LJ12623
REVIEW COPY
NOT FOR DISTRIBUTION
Measurement of the proton-air cross-section at
√
s = 57 TeV
with the Pierre Auger Observatory
P. Abreu,
1
M. Aglietta,
2
E.J. Ahn,
3
I.F.M. Albuquerque,
4
D. Allard,
5
I. Allekotte,
6
J. Allen,
7
P. Allison,
8
A. Almeda,
9, 10
J. Alvarez Castillo,
11
J. Alvarez-Mu˜niz,
12
M. Ambrosio,
13
A. Aminaei,
14
L. Anchordoqui,
15
S. Andringa,
1
T. Antiˇci´c,
16
C. Aramo,
13
E. Arganda,
17, 18
F. Arqueros,
18
H. Asorey,
6
P. Assis,
1
J. Aublin,
19
M. Ave,
20
M. Avenier,
21
G. Avila,
22
T. B¨acker,
23
M. Balzer,
24
K.B. Barber,
25
A.F. Barbosa,
26
R. Bardenet,
27
S.L.C. Barroso,
28
B. Baughman,
8
J. B¨auml,
29
J.J. Beatty,
8
B.R. Becker,
30
K.H. Becker,
31
A. Bell´etoile,
32
J.A. Bellido,
25
S. BenZvi,
33
C. Berat,
21
X. Bertou,
6
P.L. Biermann,
34
P. Billoir,
19
F. Blanco,
18
M. Blanco,
35
C. Bleve,
31
H. Bl¨umer,
20, 29
M. Boh´aˇcov´a,
36
D. Boncioli,
37
C. Bonifazi,
38, 19
R. Bonino,
2
N. Borodai,
39
J. Brack,
40
P. Brogueira,
1
W.C. Brown,
41
R. Bruijn,
42
P. Buchholz,
23
A. Bueno,
43
R.E. Burton,
44
K.S. Caballero-Mora,
45
L. Caramete,
34
R. Caruso,
46
A. Castellina,
2
O. Catalano,
47
G. Cataldi,
48
L. Cazon,
1
R. Cester,
49
J. Chauvin,
21
S.H. Cheng,
45
A. Chiavassa,
2
J.A. Chinellato,
50
J. Chirinos Diaz,
51
J. Chudoba,
36
R.W. Clay,
25
M.R. Coluccia,
48
R. Concei¸c˜ao,
1
F. Contreras,
52
H. Cook,
42
M.J. Cooper,
25
J. Coppens,
14, 53
A. Cordier,
27
S. Coutu,
45
C.E. Covault,
44
A. Creusot,
5, 54
A. Criss,
45
J. Cronin,
55
A. Curutiu,
34
S. Dagoret-Campagne,
27
R. Dallier,
32
S. Dasso,
56, 57
K. Daumiller,
29
B.R. Dawson,
25
R.M. de Almeida,
58
M. De Domenico,
46
C. De Donato,
11
S.J. de
Jong,
14, 53
G. De La Vega,
59
W.J.M. de Mello Junior,
50
J.R.T. de Mello Neto,
38
I. De Mitri,
48
V. de Souza,
60
K.D. de Vries,
61
G. Decerprit,
5
L. del Peral,
35
M. del R´ıo,
37, 52
O. Deligny,
62
H. Dembinski,
20
N. Dhital,
51
C. Di
Giulio,
63
M.L. D´ıaz Castro,
64
P.N. Diep,
65
C. Dobrigkeit,
50
W. Docters,
61
J.C. D’Olivo,
11
P.N. Dong,
65, 62
A. Dorofeev,
40
J.C. dos Anjos,
26
M.T. Dova,
17
D. D’Urso,
13
I. Dutan,
34
J. Ebr,
36
R. Engel,
29
M. Erdmann,
66
C.O. Escobar,
50
J. Espadanal,
1
A. Etchegoyen,
10, 9
P. Facal San Luis,
55
I. Fajardo Tapia,
11
H. Falcke,
14, 67
G. Farrar,
7
A.C. Fauth,
50
N. Fazzini,
3
A.P. Ferguson,
44
A. Ferrero,
10
B. Fick,
51
A. Filevich,
10
A. Filipˇciˇc,
68, 54
S. Fliescher,
66
C.E. Fracchiolla,
40
E.D. Fraenkel,
61
U. Fr¨ohlich,
23
B. Fuchs,
26
R. Gaior,
19
R.F. Gamarra,
10
S. Gambetta,
69
B. Garc´ıa,
59
D. Garcia-Gamez,
27
D. Garcia-Pinto,
18
A. Gascon,
43
H. Gemmeke,
24
K. Gesterling,
30
P.L. Ghia,
19, 2
U. Giaccari,
48
M. Giller,
70
H. Glass,
3
M.S. Gold,
30
G. Golup,
6
F. Gomez Albarracin,
17
M. G´omez
Berisso,
6
P. Gon¸calves,
1
D. Gonzalez,
20
J.G. Gonzalez,
20
B. Gookin,
40
D. G´ora,
20, 39
A. Gorgi,
2
P. Gouffon,
4
S.R. Gozzini,
42
E. Grashorn,
8
S. Grebe,
14, 53
N. Griffith,
8
M. Grigat,
66
A.F. Grillo,
71
Y. Guardincerri,
57
F. Guarino,
13
G.P. Guedes,
72
A. Guzman,
11
J.D. Hague,
30
P. Hansen,
17
D. Harari,
6
S. Harmsma,
61, 53
T.A. Harrison,
25
J.L. Harton,
40
A. Haungs,
29
T. Hebbeker,
66
D. Heck,
29
A.E. Herve,
25
C. Hojvat,
3
N. Hollon,
55
V.C. Holmes,
25
P. Homola,
39
J.R. H¨orandel,
14
A. Horneffer,
14
P. Horvath,
73
M. Hrabovsk´y,
73, 36
T. Huege,
29
A. Insolia,
46
F. Ionita,
55
A. Italiano,
46
C. Jarne,
17
S. Jiraskova,
14
M. Josebachuili,
10
K. Kadija,
16
K.H. Kampert,
31
P. Karhan,
74
P. Kasper,
3
B. K´egl,
27
B. Keilhauer,
29
A. Keivani,
75
J.L. Kelley,
14
E. Kemp,
50
R.M. Kieckhafer,
51
H.O. Klages,
29
M. Kleifges,
24
J. Kleinfeller,
29
J. Knapp,
42
D.-H. Koang,
21
K. Kotera,
55
N. Krohm,
31
O. Kr¨omer,
24
D. Kruppke-Hansen,
31
F. Kuehn,
3
D. Kuempel,
31
J.K. Kulbartz,
76
N. Kunka,
24
G. La Rosa,
47
C. Lachaud,
5
R. Lauer,
30
P. Lautridou,
32
S. Le Coz,
21
M.S.A.B. Le˜ao,
77
D. Lebrun,
21
P. Lebrun,
3
M.A. Leigui de Oliveira,
77
A. Lemiere,
62
A. Letessier-Selvon,
19
I. Lhenry-Yvon,
62
K. Link,
20
R. L´opez,
78
A. Lopez Ag¨uera,
12
K. Louedec,
21, 27
J. Lozano Bahilo,
43
L. Lu,
42
A. Lucero,
10, 2
M. Ludwig,
20
H. Lyberis,
62
C. Macolino,
19
S. Maldera,
2
D. Mandat,
36
P. Mantsch,
3
A.G. Mariazzi,
17
J. Marin,
52, 2
V. Marin,
32
I.C. Maris,
19
H.R. Marquez Falcon,
79
G. Marsella,
80
D. Martello,
48
L. Martin,
32
H. Martinez,
81
O. Mart´ınez Bravo,
78
H.J. Mathes,
29
J. Matthews,
75, 82
J.A.J. Matthews,
30
G. Matthiae,
37
D. Maurizio,
49
P.O. Mazur,
3
G. Medina-Tanco,
11
M. Melissas,
20
D. Melo,
10, 49
E. Menichetti,
49
A. Menshikov,
24
P. Mertsch,
83
C. Meurer,
66
S. Mi´canovi´c,
16
M.I. Micheletti,
84
W. Miller,
30
L. Miramonti,
85
L. Molina-Bueno,
43
S. Mollerach,
6
M. Monasor,
55
D. Monnier Ragaigne,
27
F. Montanet,
21
B. Morales,
11
C. Morello,
2
E. Moreno,
78
J.C. Moreno,
17
C. Morris,
8
M. Mostaf´a,
40
C.A. Moura,
77, 13
S. Mueller,
29
M.A. Muller,
50
G. M¨uller,
66
M. M¨unchmeyer,
19
R. Mussa,
49
G. Navarra,
2
J.L. Navarro,
43
S. Navas,
43
P. Necesal,
36
L. Nellen,
11
A. Nelles,
14, 53
J. Neuser,
31
P.T. Nhung,
65
L. Niemietz,
31
N. Nierstenhoefer,
31
D. Nitz,
51
D. Nosek,
74
L. Noˇzka,
36
M. Nyklicek,
36
J. Oehlschl¨ager,
29
A. Olinto,
55
V.M. Olmos-Gilbaja,
12
M. Ortiz,
18
N. Pacheco,
35
D. Pakk Selmi-Dei,
50
M. Palatka,
36
J. Pallotta,
86
N. Palmieri,
20
G. Parente,
12
E. Parizot,
5
A. Parra,
12
R.D. Parsons,
42
S. Pastor,
87
T. Paul,
88
M. Pech,
36
J. P¸ekala,
39
R. Pelayo,
78, 12
I.M. Pepe,
89
L. Perrone,
80
R. Pesce,
69
E. Petermann,
90
S. Petrera,
63
P. Petrinca,
37
A. Petrolini,
69
Y. Petrov,
40
J. Petrovic,
53
C. Pfendner,
33
N. Phan,
30
R. Piegaia,
57
T. Pierog,
29
P. Pieroni,
57
M. Pimenta,
1
V. Pirronello,
46
M. Platino,
10
V.H. Ponce,
6
M. Pontz,
23
P. Privitera,
55
M. Prouza,
36
E.J. Quel,
86
S. Querchfeld,
31
J. Rautenberg,
31
O. Ravel,
32
D. Ravignani,
10
2
B. Revenu,
32
J. Ridky,
36
S. Riggi,
12, 46
M. Risse,
23
P. Ristori,
86
H. Rivera,
85
V. Rizi,
63
J. Roberts,
7
C. Robledo,
78
W. Rodrigues de Carvalho,
12, 4
G. Rodriguez,
12
J. Rodriguez Martino,
52
J. Rodriguez Rojo,
52
I. Rodriguez-Cabo,
12
M.D. Rodr´ıguez-Fr´ıas,
35
G. Ros,
35
J. Rosado,
18
T. Rossler,
73
M. Roth,
29
B. Rouill´e-d’Orfeuil,
55
E. Roulet,
6
A.C. Rovero,
56
C. R¨uhle,
24
F. Salamida,
62, 63
H. Salazar,
78
F. Salesa Greus,
40
G. Salina,
37
F. S´anchez,
10
C.E. Santo,
1
E. Santos,
1
E.M. Santos,
38
F. Sarazin,
91
B. Sarkar,
31
S. Sarkar,
83
R. Sato,
52
N. Scharf,
66
V. Scherini,
85
H. Schieler,
29
P. Schiffer,
76, 66
A. Schmidt,
24
O. Scholten,
61
H. Schoorlemmer,
14, 53
J. Schovancova,
36
P. Schov´anek,
36
F. Schr¨oder,
29
S. Schulte,
66
D. Schuster,
91
S.J. Sciutto,
17
M. Scuderi,
46
A. Segreto,
47
M. Settimo,
23
A. Shadkam,
75
R.C. Shellard,
26, 64
I. Sidelnik,
10
G. Sigl,
76
H.H. Silva Lopez,
11
A.
´
Smia lkowski,
70
R.
ˇ
Sm´ıda,
29, 36
G.R. Snow,
90
P. Sommers,
45
J. Sorokin,
25
H. Spinka,
92, 3
R. Squartini,
52
S. Stanic,
54
J. Stapleton,
8
J. Stasielak,
39
M. Stephan,
66
A. Stutz,
21
F. Suarez,
10
T. Suomij¨arvi,
62
A.D. Supanitsky,
56, 11
T.
ˇ
Suˇsa,
16
M.S. Sutherland,
75, 8
J. Swain,
88
Z. Szadkowski,
70
M. Szuba,
29
A. Tamashiro,
56
A. Tapia,
10
M. Tartare,
21
O. Ta¸sc˘au,
31
C.G. Tavera Ruiz,
11
R. Tcaciuc,
23
D. Tegolo,
46, 93
N.T. Thao,
65
D. Thomas,
40
J. Tiffenberg,
57
C. Timmermans,
53, 14
D.K. Tiwari,
79
W. Tkaczyk,
70
C.J. Todero Peixoto,
60, 77
B. Tom´e,
1
A. Tonachini,
49
P. Travnicek,
36
D.B. Tridapalli,
4
G. Tristram,
5
E. Trovato,
46
M. Tueros,
12, 57
R. Ulrich,
29, 45
M. Unger,
29
M. Urban,
27
J.F. Vald´es Galicia,
11
I. Vali˜no,
12
L. Valore,
13
A.M. van den Berg,
61
E. Varela,
78
B. Vargas
C´ardenas,
11
J.R. V´azquez,
18
R.A. V´azquez,
12
D. Veberiˇc,
54, 68
V. Verzi,
37
J. Vicha,
36
M. Videla,
59
L. Villase˜nor,
79
H. Wahlberg,
17
P. Wahrlich,
25
O. Wainberg,
10, 9
D. Walz,
66
D. Warner,
40
A.A. Watson,
42
M. Weber,
24
K. Weidenhaupt,
66
A. Weindl,
29
S. Westerhoff,
33
B.J. Whelan,
25
G. Wieczorek,
70
L. Wiencke,
91
B. Wilczy´nska,
39
H. Wilczy´nski,
39
M. Will,
29
C. Williams,
55
T. Winchen,
66
M.G. Winnick,
25
M. Wommer,
29
B. Wundheiler,
10
T. Yamamoto,
55
T. Yapici,
51
P. Younk,
23, 94
G. Yuan,
75
A. Yushkov,
12, 13
B. Zamorano,
43
E. Zas,
12
D. Zavrtanik,
54, 68
M. Zavrtanik,
68, 54
I. Zaw,
7
A. Zepeda,
81
Y. Zhu,
24
M. Zimbres Silva,
31, 50
and M. Ziolkowski
23
(The Pierre Auger Collaboration)
1
LIP and Instituto Superior T´ecnico, Technical University of Lisbon, Portugal
2
Istituto di Fisica dello Spazio Interplanetario (INAF),
Universit`a di Torino and Sezione INFN, Torino, Italy
3
Fermilab, Batavia, IL, USA
4
Universidade de S˜ao Paulo, Instituto de F´ısica, S˜ao Paulo, SP, Brazil
5
Laboratoire AstroParticule et Cosmologie (APC),
Universit´e Paris 7, CNRS-IN2P3, Paris, France
6
Centro At´omico Bariloche and Instituto Balseiro (CNEA-UNCuyo-CONICET), San Carlos de Bariloche, Argentina
7
New York University, New York, NY, USA
8
Ohio State University, Columbus, OH, USA
9
Universidad Tecnol´ogica Nacional - Facultad Regional Buenos Aires, Buenos Aires, Argentina
10
Instituto de Tecnolog´ıas en Detecci´on y Astropart´ıculas (CNEA, CONICET, UNSAM), Buenos Aires, Argentina
11
Universidad Nacional Autonoma de Mexico, Mexico, D.F., Mexico
12
Universidad de Santiago de Compostela, Spain
13
Universit`a di Napoli ”Federico II” and Sezione INFN, Napoli, Italy
14
IMAPP, Radboud University Nijmegen, Netherlands
15
University of Wisconsin, Milwaukee, WI, USA
16
Rudjer Boˇskovi´c Institute, 10000 Zagreb, Croatia
17
IFLP, Universidad Nacional de La Plata and CONICET, La Plata, Argentina
18
Universidad Complutense de Madrid, Madrid, Spain
19
Laboratoire de Physique Nucl´eaire et de Hautes Energies (LPNHE),
Universit´es Paris 6 et Paris 7, CNRS-IN2P3, Paris, France
20
Karlsruhe Institute of Technology - Campus South - Institut f¨ur Experimentelle Kernphysik (IEKP), Karlsruhe, Germany
21
Laboratoire de Physique Subatomique et de Cosmologie (LPSC),
Universit´e Joseph Fourier, INPG, CNRS-IN2P3, Grenoble, France
22
Observatorio Pierre Auger and Comisi´on Nacional de Energ´ıa At´omica, Malarg¨ue, Argentina
23
Universit¨at Siegen, Siegen, Germany
24
Karlsruhe Institute of Technology - Campus North - Institut f¨ur Prozessdatenverarbeitung und Elektronik, Karlsruhe, Germany
25
University of Adelaide, Adelaide, S.A., Australia
26
Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro, RJ, Brazil
27
Laboratoire de l’Acc´el´erateur Lin´eaire (LAL), Universit´e Paris 11, CNRS-IN2P3, Orsay, France
28
Universidade Estadual do Sudoeste da Bahia, Vitoria da Conquista, BA, Brazil
29
Karlsruhe Institute of Technology - Campus North - Institut f¨ur Kernphysik, Karlsruhe, Germany
30
University of New Mexico, Albuquerque, NM, USA
31
Bergische Universit¨at Wuppertal, Wuppertal, Germany
32
SUBATECH,
´
Ecole des Mines de Nantes, CNRS-IN2P3, Universit´e de Nantes, Nantes, France
3
33
University of Wisconsin, Madison, WI, USA
34
Max-Planck-Institut f¨ur Radioastronomie, Bonn, Germany
35
Universidad de Alcal´a, Alcal´a de Henares (Madrid), Spain
36
Institute of Physics of the Academy of Sciences of the Czech Republic, Prague, Czech Republic
37
Universit`a di Roma II ”Tor Vergata” and Sezione INFN, Roma, Italy
38
Universidade Federal do Rio de Janeiro, Instituto de F´ısica, Rio de Janeiro, RJ, Brazil
39
Institute of Nuclear Physics PAN, Krakow, Poland
40
Colorado State University, Fort Collins, CO, USA
41
Colorado State University, Pueblo, CO, USA
42
School of Physics and Astronomy, University of Leeds, United Kingdom
43
Universidad de Granada & C.A.F.P.E., Granada, Spain
44
Case Western Reserve University, Cleveland, OH, USA
45
Pennsylvania State University, University Park, PA, USA
46
Universit`a di Catania and Sezione INFN, Catania, Italy
47
Istituto di Astrofisica Spaziale e Fisica Cosmica di Palermo (INAF), Palermo, Italy
48
Dipartimento di Fisica dell’Universit`a del Salento and Sezione INFN, Lecce, Italy
49
Universit`a di Torino and Sezione INFN, Torino, Italy
50
Universidade Estadual de Campinas, IFGW, Campinas, SP, Brazil
51
Michigan Technological University, Houghton, MI, USA
52
Observatorio Pierre Auger, Malarg¨ue, Argentina
53
Nikhef, Science Park, Amsterdam, Netherlands
54
Laboratory for Astroparticle Physics, University of Nova Gorica, Slovenia
55
University of Chicago, Enrico Fermi Institute, Chicago, IL, USA
56
Instituto de Astronom´ıa y F´ısica del Espacio (CONICET-UBA), Buenos Aires, Argentina
57
Departamento de F´ısica, FCEyN, Universidad de Buenos Aires y CONICET, Argentina
58
Universidade Federal Fluminense, EEIMVR, Volta Redonda, RJ, Brazil
59
National Technological University, Faculty Mendoza (CONICET/CNEA), Mendoza, Argentina
60
Universidade de S˜ao Paulo, Instituto de F´ısica, S˜ao Carlos, SP, Brazil
61
Kernfysisch Versneller Instituut, University of Groningen, Groningen, Netherlands
62
Institut de Physique Nucl´eaire d’Orsay (IPNO),
Universit´e Paris 11, CNRS-IN2P3, Orsay, France
63
Universit`a dell’Aquila and INFN, L’Aquila, Italy
64
Pontif´ıcia Universidade Cat´olica, Rio de Janeiro, RJ, Brazil
65
Institute for Nuclear Science and Technology (INST), Hanoi, Vietnam
66
RWTH Aachen University, III. Physikalisches Institut A, Aachen, Germany
67
ASTRON, Dwingeloo, Netherlands
68
J. Stefan Institute, Ljubljana, Slovenia
69
Dipartimento di Fisica dell’Universit`a and INFN, Genova, Italy
70
University of L´od´z, L´od´z, Poland
71
INFN, Laboratori Nazionali del Gran Sasso, Assergi (L’Aquila), Italy
72
Universidade Estadual de Feira de Santana, Brazil
73
Palacky University, RCPTM, Olomouc, Czech Republic
74
Charles University, Faculty of Mathematics and Physics,
Institute of Particle and Nuclear Physics, Prague, Czech Republic
75
Louisiana State University, Baton Rouge, LA, USA
76
Universit¨at Hamburg, Hamburg, Germany
77
Universidade Federal do ABC, Santo Andr´e, SP, Brazil
78
Benem´erita Universidad Aut´onoma de Puebla, Puebla, Mexico
79
Universidad Michoacana de San Nicolas de Hidalgo, Morelia, Michoacan, Mexico
80
Dipartimento di Ingegneria dell’Innovazione dell’Universit`a del Salento and Sezione INFN, Lecce, Italy
81
Centro de Investigaci´on y de Estudios Avanzados del IPN (CINVESTAV), M´exico, D.F., Mexico
82
Southern University, Baton Rouge, LA, USA
83
Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford, United Kingdom
84
Instituto de F´ısica de Rosario (IFIR) - CONICET/U.N.R. and Facultad
de Ciencias Bioqu´ımicas y Farmac´euticas U.N.R., Rosario, Argentina
85
Universit`a di Milano and Sezione INFN, Milan, Italy
86
Centro de Investigaciones en L´aseres y Aplicaciones, CITEFA and CONICET, Argentina
87
Instituto de F´ısica Corpuscular, CSIC-Universitat de Val`encia, Valencia, Spain
88
Northeastern University, Boston, MA, USA
89
Universidade Federal da Bahia, Salvador, BA, Brazil
90
University of Nebraska, Lincoln, NE, USA
91
Colorado School of Mines, Golden, CO, USA
92
Argonne National Laboratory, Argonne, IL, USA
93
Universit`a di Palermo and Sezione INFN, Catania, Italy
4
94
Los Alamos National Laboratory, Los Alamos, NM, USA
We report a measurement of the proton-air cross-section for particle production at t he center-of-
mass energy per nucleon of 57 TeV. This is derived from the distribution of the depths of shower
maxima observed with the Pierre Auger Observatory: systematic un certainties are studied in d e-
tail. Analysing the tail of the distribution of the shower maxima, a proton-air cross-section of
505 ± 22(stat)
+28
−36
(sys)
mb is found.
PACS numbers:
INTRODUCTION
We present the first analysis of the proton-air cross-
section based on measurements made at the Pierre Auger
Observatory [1]. For this purpose we analyse the shape
of the distribution of the largest values of the depth of
shower maximum, X
max
, the position at which an air
shower deposits the maximum energy per unit of mass of
atmosphere traversed. The tail of the X
max
-distribution
is sensitive to the proton-air cross-section, a fact first
exploited in the pioneering work of the Fly’s Eye Collab-
oration [2]. To obtain accurate measurements of X
max
,
timing data from the fluorescence telescopes is combined
with that from the surface detector array for a precise
hybrid reconstruction of the geometry of events [3].
We place particular emphasis on studying systematic
uncertainties in the cross-section analysis. The unknown
mass composition of cosmic-rays [4] is identified to be
the major source of systematic uncertainty and accord-
ingly the analysis has been optimised to minimise the
impact of particles other than protons in the primary
beam. This begins with restricting the analysis to the
energy interval 10
18
to 10
18.5
eV, where the shape of the
X
max
-distribution is compatible with there being a sub-
stantial fraction of protons; also there are a large number
of events recorded in this energy range. The correspond-
ing average center-of-mass energy of a proton interacting
with a nucleon is 57 TeV, significantly above the reach of
the Large Hadron Collider.
ANALYSIS APPROACH
The proton-air cross-section is derived in a two-step
process. Firstly, we measure an air-shower observable
with high sensitivity to the cross-section. Secondly, we
convert this measurement to a value of the proton-air
cross-section for particle production (c.f. [5]). This is
the cross-section that accounts for all interactions which
produce particles and thus contribute to the air-shower
development; it implicitly also includes diffractive inter-
actions. As the primary observable we define Λ
η
via the
exponential shape of the tail of the X
max
-distribution,
dN/dX
max
∝ exp(−X
max
/Λ
η
), where η denotes the frac-
tion of most deeply penetrating air showers used. Con-
sidering only these events enhances the contribution of
protons in the sample, since the depth at which proton-
induced showers maximise is deeper in the atmosphere
than for showers from heavier nuclei. Thus, η is a key
parameter: a small value enhances the proton fraction,
but reduces the number of events available for the analy-
sis. We have chosen η = 0.2 so that, for helium-fractions
up to 25 %, biases introduced by the possible presence of
helium and heavier nuclei do not exceed the level of the
statistical uncertainty. This was chosen after a Monte
Carlo study that probed the sensitivity of the analysis to
the mass composition depending on the choice of different
values of η.
THE MEASUREMENT OF Λ
η
We use events collected between 1 Dec 2004 and 20
Sept 2010. The atmospheric and event-quality cuts ap-
plied are identical to those used for the analysis of hX
max
i
and RMS(X
max
) [6] yielding 11628 high-quality events.
The X
max
-distribution of these data is affected by the
known geometrical acceptance of the fluorescence tele-
scopes as well as by limitations related to atmospheric
light transmission. We use the strategy developed for
the measurement of hX
max
i and RMS(X
max
) to extract
a sample that has an unbiased X
max
-distribution: a fidu-
cial volume selection, which requires event geometries
that allow, for each individual shower, the complete ob-
servation of a defined slant depth range.
Firstly, we derive the range of values of X
max
that
corresponds to the fraction η = 0.2 of the most deeply
penetrating showers. For this we need an unbiased distri-
bution of X
max
over the entire depth range of observed
values of X
max
. To achieve this we perform a fiducial
event selection of the slant depth range containing 99.8 %
of the observed X
max
-distribution, which corresponds to
the range from 550 to 1004 g/cm
2
. This reduces the
data sample to 1635 events providing an unbiased X
max
-
distribution that is used to find the range of values of
X
max
corresponding to η = 0.2, identified to extend from
768 to 1004 g/cm
2
.
Secondly, we select those events from the original
11628 that have geometries allowing the complete ob-
servation of values of X
max
from 768 to 1004 g/cm
2
, the
tail of the unbiased distribution. This fiducial cut max-
imises the statistics of an unbiased X
max
-distribution in
the range of interest. In total 3082 events pass the fidu-
5
]
2
[g/cm
max
X
500 600 700 800 900 1000 1100 1200
/g]
2
[cm
max
dN/dX
-1
10
1
10
2
2.3 g/cm± = 55.8
η
Λ
FIG. 1: Unbinned likelihood fit to obtain Λ
η
(thick line).
The X
max
-distribution is unbiased by the fiducial geometry
selection applied in the range of the fit.
cial volume cuts, of which 783 events have their X
max
in the selected range and thus contribute directly to the
measurement of Λ
η
. In Fig. 1 we show the 3082 selected
events and the result of an unbinned maximum likeli-
hood fit of an exponential function over the range 768 to
1004 g/cm
2
. Values of Λ
η
have been re-calculated for sub-
samples of the full dataset selected according to zenith-
angle, shower-to-telescope distance and energy: the dif-
ferent values obtained for Λ
η
are consistent with statisti-
cal fluctuations. The re-analyses of the data for changes
of fiducial event selection, modified values of η and for
different ranges of atmospheric depths yield changes of
Λ
η
that are distributed around zero with a root-mean-
square of 1.6 g/cm
2
. We use this root-mean-square as an
estimate of the systematic uncertainties associated to the
measurement. This yields
Λ
η
= [55.8 ± 2.3(stat) ± 1.6(sys)] g/cm
2
, (1)
with the average energy of these events being
10
18.24 ±0.005(stat)
eV. The differential energy distribution
for these events follows a power-law with index −1.9. The
average energy corresponds to a center-of-mass energy of
√
s = 57 ± 0.3(stat) TeV in proton-proton collisions.
DETERMINATION OF THE CROSS-SECTION
The determination of the proton-air cross-section for
particle production requires the use of air-shower sim-
ulations, which inherently introduces some dependence
on model assumptions. We emulate the measurement of
Λ
η
with Monte Carlo simulations to derive predictions of
the slope, Λ
MC
η
. It is known from previous work that the
values of Λ
MC
η
are directly linked to the hadronic cross-
sections used in the simulations [2]. Accordingly we can
explore the effect of changing cross-sections empirically
by multiplying all hadronic cross-sections input to the
simulations by an energy-dependent factor [7]
f(E, f
19
) = 1 + (f
19
− 1)
lg
E/10
15
eV
lg (10
19
eV/10
15
eV)
, (2)
where E denotes the shower energy and f
19
is the factor
by which the cross-section is rescaled at 10
19
eV. This
factor is unity below 10
15
eV reflecting the fact that mea-
surements of the cross-section at the Tevatron were used
to tune the interaction models. This technique of modi-
fying the original predictions of the cross-sections during
the simulation process assures a smooth transition from
accelerator data up to the energies of our analysis.
For each hadronic interaction model, the value of f
19
is
obtained that reproduces the measured value of Λ
η
. The
modified cross-section is then deduced by multiplying the
original cross-section used in the model by the factor
f(E, f
19
) of Eq. (2) using E = 10
18.24
eV. For the conver-
sion of Λ
η
into cross-section, we have used the four high-
energy hadronic interaction models commonly adopted
for air shower simulations: QGSJet01 [8], QGSJetII.3 [9],
SIBYLL 2.1 [10] and EPOS1.99 [11]. While in general
no model gives a completely accurate representation of
cosmic-ray data in all respects, these have been found to
give reasonably good descriptions of many of the main
features. It has been shown [12] that the differences be-
tween the models used in the analysis are typically bigger
than the variations obtained within one model by param-
eter variation. Therefore we use the model differences for
estimating the systematic model dependence.
The proton-air cross-sections for particle production
derived for QGSJet01, QGSJetII, SIBYLL and EPOS
are 523.7, 502.9, 496.7 and 497.7 mb respectively, with
the statistical uncertainty for each of these values being
22 mb. The difference of these cross-sections from the
original model predictions are < 5 %, with the exception
of the result obtained with the SIBYLL model, which is
12 % smaller than the original SIBYLL prediction. We
use the maximum deviations derived from using the four
models, relative to the average result of 505 mb, to es-
timate a systematic uncertainty of (−8, +19) mb related
to the difficulties of modelling high energy interactions.
This procedure relies on the coverage of the underlying
theoretical uncertainties by the available models. For
example diffraction, fragmentation, inelastic intermedi-
ate states, nuclear effects, QCD saturation, etc. are all
described at different levels using different phenomeno-
logical, but self-consistent, approaches in these models.
It is thus possible that the true range of the uncertainties
for air-shower analyses is larger, but this cannot be esti-
mated with these models. Furthermore, certain features
of hadronic particle production, such as the multiplic-
ity, elasticity and pion-charge ratio, have an especially
important impact on air shower development [13, 14]; of
these we found that only the elasticity can have a relevant
impact on Λ
η
. The identified systematic uncertainty of
![FIG. 3: Correlation of elastic slope parameter, Bel, and the inelastic proton-proton cross-section in the Glauber framework. The solid line indicates the parameter combinations yielding the observed proton-air production cross-section, and the dotted lines are the statistical uncertainties. The hatched area corresponds to the predictions by SIBYLL, QGSJet, QGSJetII and EPOS. See also Ref. [5].](/figures/fig-3-correlation-of-elastic-slope-parameter-bel-and-the-yjkkn3rd.webp)