection less than 2° for protons of 100 EeV, even at source distances of

50 Mpc. In line with the previously reported results (The IceCube Collaboration et al. 2016), the deflection outside of our Galaxy is assumed to be generally weaker than within our Galaxy, and it is not modeled explicitly but benchmarked within the uncertainty of the deflection by GMF and the uncertainty of the rigidity due to the unknown composition.

The measurement and modeling of the GMF is a complex task and the subject of ongoing discussion.<fnref n="240" /> Among different proposed models, we use the JF2012 (Jansson & Farrar 2012) model and the PT2011 (Pshirkov et al. 2011) model to estimate the deflection of the UHECRs in the GMF. Both models consist of a disk and a halo component, while the JF2012 model has an additional X-shaped field component perpendicular to the disk. A propagation simulation has been conducted with a Monte Carlo approach to estimate the deflection of protons with an energy of 100 EeV that are distributed isotropically outside of the GMF. The resulting deflections are shown in Figure 4, split into the Galactic northern and southern hemispheres based on the arrival direction of the proton. The estimated deflection shows a complex structure that differs considerably for the two models. However, the mean deflection is about 3° in both cases. Due to the heavy tails of the distributions, the mean is consistently larger than the median; thus, we choose the mean as a conservative deflection estimate. The split into the two hemispheres shows a considerably smaller deflection in the Galactic north than in the Galactic south owing to north–south asymmetries present in both models of the GMF. For this work we have chosen a robust benchmark modeling of the effect of the GMF, thus avoiding biases of detailed model uncertainties. The deflection process is assumed to be random, resulting in a symmetric 2D Gaussian distribution of UHECR arrival directions around a given source direction. In reverse, the source position is assumed to be located within the 2D Gaussian distribution around the arrival direction of a UHECR event. The standard deviation of the Gaussian deflection, $\sigma_{MD}$, depends on a scaling factor, C, and inversely on the energy of the UHECR event, $E_{CR}$,

$$ \sigma_{MD} = \frac{D_0 \cdot C}{E_{CR}/100 \, \text{EeV}} \quad (1) $$

The default deflection, $D_0$, for $C=1$ and $E_{CR}=100$ EeV is derived from the mean of the deflection values obtained with the simulation, which are shown in Figure 4. Note that the deflection is usually a 2D quantity with (x, y) coordinates, which in Figure 4 is projected to the absolute value, i.e., $\sqrt{x^2 + y^2}$. Furthermore, larger and thus more conservative values of the deflection will be tested to include an uncertainty of the GMF model. The scaling factor C thus accounts for uncertainties of both GMF model and UHECR charge, which is not known on an event-by-event basis at highest energies.

Analysis 1 uses a default deflection of $D_0 = 3°$ over the whole sky, while analysis 2 uses $D_{North/South} = (2°.4, 3°.7)$ for UHECRs with arrival directions from the Galactic northern and southern hemispheres, respectively. The different choices have been made based on the respectively better sensitivity for the two analyses. Analyses 1 and 2 implement the deflection into their methods as described in the respective Sections 4.1 and 4.2. Analysis 3 employs a model-independent approach with respect to the UHECR deflection.

<footnote n="240">²⁴⁰ See https://icrc2021-venue.desy.de for a recent review by Tess Jaffe on the GMF at ICRC2021: “Constraining Magnetic Fields at Galactic Scales”.</footnote>
<figure><img src="image_6.png" /><figcaption>**Figure 3.** Directional exposures of TA, Auger, and their sum. The underlying geometric exposure functions are derived from Sommers (2001). See also Figure 1 of Biteau et al. (2019).</figcaption></figure>

consists of a surface detector (SD) array, composed of 507 plastic scintillation detectors of $3\,m^2$ each. The SD stations are located on a square grid with 1.2 km separation, which extends over an area of $700\,km^2$ (Abu-Zayyad et al. 2012). The TA SD array observes UHECRs with a duty cycle near 100%. With its wide field of view, the SD array covers a range from -15° to 90° in decl. In addition to the SD, there are three fluorescence telescope stations, instrumented with 12–14 telescopes each (Tokuno et al. 2012). The telescope stations observe the sky above the SD array and measure the longitudinal development of the air showers as they traverse the atmosphere.

The data set for this analysis follows the selection in Abbasi et al. (2014), which has been updated to 9 yr of data in Abbasi et al. (2018). The data set is identical to the data set used for anisotropy analyses presented in Troitsky et al. (2017). It consists of 143 events observed with the SD between 2008 May and 2017 May with reconstructed energies ≥57 EeV and zenith angles ≤55°. At these energies, the angular uncertainty is about 1°.5. The statistical uncertainty on the reconstructed energy is about 15%–20%, with an additional systematic uncertainty on the energy scale of 21% (Abbasi et al. 2018). All energies are scaled down by -14% to match both Auger and TA energies on a common energy scale (see Section 2.3), following the recommendation of the Auger–TA joint working group (Biteau et al. 2019).

The celestial distribution of selected events is shown in Figure 1. The right-hand side of the figure is a histogram of the sine of decl. of all UHECR events, showing that TA data substantially contributes to the full-sky exposure of the combined UHECR data set. Figure 3 shows the directional exposure as a function of decl., with an integrated total exposure of 11,600 $km^2$ yr sr (Biteau et al. 2019).

### 3. Magnetic Deflections

Despite their extremely high rigidity, UHECRs are deflected in galactic and in extragalactic magnetic fields (GMF/EGMF) by a nonnegligible amount. Neither the strength and correlation length of the extragalactic magnetic fields (Durrer & Neronov 2013; Kronberg 1994) nor the distances of the UHECR sources are known well. Measurements of the Faraday rotation of extragalactic sources indicate that the extragalactic magnetic fields are weaker than 1 nG (Pshirkov et al. 2016). Assuming a correlation length of ~1 Mpc, this results in a deflection less than 2° for protons of 100 EeV, even at source distances of

50 Mpc. In line with the previously reported results (The IceCube Collaboration et al. 2016), the deflection outside of our Galaxy is assumed to be generally weaker than within our Galaxy, and it is not modeled explicitly but benchmarked within the uncertainty of the deflection by GMF and the uncertainty of the rigidity due to the unknown composition.

The measurement and modeling of the GMF is a complex task and the subject of ongoing discussion.<fnref n="240" /> Among different proposed models, we use the JF2012 (Jansson & Farrar 2012) model and the PT2011 (Pshirkov et al. 2011) model to estimate the deflection of the UHECRs in the GMF. Both models consist of a disk and a halo component, while the JF2012 model has an additional X-shaped field component perpendicular to the disk. A propagation simulation has been conducted with a Monte Carlo approach to estimate the deflection of protons with an energy of 100 EeV that are distributed isotropically outside of the GMF. The resulting deflections are shown in Figure 4, split into the Galactic northern and southern hemispheres based on the arrival direction of the proton. The estimated deflection shows a complex structure that differs considerably for the two models. However, the mean deflection is about 3° in both cases. Due to the heavy tails of the distributions, the mean is consistently larger than the median; thus, we choose the mean as a conservative deflection estimate. The split into the two hemispheres shows a considerably smaller deflection in the Galactic north than in the Galactic south owing to north–south asymmetries present in both models of the GMF. For this work we have chosen a robust benchmark modeling of the effect of the GMF, thus avoiding biases of detailed model uncertainties. The deflection process is assumed to be random, resulting in a symmetric 2D Gaussian distribution of UHECR arrival directions around a given source direction. In reverse, the source position is assumed to be located within the 2D Gaussian distribution around the arrival direction of a UHECR event. The standard deviation of the Gaussian deflection, $\sigma_{MD}$, depends on a scaling factor, C, and inversely on the energy of the UHECR event, $E_{CR}$,

$$ \sigma_{MD} = \frac{D_0 \cdot C}{E_{CR}/100 \, \text{EeV}} \quad (1) $$

The default deflection, $D_0$, for $C=1$ and $E_{CR}=100$ EeV is derived from the mean of the deflection values obtained with the simulation, which are shown in Figure 4. Note that the deflection is usually a 2D quantity with (x, y) coordinates, which in Figure 4 is projected to the absolute value, i.e., $\sqrt{x^2 + y^2}$. Furthermore, larger and thus more conservative values of the deflection will be tested to include an uncertainty of the GMF model. The scaling factor C thus accounts for uncertainties of both GMF model and UHECR charge, which is not known on an event-by-event basis at highest energies.

Analysis 1 uses a default deflection of $D_0 = 3°$ over the whole sky, while analysis 2 uses $D_{North/South} = (2°.4, 3°.7)$ for UHECRs with arrival directions from the Galactic northern and southern hemispheres, respectively. The different choices have been made based on the respectively better sensitivity for the two analyses. Analyses 1 and 2 implement the deflection into their methods as described in the respective Sections 4.1 and 4.2. Analysis 3 employs a model-independent approach with respect to the UHECR deflection.

<footnote n="240">²⁴⁰ See https://icrc2021-venue.desy.de for a recent review by Tess Jaffe on the GMF at ICRC2021: “Constraining Magnetic Fields at Galactic Scales”.</footnote>
<figure><img src="image_7.png" /><figcaption>**Figure 4.** Deflection simulation for protons of 100 EeV and two different galactic magnetic models: PT2011 (Pshirkov et al. 2011) and JF2012 (Jansson & Farrar 2012). Shaded areas in blue and orange show the histogram split into Galactic northern and southern hemispheres, respectively, with their sum shown as a black line. Additionally, Rayleigh pdfs are shown in the same colors; a Rayleigh function with mode σ is the 1D projection of a symmetric 2D Gaussian function with width σ, with the projection being (x, y) → √(x² + y²). The position of the markers indicate the mode of the Rayleigh pdf and thus the value of the default deflection, D. Note that the integrals of the split pdfs are normalized to 0.5, while the integrals of the full pdfs are normalized to 1.</figcaption></figure>

<figure><img src="image_8.png" /></figure>

## 4. Analysis Methods

### 4.1. Unbinned Neutrino Point-source Search with UHECR Information

The goal of analysis 1 is to find point-like neutrino sources that are spatially correlated with UHECR arrival directions within a region derived from their magnetic deflection estimate. The search for neutrino sources utilizes the unbinned maximum likelihood analysis commonly used in IceCube (Aartsen et al. 2017b, 2019, 2020c). This enables an easy combination of the high-statistics, full-sky neutrino data sets, which are the PS data sets of IceCube and ANTARES described in Sections 2.1 and 2.2, respectively. In addition to this standard method, the magnetic deflection regions, defined by the UHECR arrival directions, energy, and scaling factor, are used for constraining the possible source regions.

The unbinned neutrino likelihood in the source direction $\vec{\Omega} = (\alpha, \delta)$ in R.A. and decl. consists of the sum of a signal probability density function (pdf), S, and a background pdf, B,

$$ \mathcal{L}(n_s, \gamma) = \prod_{j=1}^{7} \prod_{i=1}^{N_j} f_j(\sin \delta, \gamma) \frac{n_s}{N_j} \cdot S(\vec{\Omega}_i, E_i, \sigma_i|\gamma, \vec{\Omega}) + \left(1 - f_j(\sin \delta, \gamma) \frac{n_s}{N_j}\right) \cdot B(\delta_i, E_i). \quad (2) $$

The likelihood product with index i runs over all neutrino events within each data set with index j (see Sections 2.1 and 2.2). The likelihood product with index j runs over all data sets to yield the final likelihood function. The likelihood combination of the seven data sets by ANTARES and IceCube is thus handled the same way as described in Aartsen et al. (2017b). The signal and background pdfs, S and B, are evaluated for four observables per neutrino event, weighted with the number of signal events over total number of events per data set, $n_s/N_j$. The observables are the reconstructed R.A. and decl., summarized as $\vec{\Omega}_i = (\alpha_i, \delta_i)$, reconstructed energy, $E_i$, and angular error estimator, $\sigma_i$. The signal pdf consists of two terms: the first term is a decl.-dependent reconstructed energy

distribution, where the underlying neutrino flux is modeled as a power law,

$$ \frac{d\Phi^{\nu}}{dE} = \frac{dN^{\nu}}{dE \, dA \, dt \, d\Omega} = \Phi_{0}^{\nu} \cdot \left( \frac{E}{1 \, \text{GeV}} \right)^{-\gamma}. \quad (3) $$

Here $\Phi_0^\nu$ is the flux normalization, 1 GeV is the corresponding pivot energy, and $\gamma$ is the spectral index of the power law. The second term of the signal pdf is a spatial term modeled as a Gaussian, with the width, $\sigma_i$, given by the angular error estimator of each neutrino candidate on an event-by-event basis. The background pdf is determined as a function of reconstructed energy, $E_i$, and decl., $\delta_i$, from randomized experimental data. A full description of the signal and background pdfs, S and B, can be found in Aartsen et al. (2017b). The proportionality factor, $f_j(\sin \delta, \gamma)$, is the relative signal acceptance per neutrino data set calculated from the expected number of signal events via

$$ \begin{aligned} f_j(\sin \delta, \gamma) &= \frac{n_j}{n_{\text{tot}}} \\ &= \frac{T^j \int (d\Phi^\nu/dE) A_{\text{eff}}^j(E, \sin \delta) dE}{\sum_{j=1}^7 T^j \int (d\Phi^\nu/dE) A_{\text{eff}}^j(E, \sin \delta) dE}. \end{aligned} \quad (4) $$

This factor is used to correctly weight the signal contribution of each data set j for a given source decl. and spectral index. Per data set j, the live time is denoted with $T^j$ and the effective area is denoted with $A_{\text{eff}}^j$. Figure 5 shows the proportionality factor of each data set as a function of decl. for two spectral indices, $\gamma=2.0$ and $2.5$. Note that all data sets are evaluated with the same formulation of the likelihood function as used by IceCube, instead of combining different formulations as described in Albert et al. (2020).

The best-fit signal parameters, $\hat{n}_s$ and $\hat{\gamma}$, at a given source position, $\vec{\Omega}$, are obtained with the maximum likelihood method. The number of events and the proportionality factor are related to the neutrino flux using the respective live time and effective
<figure><img src="image_7.png" /><figcaption>**Figure 4.** Deflection simulation for protons of 100 EeV and two different galactic magnetic models: PT2011 (Pshirkov et al. 2011) and JF2012 (Jansson & Farrar 2012). Shaded areas in blue and orange show the histogram split into Galactic northern and southern hemispheres, respectively, with their sum shown as a black line. Additionally, Rayleigh pdfs are shown in the same colors; a Rayleigh function with mode $\sigma$ is the 1D projection of a symmetric 2D Gaussian function with width $\sigma$, with the projection being $(x, y) \rightarrow \sqrt{x^2 + y^2}$. The position of the markers indicate the mode of the Rayleigh pdf and thus the value of the default deflection, $D$. Note that the integrals of the split pdfs are normalized to 0.5, while the integrals of the full pdfs are normalized to 1.</figcaption></figure>

<figure><img src="image_8.png" /></figure>

# 4. Analysis Methods

## 4.1. Unbinned Neutrino Point-source Search with UHECR Information

The goal of analysis 1 is to find point-like neutrino sources that are spatially correlated with UHECR arrival directions within a region derived from their magnetic deflection estimate. The search for neutrino sources utilizes the unbinned maximum likelihood analysis commonly used in IceCube (Aartsen et al. 2017b, 2019, 2020c). This enables an easy combination of the high-statistics, full-sky neutrino data sets, which are the PS data sets of IceCube and ANTARES described in Sections 2.1 and 2.2, respectively. In addition to this standard method, the magnetic deflection regions, defined by the UHECR arrival directions, energy, and scaling factor, are used for constraining the possible source regions.

The unbinned neutrino likelihood in the source direction $\vec{\Omega} = (\alpha, \delta)$ in R.A. and decl. consists of the sum of a signal probability density function (pdf), $S$, and a background pdf, $B$,

$$ \mathcal{L}(n_s, \gamma) = \prod_{j=1}^{7} \prod_{i=1}^{N_j} f_j(\sin \delta, \gamma) \frac{n_s}{N_j} \cdot S(\vec{\Omega}_i, E_i, \sigma_i|\gamma, \vec{\Omega}) + \left(1 - f_j(\sin \delta, \gamma) \frac{n_s}{N_j}\right) \cdot B(\delta_i, E_i). \quad (2) $$

The likelihood product with index $i$ runs over all neutrino events within each data set with index $j$ (see Sections 2.1 and 2.2). The likelihood product with index $j$ runs over all data sets to yield the final likelihood function. The likelihood combination of the seven data sets by ANTARES and IceCube is thus handled the same way as described in Aartsen et al. (2017b). The signal and background pdfs, $S$ and $B$, are evaluated for four observables per neutrino event, weighted with the number of signal events over total number of events per data set, $n_s/N_j$. The observables are the reconstructed R.A. and decl., summarized as $\vec{\Omega}_i = (\alpha_i, \delta_i)$, reconstructed energy, $E_i$, and angular error estimator, $\sigma_i$. The signal pdf consists of two terms: the first term is a decl.-dependent reconstructed energy

distribution, where the underlying neutrino flux is modeled as a power law,

$$ \frac{d\Phi^{\nu}}{dE} = \frac{dN^{\nu}}{dE \, dA \, dt \, d\Omega} = \Phi_{0}^{\nu} \cdot \left( \frac{E}{1 \, \text{GeV}} \right)^{-\gamma}. \quad (3) $$

Here $\Phi_0^\nu$ is the flux normalization, 1 GeV is the corresponding pivot energy, and $\gamma$ is the spectral index of the power law. The second term of the signal pdf is a spatial term modeled as a Gaussian, with the width, $\sigma_i$, given by the angular error estimator of each neutrino candidate on an event-by-event basis. The background pdf is determined as a function of reconstructed energy, $E_i$, and decl., $\delta_i$, from randomized experimental data. A full description of the signal and background pdfs, $S$ and $B$, can be found in Aartsen et al. (2017b). The proportionality factor, $f_j(\sin \delta, \gamma)$, is the relative signal acceptance per neutrino data set calculated from the expected number of signal events via

$$ f_j(\sin \delta, \gamma) = \frac{n_j}{n_{\text{tot}}} = \frac{T^j \int (d\Phi^\nu/dE) A_{\text{eff}}^j(E, \sin \delta) dE}{\sum_{j=1}^7 T^j \int (d\Phi^\nu/dE) A_{\text{eff}}^j(E, \sin \delta) dE}. \quad (4) $$

This factor is used to correctly weight the signal contribution of each data set $j$ for a given source decl. and spectral index. Per data set $j$, the live time is denoted with $T^j$ and the effective area is denoted with $A_{\text{eff}}^j$. Figure 5 shows the proportionality factor of each data set as a function of decl. for two spectral indices, $\gamma=2.0$ and $2.5$. Note that all data sets are evaluated with the same formulation of the likelihood function as used by IceCube, instead of combining different formulations as described in Albert et al. (2020).

The best-fit signal parameters, $\hat{n}_s$ and $\hat{\gamma}$, at a given source position, $\vec{\Omega}$, are obtained with the maximum likelihood method. The number of events and the proportionality factor are related to the neutrino flux using the respective live time and effective
<figure><img src="image_9.png" /><figcaption>**Figure 5.** The relative signal contribution, $n_j/n_{tot}$, for analysis 1 of the different configurations of IceCube and the ANTARES point-source data set as a function of the decl. $\delta$. The left panel shows a hard underlying power-law signal spectrum (Equation (3)) with index $\gamma = 2.0$, while the right panel shows a softer spectrum with $\gamma = 2.5$. The relative signal contributions from partial detector configurations of IceCube are shown in blue, while the contributions of the full detector configuration of IceCube with 86 strings are shown in orange. The ANTARES contribution is shown in gray. The lines as listed in the rightmost legend indicate the contributions from each data set individually.</figcaption></figure>

area of each data set via

$$n_s = \sum_{j=1}^{7} T^j \int \frac{d\Phi^\nu}{dE} A_{\text{eff}}^j(E, \sin \delta) dE. \qquad (5)$$

The corresponding significance of a source at position $\vec{\Omega}$ is evaluated using a likelihood ratio test, which yields the test statistic (TS)

$$\text{TS}_{\nu}(\vec{\Omega}) = 2 \log \left( \frac{\mathcal{L}(\hat{n}_s, \hat{\gamma})}{\mathcal{L}(n_s = 0)} \right) \bigg|_{\vec{\Omega}}. \qquad (6)$$

Here the null hypothesis is defined via $n_s = 0$, representing the case of no neutrino sources in the vicinity of UHECR events. Instead of searching for a single neutrino source, the whole sky is searched on a HEALpix (Górski et al. 2005) grid with a resolution of approximately 0.2. This results in a sky map of the TS values at each grid center, $\vec{\Omega}_{\text{grid}}$.

The second step is to combine the neutrino likelihood with the information provided by UHECR events and their deflection estimate. The deflection estimate for one UHECR with index $k$ and arrival direction $\vec{\Omega}_k^{\text{CR}}$, which possibly originated in the direction of $\vec{\Omega}_{\text{grid}}$, is defined as a 2D Gaussian. Its width is the quadratic sum of the magnetic deflection estimate, $\sigma_{\text{MD}}^2$ of Equation (1), and the UHECR angular reconstruction error, $\sigma_{\text{CR}}^2$, which is 0.9 for Auger events and 1.5 for TA events,

$$P_k \propto \exp\left(-\frac{(\vec{\Omega}_{\text{grid}} - \vec{\Omega}_k^{\text{CR}})^2}{2(\sigma_{\text{MD}}^2 + \sigma_{\text{CR}}^2)}\right). \qquad (7)$$

The term $P_k$ acts as a spatial constraint, which is multiplied by the neutrino likelihood function defined in Equation (2) via $\mathcal{L} \rightarrow \mathcal{L} \cdot P_k$. Effectively, the constraint is added as a logarithmic term to the neutrino TS defined for each grid point in Equation (6) via $\text{TS} \rightarrow \text{TS} + 2 \cdot \log(P_k)$. Finally, the maximum of the combined UHECR–neutrino TS is found at the best-fit grid point, $\hat{\Omega}_{\text{grid}}$, via

$$\text{TS}_{k}^{\text{comb}}(\hat{\Omega}_{\text{grid}}) = \max \left( \text{TS}_{\nu}(\hat{\Omega}_{\text{grid}}) - \frac{(\hat{\Omega}_{\text{grid}} - \vec{\Omega}_{k}^{\text{CR}})^2}{\sigma_{\text{MD}}^2 + \sigma_{\text{CR}}^2} \right). \qquad (8)$$

The maximum marks the neutrino source candidate that is the most likely counterpart to the respective UHECR event used to calculate the spatial constraint. The normalization in Equation (7) adds only a constant term to the TS, which can be omitted when calculating significances and p-values based on pseudo-experiments (see below). As a third, final step, this procedure is repeated for all UHECRs, and all obtained TS values are added up, yielding the final test statistic. This search strategy was first developed in the context of this point-source correlation analysis and first outlined in Section 4 of Schumacher (2019). It has already been applied also to other neutrino correlation searches with spatially extended source localization, namely, the correlation with ANITA events (Aartsen et al. 2020a) and with gravitational wave events (Aartsen et al. 2020b). Note that the analysis described here is improved with respect to the previous search (The IceCube Collaboration et al. 2016, Section 5), as it models the displacement of a point-like neutrino source with respect to the UHECR arrival direction based on the assumed magnetic deflection. Here the position and flux of a point-like source are fit in the vicinity of the UHECR direction, while in the previous search a spatially extended neutrino emission around the UHECR direction was fit.

Six different signal hypotheses are tested, which are characterized by a lower cut on the UHECR energy, $E_{\text{CR}} > E_{\text{cut}}$ with $E_{\text{cut}} \in \{70, 85, 100\}$ EeV, and the scaling factor of the deflection estimate, $C \in \{1, 2\}$. The lower energy cut is a threshold for selecting only the highest-energy UHECRs with the lowest deflection for this analysis. No analogous energy cut is applied to the neutrino data. The scaling factor $C$ is a model parameter for scaling the baseline magnetic deflection, and it is not derived from the UHECR data. The baseline magnetic deflection is $D_0 = 3^\circ$ for all UHECRs, which is then converted into the spatial constraint of an individual UHECR event using Equations (1) and (7). The corresponding sensitivity and $3\sigma$-discovery potential are evaluated based on the normalization of the neutrino flux per source, $\Phi_0^\nu$ (see Equation (3)). The $3\sigma$ threshold is chosen since the background expectation needs to be calculated based on pseudo-experiments, as described in the next paragraph. A $5\sigma$-discovery potential is computationally too expensive to be calculated for the various hypotheses. Sensitivity is defined as
<figure><img src="image_9.png" /><figcaption>**Figure 5.** The relative signal contribution, $n_j/n_{tot}$, for analysis 1 of the different configurations of IceCube and the ANTARES point-source data set as a function of the decl. $\delta$. The left panel shows a hard underlying power-law signal spectrum (Equation (3)) with index $\gamma = 2.0$, while the right panel shows a softer spectrum with $\gamma = 2.5$. The relative signal contributions from partial detector configurations of IceCube are shown in blue, while the contributions of the full detector configuration of IceCube with 86 strings are shown in orange. The ANTARES contribution is shown in gray. The lines as listed in the rightmost legend indicate the contributions from each data set individually.</figcaption></figure>

area of each data set via

$$n_s = \sum_{j=1}^{7} T^j \int \frac{d\Phi^\nu}{dE} A_{\text{eff}}^j(E, \sin \delta) dE. \qquad (5)$$

The corresponding significance of a source at position $\vec{\Omega}$ is evaluated using a likelihood ratio test, which yields the test statistic (TS)

$$TS_{\nu}(\vec{\Omega}) = 2 \log \left( \frac{\mathcal{L}(\hat{n}_s, \hat{\gamma})}{\mathcal{L}(n_s = 0)} \right) \bigg|_{\vec{\Omega}}. \qquad (6)$$

Here the null hypothesis is defined via $n_s = 0$, representing the case of no neutrino sources in the vicinity of UHECR events. Instead of searching for a single neutrino source, the whole sky is searched on a HEALpix (Górski et al. 2005) grid with a resolution of approximately 0.2. This results in a sky map of the TS values at each grid center, $\vec{\Omega}_{\text{grid}}$.

The second step is to combine the neutrino likelihood with the information provided by UHECR events and their deflection estimate. The deflection estimate for one UHECR with index $k$ and arrival direction $\vec{\Omega}_k^{\text{CR}}$, which possibly originated in the direction of $\vec{\Omega}_{\text{grid}}$, is defined as a 2D Gaussian. Its width is the quadratic sum of the magnetic deflection estimate, $\sigma_{\text{MD}}^2$ of Equation (1), and the UHECR angular reconstruction error, $\sigma_{\text{CR}}^2$, which is 0.9 for Auger events and 1.5 for TA events,

$$P_k \propto \exp \left( - \frac{(\vec{\Omega}_{\text{grid}} - \vec{\Omega}_k^{\text{CR}})^2}{2(\sigma_{\text{MD}}^2 + \sigma_{\text{CR}}^2)} \right). \qquad (7)$$

The term $P_k$ acts as a spatial constraint, which is multiplied by the neutrino likelihood function defined in Equation (2) via $\mathcal{L} \rightarrow \mathcal{L} \cdot P_k$. Effectively, the constraint is added as a logarithmic term to the neutrino TS defined for each grid point in Equation (6) via $\text{TS} \rightarrow \text{TS} + 2 \cdot \log(P_k)$. Finally, the maximum of the combined UHECR-neutrino TS is found at the best-fit grid point, $\hat{\Omega}_{\text{grid}}$, via

$$TS_k^{\text{comb}}(\hat{\Omega}_{\text{grid}}) = \max \left( TS_{\nu}(\hat{\Omega}_{\text{grid}}) - \frac{(\hat{\Omega}_{\text{grid}} - \vec{\Omega}_k^{\text{CR}})^2}{\sigma_{\text{MD}}^2 + \sigma_{\text{CR}}^2} \right). \qquad (8)$$

The maximum marks the neutrino source candidate that is the most likely counterpart to the respective UHECR event used to calculate the spatial constraint. The normalization in Equation (7) adds only a constant term to the TS, which can be omitted when calculating significances and p-values based on pseudo-experiments (see below). As a third, final step, this procedure is repeated for all UHECRs, and all obtained TS values are added up, yielding the final test statistic. This search strategy was first developed in the context of this point-source correlation analysis and first outlined in Section 4 of Schumacher (2019). It has already been applied also to other neutrino correlation searches with spatially extended source localization, namely, the correlation with ANITA events (Aartsen et al. 2020a) and with gravitational wave events (Aartsen et al. 2020b). Note that the analysis described here is improved with respect to the previous search (The IceCube Collaboration et al. 2016, Section 5), as it models the displacement of a point-like neutrino source with respect to the UHECR arrival direction based on the assumed magnetic deflection. Here the position and flux of a point-like source are fit in the vicinity of the UHECR direction, while in the previous search a spatially extended neutrino emission around the UHECR direction was fit.

Six different signal hypotheses are tested, which are characterized by a lower cut on the UHECR energy, $E_{\text{CR}} > E_{\text{cut}}$ with $E_{\text{cut}} \in \{70, 85, 100\}$ EeV, and the scaling factor of the deflection estimate, $C \in \{1, 2\}$. The lower energy cut is a threshold for selecting only the highest-energy UHECRs with the lowest deflection for this analysis. No analogous energy cut is applied to the neutrino data. The scaling factor $C$ is a model parameter for scaling the baseline magnetic deflection, and it is not derived from the UHECR data. The baseline magnetic deflection is $D_0 = 3^\circ$ for all UHECRs, which is then converted into the spatial constraint of an individual UHECR event using Equations (1) and (7). The corresponding sensitivity and $3\sigma$-discovery potential are evaluated based on the normalization of the neutrino flux per source, $\Phi_0^\nu$ (see Equation (3)). The $3\sigma$ threshold is chosen since the background expectation needs to be calculated based on pseudo-experiments, as described in the next paragraph. A $5\sigma$-discovery potential is computationally too expensive to be calculated for the various hypotheses. Sensitivity is defined as
<table><caption>**Table 2**
Sensitivity, Discovery Potential, and 90% C.L. Upper Limit for the Different Analysis Parameters, Which Are The UHECR Energy Cut, $E_{\text{cut}}$, Magnetic Deflection, $D_0 \cdot C$, and Correlation Fraction, $f_{\text{corr}}$, for the Point-source Correlation Analysis</caption><thead><tr><th>Magnetic deflection, $D_0 \cdot C$<br>Energy cut, $E_{\text{cut}}$</th><th>3°<br>70 EeV</th><th>3°<br>85 EeV</th><th>3°<br>100 EeV</th><th>6°<br>70 EeV</th><th>6°<br>85 EeV</th><th>6°<br>100 EeV</th></tr></thead><tbody><tr><td colspan="7"><strong>Number of neutrino sources, $N_{\nu}^{\text{src}}$</strong></td></tr><tr><td>$f_{\text{corr}} = 0.1$</td><td>22</td><td>9</td><td>4</td><td>22</td><td>9</td><td>4</td></tr><tr><td>$f_{\text{corr}} = 0.5$</td><td>106</td><td>41</td><td>20</td><td>106</td><td>41</td><td>20</td></tr><tr><td>$f_{\text{corr}} = 1$</td><td>211</td><td>82</td><td>40</td><td>211</td><td>82</td><td>40</td></tr><tr><td colspan="7"><strong>Sensitivity</strong></td></tr><tr><td>$f_{\text{corr}} = 0.1$</td><td>5.6</td><td>7.1</td><td>9.8</td><td>5.4</td><td>8.5</td><td>9.5</td></tr><tr><td>$f_{\text{corr}} = 0.5$</td><td>2.4</td><td>2.6</td><td>3.1</td><td>2.6</td><td>3.4</td><td>4.2</td></tr><tr><td>$f_{\text{corr}} = 1$</td><td>1.4</td><td>1.7</td><td>2.1</td><td>1.8</td><td>2.2</td><td>2.9</td></tr><tr><td colspan="7"><strong>3σ disc. potential</strong></td></tr><tr><td>$f_{\text{corr}} = 0.1$</td><td>7.2</td><td>8.5</td><td>11.6</td><td>6.7</td><td>8.0</td><td>10.8</td></tr><tr><td>$f_{\text{corr}} = 0.5$</td><td>3.7</td><td>3.8</td><td>4.9</td><td>3.7</td><td>4.1</td><td>5.1</td></tr><tr><td>$f_{\text{corr}} = 1$</td><td>2.6</td><td>2.6</td><td>3.2</td><td>2.7</td><td>3.1</td><td>3.7</td></tr><tr><td colspan="7"><strong>90% C.L. upper limit</strong></td></tr><tr><td>$f_{\text{corr}} = 0.1$</td><td>6.4</td><td>9.2</td><td>9.8</td><td>6.7</td><td>10.9</td><td>10.6</td></tr><tr><td>$f_{\text{corr}} = 0.5$</td><td>2.8</td><td>3.6</td><td>3.1</td><td>3.3</td><td>4.7</td><td>4.5</td></tr><tr><td>$f_{\text{corr}} = 1$</td><td>1.7</td><td>2.3</td><td>2.1</td><td>2.3</td><td>3.2</td><td>3.1</td></tr><tr><td><strong>Pretrial p-value</strong></td><td>0.33</td><td>0.23</td><td>&gt;0.5</td><td>0.19</td><td>0.097</td><td>0.43</td></tr></tbody></table>

Note. The values of sensitivity, discovery potential, and limit are given as normalization of the neutrino flux per source in units of $10^{-10} \text{ GeV}^{-1} \text{ cm}^{-2} \text{ s}^{-1}$. The neutrino flux per source is modeled with a power law of the form $d\Phi/dE = \Phi_0 \cdot (E/1 \text{ GeV})^{-2}$. The last row states all six experimentally obtained pretrial p-values with respect to the null hypothesis of an isotropic neutrino flux.

the expected median upper limit at 90% confidence level on $\Phi_0'$ in the case of a null measurement, while the discovery potential is defined as the median of $\Phi_0'$ required for a rejection of the background hypothesis with 3σ significance.

Both sensitivity and discovery potential are calculated based on data challenges, for which signal-like and background-like pseudo-experiments (PEs) are generated. Since the calculation is based on PEs, the constant term in Equations (7) and (8), i.e., the normalization of the constraint term, can be omitted. For all signal and background PEs, the UHECR arrival directions are kept fixed. The background PEs are obtained by randomizing the experimental neutrino data in R.A. The signal PEs reflect the six different signal hypotheses; one signal PE is based on experimental neutrino data randomized in R.A., to which Monte Carlo neutrino events, representing a neutrino source, are added. The location of this neutrino source is chosen randomly within the spatial deflection constraint of one UHECR event. This way, we mimic a neutrino source that is displaced with respect to the UHECR direction owing to the UHECR deflection. In the baseline model of the signal PEs, all UHECR events have such an artificial neutrino source in their vicinity. For one PE, all neutrino sources have an $E^{-2}$ power-law spectrum and the same flux normalization. Note that the UHECR energy cut and scaling factor used to generate the spatial constraints for the likelihood function are the same as for determining the neutrino source location.

We generate additional signal PEs by varying the fraction of UHECR events, $f_{\text{corr}}$, with a neutrino source in their vicinity. This mimics signal models where not all UHECRs have a corresponding neutrino source in the vicinity of their arrival direction. Thus, the number of neutrino sources is determined by the rounded-up product of the correlation fraction with the numbers of UHECRs, $N_{\nu}^{\text{src}} = [f_{\text{corr}} N_{\text{CR}}]$, where $N_{\text{CR}}$ is the number of UHECRs passing the energy cut. We choose the correlation

fraction from three discrete values, $f_{\text{corr}} \in \{1, 0.5, 0.1\}$, where $f_{\text{corr}} = 1$ represents the baseline model. Note that the correlation fraction is only used to choose the number of neutrino sources in the signal PEs, while in real experimental data it is not known a priori which UHECRs have a correlated neutrino source and which do not. For $f_{\text{corr}} < 1$, only a number $N_{\nu}^{\text{src}} < N_{\text{CR}}$ of randomly selected UHECRs will have a correlated neutrino source in the signal PEs. Independent of the correlation fraction, the TS values of all UHECR constraints are added up, since in real experimental data it would not be known which UHECRs have a correlated neutrino source. The sensitivity and 3σ-discovery potential in terms of the flux normalization per neutrino source for all signal models and hypotheses are reported in Table 2 and also shown in Figure 6. We see that a smaller correlation fraction increases the flux normalization per neutrino source for both sensitivity and 3σ-discovery potential. This is expected since a smaller correlation fraction corresponds to fewer neutrino sources, which in turn need to have a higher flux normalization to be detected by the analysis.

It is noticeable that for a small correlation fraction of $f_{\text{corr}} = 0.1$, the signal model with a larger scaling factor of $C=2$ yields a better 3σ-discovery potential than the smaller scaling factor of $C=1$. This is unexpected, as a smaller UHECR deflection and thus a smaller spatial constraint should yield more accuracy in detecting the corresponding neutrino sources. However, the amount of sky covered by the constraints does not grow uniformly with the scaling factor and numbers of UHECRs, since the UHECR spatial constraints can overlap. In the case of a small correlation fraction, a high neutrino flux per source is required to reach the 3σ-discovery potential owing to the small number of sources present.<fnref n="241" /> A

<footnote n="241">²⁴¹ There are 22, 9, and 4 sources for $f_{\text{corr}} = 0.1$ and the energy cuts of $E_{\text{cut}} \in \{70, 85, 100\}$ EeV, respectively.</footnote>
<table><caption>**Table 2**
Sensitivity, Discovery Potential, and 90% C.L. Upper Limit for the Different Analysis Parameters, Which Are The UHECR Energy Cut, $E_{cut}$, Magnetic Deflection, $D_0 \cdot C$, and Correlation Fraction, $f_{corr}$, for the Point-source Correlation Analysis</caption><thead><tr><th>Magnetic deflection, D<sub>0</sub> &middot; C</th><th>3&deg;</th><th>3&deg;</th><th>3&deg;</th><th>6&deg;</th><th>6&deg;</th><th>6&deg;</th></tr><tr><th>Energy cut, E<sub>cut</sub></th><th>70 EeV</th><th>85 EeV</th><th>100 EeV</th><th>70 EeV</th><th>85 EeV</th><th>100 EeV</th></tr></thead><tbody><tr><td colspan="7"><strong>Number of neutrino sources, N<sub>&nu;</sub><sup>src</sup></strong></td></tr><tr><td>f<sub>corr</sub> = 0.1</td><td>22</td><td>9</td><td>4</td><td>22</td><td>9</td><td>4</td></tr><tr><td>f<sub>corr</sub> = 0.5</td><td>106</td><td>41</td><td>20</td><td>106</td><td>41</td><td>20</td></tr><tr><td>f<sub>corr</sub> = 1</td><td>211</td><td>82</td><td>40</td><td>211</td><td>82</td><td>40</td></tr><tr><td colspan="7"><strong>Sensitivity</strong></td></tr><tr><td>f<sub>corr</sub> = 0.1</td><td>5.6</td><td>7.1</td><td>9.8</td><td>5.4</td><td>8.5</td><td>9.5</td></tr><tr><td>f<sub>corr</sub> = 0.5</td><td>2.4</td><td>2.6</td><td>3.1</td><td>2.6</td><td>3.4</td><td>4.2</td></tr><tr><td>f<sub>corr</sub> = 1</td><td>1.4</td><td>1.7</td><td>2.1</td><td>1.8</td><td>2.2</td><td>2.9</td></tr><tr><td colspan="7"><strong>3&sigma; disc. potential</strong></td></tr><tr><td>f<sub>corr</sub> = 0.1</td><td>7.2</td><td>8.5</td><td>11.6</td><td>6.7</td><td>8.0</td><td>10.8</td></tr><tr><td>f<sub>corr</sub> = 0.5</td><td>3.7</td><td>3.8</td><td>4.9</td><td>3.7</td><td>4.1</td><td>5.1</td></tr><tr><td>f<sub>corr</sub> = 1</td><td>2.6</td><td>2.6</td><td>3.2</td><td>2.7</td><td>3.1</td><td>3.7</td></tr><tr><td colspan="7"><strong>90% C.L. upper limit</strong></td></tr><tr><td>f<sub>corr</sub> = 0.1</td><td>6.4</td><td>9.2</td><td>9.8</td><td>6.7</td><td>10.9</td><td>10.6</td></tr><tr><td>f<sub>corr</sub> = 0.5</td><td>2.8</td><td>3.6</td><td>3.1</td><td>3.3</td><td>4.7</td><td>4.5</td></tr><tr><td>f<sub>corr</sub> = 1</td><td>1.7</td><td>2.3</td><td>2.1</td><td>2.3</td><td>3.2</td><td>3.1</td></tr><tr><td><strong>Pretrial p-value</strong></td><td>0.33</td><td>0.23</td><td>&gt;0.5</td><td>0.19</td><td>0.097</td><td>0.43</td></tr></tbody></table>

**Note.** The values of sensitivity, discovery potential, and limit are given as normalization of the neutrino flux per source in units of $10^{-10} \text{ GeV}^{-1} \text{ cm}^{-2} \text{ s}^{-1}$. The neutrino flux per source is modeled with a power law of the form $d\Phi/dE = \Phi_0 \cdot (E/1 \text{ GeV})^{-2}$. The last row states all six experimentally obtained pretrial p-values with respect to the null hypothesis of an isotropic neutrino flux.

the expected median upper limit at 90% confidence level on $\Phi_0'$ in the case of a null measurement, while the discovery potential is defined as the median of $\Phi_0'$ required for a rejection of the background hypothesis with $3\sigma$ significance.

Both sensitivity and discovery potential are calculated based on data challenges, for which signal-like and background-like pseudo-experiments (PEs) are generated. Since the calculation is based on PEs, the constant term in Equations (7) and (8), i.e., the normalization of the constraint term, can be omitted. For all signal and background PEs, the UHECR arrival directions are kept fixed. The background PEs are obtained by randomizing the experimental neutrino data in R.A. The signal PEs reflect the six different signal hypotheses; one signal PE is based on experimental neutrino data randomized in R.A., to which Monte Carlo neutrino events, representing a neutrino source, are added. The location of this neutrino source is chosen randomly within the spatial deflection constraint of one UHECR event. This way, we mimic a neutrino source that is displaced with respect to the UHECR direction owing to the UHECR deflection. In the baseline model of the signal PEs, all UHECR events have such an artificial neutrino source in their vicinity. For one PE, all neutrino sources have an $E^{-2}$ power-law spectrum and the same flux normalization. Note that the UHECR energy cut and scaling factor used to generate the spatial constraints for the likelihood function are the same as for determining the neutrino source location.

We generate additional signal PEs by varying the fraction of UHECR events, $f_{corr}$, with a neutrino source in their vicinity. This mimics signal models where not all UHECRs have a corresponding neutrino source in the vicinity of their arrival direction. Thus, the number of neutrino sources is determined by the rounded-up product of the correlation fraction with the numbers of UHECRs, $N_{\nu}^{src} = [f_{corr} N_{CR}]$, where $N_{CR}$ is the number of UHECRs passing the energy cut. We choose the correlation

fraction from three discrete values, $f_{corr} \in \{1, 0.5, 0.1\}$, where $f_{corr} = 1$ represents the baseline model. Note that the correlation fraction is only used to choose the number of neutrino sources in the signal PEs, while in real experimental data it is not known a priori which UHECRs have a correlated neutrino source and which do not. For $f_{corr} < 1$, only a number $N_{\nu}^{src} < N_{CR}$ of randomly selected UHECRs will have a correlated neutrino source in the signal PEs. Independent of the correlation fraction, the TS values of all UHECR constraints are added up, since in real experimental data it would not be known which UHECRs have a correlated neutrino source. The sensitivity and $3\sigma$-discovery potential in terms of the flux normalization per neutrino source for all signal models and hypotheses are reported in Table 2 and also shown in Figure 6. We see that a smaller correlation fraction increases the flux normalization per neutrino source for both sensitivity and $3\sigma$-discovery potential. This is expected since a smaller correlation fraction corresponds to fewer neutrino sources, which in turn need to have a higher flux normalization to be detected by the analysis.

It is noticeable that for a small correlation fraction of $f_{corr} = 0.1$, the signal model with a larger scaling factor of $C=2$ yields a better $3\sigma$-discovery potential than the smaller scaling factor of $C=1$. This is unexpected, as a smaller UHECR deflection and thus a smaller spatial constraint should yield more accuracy in detecting the corresponding neutrino sources. However, the amount of sky covered by the constraints does not grow uniformly with the scaling factor and numbers of UHECRs, since the UHECR spatial constraints can overlap. In the case of a small correlation fraction, a high neutrino flux per source is required to reach the $3\sigma$-discovery potential owing to the small number of sources present.<fnref n="241" /> A

<footnote n="241">²⁴¹ There are 22, 9, and 4 sources for $f_{corr} = 0.1$ and the energy cuts of $E_{cut} \in \{70, 85, 100\}$ EeV, respectively.</footnote>
<figure><img src="image_10.png" /><figcaption>Figure 6. Sensitivity (left) and 3σ-discovery potential (right) of the neutrino flux per source as a function of the UHECR energy cut in EeV, for the three correlation fractions and for the two scaling factors we used.</figcaption></figure>

closer study showed that in the case of few, but strong, neutrino sources, it is beneficial to search a larger area of the sky for sources so that fewer of them are missed. This effect is thus caused by an interplay between deflection size, number of UHECRs, and relatively high neutrino flux per source in this particular case.

## 4.2. Unbinned Likelihood-stacking Analysis of UHECRs and High-energy Neutrinos

The second analysis is based on using the neutrinos with a high probability of astrophysical origin as markers for the location of the sources of UHECRs and neutrinos. The UHECR events are stacked using an unbinned likelihood analysis with the arrival directions of the high-energy neutrinos as the source positions. The signal hypothesis is defined by the number of UHECR events, which are clustered around the neutrino arrival directions, as well as the size of the magnetic deflection. The background hypothesis is defined by an isotropic flux of UHECRs. This approach is thus complementary to the point-source correlation analysis, where the UHECR events are used as source markers and an isotropic flux of neutrinos defines the background hypothesis. The logarithm of the likelihood function is defined as

$$ \log \mathcal{L}(n_s) = \sum_{i=1}^{N_{\text{Auger}}} \log \left( \frac{n_s}{N_{\text{CR}}} S_{\text{Auger}}^i + \frac{N_{\text{CR}} - n_s}{N_{\text{CR}}} B_{\text{Auger}}^i \right) + \sum_{i=1}^{N_{\text{TA}}} \log \left( \frac{n_s}{N_{\text{CR}}} S_{\text{TA}}^i + \frac{N_{\text{CR}} - n_s}{N_{\text{CR}}} B_{\text{TA}}^i \right), \quad (9) $$

where the free parameter is the total number of UHECR signal events, $n_s$. The sums run over all UHECR events in the respective data set, i.e., $N_{\text{Auger}}$ and $N_{\text{TA}}$, where the total number of UHECR events is $N_{\text{CR}} = N_{\text{Auger}} + N_{\text{TA}}$. In contrast to the likelihood function of analysis 1, the signal pdfs, $S_{\text{Auger}}^i$ and $S_{\text{TA}}^i$, and the background pdfs, $B_{\text{Auger}}^i$ and $B_{\text{TA}}^i$, are stacked for all

high-energy neutrinos such that $n_s$ is a global parameter. The background pdfs per UHECR detector, $B_{\text{det}}^i$, are the normalized expectations for an isotropic UHECR flux as a function of decl., which correspond to the normalized detector exposures (see Figure 3). The signal pdf per UHECR detector is defined as

$$ S_{\text{det}}^i(\vec{\Omega}_i, E_i) = R_{\text{det}}(\delta_i) \cdot \sum_{j=1}^{N_i} S_j(\vec{\Omega}_i, \sigma(E_i)) \\ \text{with } \sigma^2 = \sigma_{\text{MD}}^2 + \sigma_{\text{CR}}^2, \qquad (10) $$

as a function of the UHECR arrival direction, $\vec{\Omega}_i$, and energy, $E_i$. The term $R_{\text{det}}$ is the relative exposure of each detector as a function of the decl. per UHECR event, $\delta_i$. Figure 3 shows the absolute directional exposures, which are each normalized to 1 over the whole sky to obtain $R_{\text{det}}$. The sum runs over all neutrino events, $N_i$, where the terms of $S_j(\vec{\Omega}_i, \sigma(E_i))$ are the spatial likelihood maps obtained from the neutrino directional reconstruction smeared with the UHECR uncertainty with width $\sigma(E_i)$ (see Equation (7)). Thus, these terms are pdfs representing the total uncertainty on the common source position of UHECR event $i$ and neutrino event $j$, evaluated at the UHECR arrival direction, $\vec{\Omega}_i$. Similar to analysis 1, three different deflections corresponding to scaling factors of $C \in \{1, 2, 3\}$ are represented in the signal terms of the likelihood function. Again, $C$ is a model parameter that scales the magnitude of the deflection and it is not derived from UHECR data. Depending on whether the UHECR arrival direction is in the Galactic northern hemisphere or southern hemisphere (see Figure 4), the baseline magnetic deflection is thus

$$ \begin{aligned} D &= C \cdot (D_{\text{North}}, D_{\text{South}}) \\ &= C \cdot (2.4, 3.7) \text{ with } C \in \{1, 2, 3\}. \end{aligned} \qquad (11) $$

The final value of the deflection, $\sigma_{\text{MD}}$, is then calculated based on the UHECR energy using Equation (1).