Strandberg${}^{\textrm{\scriptsize 45a,45b}}$, M.
Strauss${}^{\textrm{\scriptsize 128}}$, P. Strizenec${}^{\textrm{\scriptsize
28b}}$, R. Ströhmer${}^{\textrm{\scriptsize 177}}$, D.M.
Strom${}^{\textrm{\scriptsize 131}}$, R. Stroynowski${}^{\textrm{\scriptsize
42}}$, A. Strubig${}^{\textrm{\scriptsize 50}}$, S.A.
Stucci${}^{\textrm{\scriptsize 29}}$, B. Stugu${}^{\textrm{\scriptsize 17}}$,
J. Stupak${}^{\textrm{\scriptsize 128}}$, N.A. Styles${}^{\textrm{\scriptsize
46}}$, D. Su${}^{\textrm{\scriptsize 153}}$, S. Suchek${}^{\textrm{\scriptsize
61a}}$, V.V. Sulin${}^{\textrm{\scriptsize 110}}$, M.J.
Sullivan${}^{\textrm{\scriptsize 90}}$, D.M.S. Sultan${}^{\textrm{\scriptsize
54}}$, S. Sultansoy${}^{\textrm{\scriptsize 4c}}$, T.
Sumida${}^{\textrm{\scriptsize 85}}$, S. Sun${}^{\textrm{\scriptsize 105}}$,
X. Sun${}^{\textrm{\scriptsize 3}}$, K. Suruliz${}^{\textrm{\scriptsize
156}}$, C.J.E. Suster${}^{\textrm{\scriptsize 157}}$, M.R.
Sutton${}^{\textrm{\scriptsize 156}}$, S. Suzuki${}^{\textrm{\scriptsize
81}}$, M. Svatos${}^{\textrm{\scriptsize 141}}$, M.
Swiatlowski${}^{\textrm{\scriptsize 37}}$, S.P. Swift${}^{\textrm{\scriptsize
2}}$, T. Swirski${}^{\textrm{\scriptsize 177}}$, A.
Sydorenko${}^{\textrm{\scriptsize 99}}$, I. Sykora${}^{\textrm{\scriptsize
28a}}$, M. Sykora${}^{\textrm{\scriptsize 143}}$, T.
Sykora${}^{\textrm{\scriptsize 143}}$, D. Ta${}^{\textrm{\scriptsize 99}}$, K.
Tackmann${}^{\textrm{\scriptsize 46,aa}}$, J. Taenzer${}^{\textrm{\scriptsize
161}}$, A. Taffard${}^{\textrm{\scriptsize 171}}$, R.
Tafirout${}^{\textrm{\scriptsize 168a}}$, H. Takai${}^{\textrm{\scriptsize
29}}$, R. Takashima${}^{\textrm{\scriptsize 86}}$, K.
Takeda${}^{\textrm{\scriptsize 82}}$, T. Takeshita${}^{\textrm{\scriptsize
150}}$, E.P. Takeva${}^{\textrm{\scriptsize 50}}$, Y.
Takubo${}^{\textrm{\scriptsize 81}}$, M. Talby${}^{\textrm{\scriptsize 101}}$,
A.A. Talyshev${}^{\textrm{\scriptsize 122b,122a}}$, N.M.
Tamir${}^{\textrm{\scriptsize 161}}$, J. Tanaka${}^{\textrm{\scriptsize
163}}$, M. Tanaka${}^{\textrm{\scriptsize 165}}$, R.
Tanaka${}^{\textrm{\scriptsize 132}}$, S. Tapia Araya${}^{\textrm{\scriptsize
173}}$, S. Tapprogge${}^{\textrm{\scriptsize 99}}$, A. Tarek Abouelfadl
Mohamed${}^{\textrm{\scriptsize 136}}$, S. Tarem${}^{\textrm{\scriptsize
160}}$, K. Tariq${}^{\textrm{\scriptsize 60b}}$, G.
Tarna${}^{\textrm{\scriptsize 27b,c}}$, G.F.
Tartarelli${}^{\textrm{\scriptsize 68a}}$, P. Tas${}^{\textrm{\scriptsize
143}}$, M. Tasevsky${}^{\textrm{\scriptsize 141}}$, T.
Tashiro${}^{\textrm{\scriptsize 85}}$, E. Tassi${}^{\textrm{\scriptsize
41b,41a}}$, A. Tavares Delgado${}^{\textrm{\scriptsize 140a,140b}}$, Y.
Tayalati${}^{\textrm{\scriptsize 35e}}$, A.J. Taylor${}^{\textrm{\scriptsize
50}}$, G.N. Taylor${}^{\textrm{\scriptsize 104}}$, W.
Taylor${}^{\textrm{\scriptsize 168b}}$, A.S. Tee${}^{\textrm{\scriptsize
89}}$, R. Teixeira De Lima${}^{\textrm{\scriptsize 153}}$, P. Teixeira-
Dias${}^{\textrm{\scriptsize 93}}$, H. Ten Kate${}^{\textrm{\scriptsize 36}}$,
J.J. Teoh${}^{\textrm{\scriptsize 120}}$, S. Terada${}^{\textrm{\scriptsize
81}}$, K. Terashi${}^{\textrm{\scriptsize 163}}$, J.
Terron${}^{\textrm{\scriptsize 98}}$, S. Terzo${}^{\textrm{\scriptsize 14}}$,
M. Testa${}^{\textrm{\scriptsize 51}}$, R.J. Teuscher${}^{\textrm{\scriptsize
167,ae}}$, S.J. Thais${}^{\textrm{\scriptsize 183}}$, T. Theveneaux-
Pelzer${}^{\textrm{\scriptsize 46}}$, F. Thiele${}^{\textrm{\scriptsize 40}}$,
D.W. Thomas${}^{\textrm{\scriptsize 93}}$, J.O. Thomas${}^{\textrm{\scriptsize
42}}$, J.P. Thomas${}^{\textrm{\scriptsize 21}}$, A.S.
Thompson${}^{\textrm{\scriptsize 57}}$, P.D. Thompson${}^{\textrm{\scriptsize
21}}$, L.A. Thomsen${}^{\textrm{\scriptsize 183}}$, E.
Thomson${}^{\textrm{\scriptsize 137}}$, E.J. Thorpe${}^{\textrm{\scriptsize
92}}$, Y. Tian${}^{\textrm{\scriptsize 39}}$, R.E. Ticse
Torres${}^{\textrm{\scriptsize 53}}$, V.O. Tikhomirov${}^{\textrm{\scriptsize
110,aq}}$, Yu.A. Tikhonov${}^{\textrm{\scriptsize 122b,122a}}$, S.
Timoshenko${}^{\textrm{\scriptsize 112}}$, P. Tipton${}^{\textrm{\scriptsize
183}}$, S. Tisserant${}^{\textrm{\scriptsize 101}}$, K.
Todome${}^{\textrm{\scriptsize 23b,23a}}$, S. Todorova-
Nova${}^{\textrm{\scriptsize 5}}$, S. Todt${}^{\textrm{\scriptsize 48}}$, J.
Tojo${}^{\textrm{\scriptsize 87}}$, S. Tokár${}^{\textrm{\scriptsize 28a}}$,
K. Tokushuku${}^{\textrm{\scriptsize 81}}$, E. Tolley${}^{\textrm{\scriptsize
126}}$, K.G. Tomiwa${}^{\textrm{\scriptsize 33c}}$, M.
Tomoto${}^{\textrm{\scriptsize 117}}$, L. Tompkins${}^{\textrm{\scriptsize
153,q}}$, B. Tong${}^{\textrm{\scriptsize 59}}$, P.
Tornambe${}^{\textrm{\scriptsize 102}}$, E. Torrence${}^{\textrm{\scriptsize
131}}$, H. Torres${}^{\textrm{\scriptsize 48}}$, E. Torró
Pastor${}^{\textrm{\scriptsize 148}}$, C. Tosciri${}^{\textrm{\scriptsize
135}}$, J. Toth${}^{\textrm{\scriptsize 101,ad}}$, D.R.
Tovey${}^{\textrm{\scriptsize 149}}$, A. Traeet${}^{\textrm{\scriptsize 17}}$,
C.J. Treado${}^{\textrm{\scriptsize 124}}$, T.
Trefzger${}^{\textrm{\scriptsize 177}}$, F. Tresoldi${}^{\textrm{\scriptsize
156}}$, A. Tricoli${}^{\textrm{\scriptsize 29}}$, I.M.
Trigger${}^{\textrm{\scriptsize 168a}}$, S. Trincaz-
Duvoid${}^{\textrm{\scriptsize 136}}$, W. Trischuk${}^{\textrm{\scriptsize
167}}$, B. Trocmé${}^{\textrm{\scriptsize 58}}$, A.
Trofymov${}^{\textrm{\scriptsize 145}}$, C. Troncon${}^{\textrm{\scriptsize
68a}}$, M. Trovatelli${}^{\textrm{\scriptsize 176}}$, F.
Trovato${}^{\textrm{\scriptsize 156}}$, L. Truong${}^{\textrm{\scriptsize
33b}}$, M. Trzebinski${}^{\textrm{\scriptsize 84}}$, A.
Trzupek${}^{\textrm{\scriptsize 84}}$, F. Tsai${}^{\textrm{\scriptsize 46}}$,
J.C-L. Tseng${}^{\textrm{\scriptsize 135}}$, P.V.
Tsiareshka${}^{\textrm{\scriptsize 107,ak}}$, A.
Tsirigotis${}^{\textrm{\scriptsize 162}}$, N.
Tsirintanis${}^{\textrm{\scriptsize 9}}$, V.
Tsiskaridze${}^{\textrm{\scriptsize 155}}$, E.G.
Tskhadadze${}^{\textrm{\scriptsize 159a}}$, M.
Tsopoulou${}^{\textrm{\scriptsize 162}}$, I.I.
Tsukerman${}^{\textrm{\scriptsize 111}}$, V. Tsulaia${}^{\textrm{\scriptsize
18}}$, S. Tsuno${}^{\textrm{\scriptsize 81}}$, D.
Tsybychev${}^{\textrm{\scriptsize 155}}$, Y. Tu${}^{\textrm{\scriptsize
63b}}$, A. Tudorache${}^{\textrm{\scriptsize 27b}}$, V.
Tudorache${}^{\textrm{\scriptsize 27b}}$, T.T. Tulbure${}^{\textrm{\scriptsize
27a}}$, A.N. Tuna${}^{\textrm{\scriptsize 59}}$, S.
Turchikhin${}^{\textrm{\scriptsize 79}}$, D. Turgeman${}^{\textrm{\scriptsize
180}}$, I. Turk Cakir${}^{\textrm{\scriptsize 4b,w}}$, R.J.
Turner${}^{\textrm{\scriptsize 21}}$, R.T. Turra${}^{\textrm{\scriptsize
68a}}$, P.M. Tuts${}^{\textrm{\scriptsize 39}}$, S.
Tzamarias${}^{\textrm{\scriptsize 162}}$, E. Tzovara${}^{\textrm{\scriptsize
99}}$, G. Ucchielli${}^{\textrm{\scriptsize 47}}$, K.
Uchida${}^{\textrm{\scriptsize 163}}$, I. Ueda${}^{\textrm{\scriptsize 81}}$,
M. Ughetto${}^{\textrm{\scriptsize 45a,45b}}$, F.
Ukegawa${}^{\textrm{\scriptsize 169}}$, G. Unal${}^{\textrm{\scriptsize 36}}$,
A. Undrus${}^{\textrm{\scriptsize 29}}$, G. Unel${}^{\textrm{\scriptsize
171}}$, F.C. Ungaro${}^{\textrm{\scriptsize 104}}$, Y.
Unno${}^{\textrm{\scriptsize 81}}$, K. Uno${}^{\textrm{\scriptsize 163}}$, J.
Urban${}^{\textrm{\scriptsize 28b}}$, P. Urquijo${}^{\textrm{\scriptsize
104}}$, G. Usai${}^{\textrm{\scriptsize 8}}$, Z. Uysal${}^{\textrm{\scriptsize
12d}}$, L. Vacavant${}^{\textrm{\scriptsize 101}}$, V.
Vacek${}^{\textrm{\scriptsize 142}}$, B. Vachon${}^{\textrm{\scriptsize
103}}$, K.O.H. Vadla${}^{\textrm{\scriptsize 134}}$, A.
Vaidya${}^{\textrm{\scriptsize 94}}$, C. Valderanis${}^{\textrm{\scriptsize
114}}$, E. Valdes Santurio${}^{\textrm{\scriptsize 45a,45b}}$, M.
Valente${}^{\textrm{\scriptsize 54}}$, S. Valentinetti${}^{\textrm{\scriptsize
23b,23a}}$, A. Valero${}^{\textrm{\scriptsize 174}}$, L.
Valéry${}^{\textrm{\scriptsize 46}}$, R.A. Vallance${}^{\textrm{\scriptsize
21}}$, A. Vallier${}^{\textrm{\scriptsize 36}}$, J.A. Valls
Ferrer${}^{\textrm{\scriptsize 174}}$, T.R. Van Daalen${}^{\textrm{\scriptsize
14}}$, P. Van Gemmeren${}^{\textrm{\scriptsize 6}}$, I. Van
Vulpen${}^{\textrm{\scriptsize 120}}$, M. Vanadia${}^{\textrm{\scriptsize
73a,73b}}$, W. Vandelli${}^{\textrm{\scriptsize 36}}$, A.
Vaniachine${}^{\textrm{\scriptsize 166}}$, D.
Vannicola${}^{\textrm{\scriptsize 72a,72b}}$, R. Vari${}^{\textrm{\scriptsize
72a}}$, E.W. Varnes${}^{\textrm{\scriptsize 7}}$, C.
Varni${}^{\textrm{\scriptsize 55b,55a}}$, T. Varol${}^{\textrm{\scriptsize
158}}$, D. Varouchas${}^{\textrm{\scriptsize 132}}$, K.E.
Varvell${}^{\textrm{\scriptsize 157}}$, M.E. Vasile${}^{\textrm{\scriptsize
27b}}$, G.A. Vasquez${}^{\textrm{\scriptsize 176}}$, J.G.
Vasquez${}^{\textrm{\scriptsize 183}}$, F. Vazeille${}^{\textrm{\scriptsize
38}}$, D. Vazquez Furelos${}^{\textrm{\scriptsize 14}}$, T. Vazquez
Schroeder${}^{\textrm{\scriptsize 36}}$, J. Veatch${}^{\textrm{\scriptsize
53}}$, V. Vecchio${}^{\textrm{\scriptsize 74a,74b}}$, M.J.
Veen${}^{\textrm{\scriptsize 120}}$, L.M. Veloce${}^{\textrm{\scriptsize
167}}$, F. Veloso${}^{\textrm{\scriptsize 140a,140c}}$, S.
Veneziano${}^{\textrm{\scriptsize 72a}}$, A. Ventura${}^{\textrm{\scriptsize
67a,67b}}$, N. Venturi${}^{\textrm{\scriptsize 36}}$, A.
Verbytskyi${}^{\textrm{\scriptsize 115}}$, V. Vercesi${}^{\textrm{\scriptsize
70a}}$, M. Verducci${}^{\textrm{\scriptsize 71a,71b}}$, C.M. Vergel
Infante${}^{\textrm{\scriptsize 78}}$, C. Vergis${}^{\textrm{\scriptsize
24}}$, W. Verkerke${}^{\textrm{\scriptsize 120}}$, A.T.
Vermeulen${}^{\textrm{\scriptsize 120}}$, J.C.
Vermeulen${}^{\textrm{\scriptsize 120}}$, M.C.
Vetterli${}^{\textrm{\scriptsize 152,ay}}$, N. Viaux
Maira${}^{\textrm{\scriptsize 147b}}$, M. Vicente Barreto
Pinto${}^{\textrm{\scriptsize 54}}$, T. Vickey${}^{\textrm{\scriptsize 149}}$,
O.E. Vickey Boeriu${}^{\textrm{\scriptsize 149}}$, G.H.A.
Viehhauser${}^{\textrm{\scriptsize 135}}$, L. Vigani${}^{\textrm{\scriptsize
61b}}$, M. Villa${}^{\textrm{\scriptsize 23b,23a}}$, M. Villaplana
Perez${}^{\textrm{\scriptsize 68a,68b}}$, E. Vilucchi${}^{\textrm{\scriptsize
51}}$, M.G. Vincter${}^{\textrm{\scriptsize 34}}$, G.S.
Virdee${}^{\textrm{\scriptsize 21}}$, A. Vishwakarma${}^{\textrm{\scriptsize
46}}$, C. Vittori${}^{\textrm{\scriptsize 23b,23a}}$, I.
Vivarelli${}^{\textrm{\scriptsize 156}}$, M. Vogel${}^{\textrm{\scriptsize
182}}$, P. Vokac${}^{\textrm{\scriptsize 142}}$, S.E. von
Buddenbrock${}^{\textrm{\scriptsize 33c}}$, E. Von
Toerne${}^{\textrm{\scriptsize 24}}$, V. Vorobel${}^{\textrm{\scriptsize
143}}$, K. Vorobev${}^{\textrm{\scriptsize 112}}$, M.
Vos${}^{\textrm{\scriptsize 174}}$, J.H. Vossebeld${}^{\textrm{\scriptsize
90}}$, M. Vozak${}^{\textrm{\scriptsize 100}}$, N.
Vranjes${}^{\textrm{\scriptsize 16}}$, M. Vranjes
Milosavljevic${}^{\textrm{\scriptsize 16}}$, V. Vrba${}^{\textrm{\scriptsize
142}}$, M. Vreeswijk${}^{\textrm{\scriptsize 120}}$, R.
Vuillermet${}^{\textrm{\scriptsize 36}}$, I. Vukotic${}^{\textrm{\scriptsize
37}}$, P. Wagner${}^{\textrm{\scriptsize 24}}$, W.
Wagner${}^{\textrm{\scriptsize 182}}$, J. Wagner-Kuhr${}^{\textrm{\scriptsize
114}}$, S. Wahdan${}^{\textrm{\scriptsize 182}}$, H.
Wahlberg${}^{\textrm{\scriptsize 88}}$, V.M. Walbrecht${}^{\textrm{\scriptsize
115}}$, J. Walder${}^{\textrm{\scriptsize 89}}$, R.
Walker${}^{\textrm{\scriptsize 114}}$, S.D. Walker${}^{\textrm{\scriptsize
93}}$, W. Walkowiak${}^{\textrm{\scriptsize 151}}$, V.
Wallangen${}^{\textrm{\scriptsize 45a,45b}}$, A.M.
Wang${}^{\textrm{\scriptsize 59}}$, C. Wang${}^{\textrm{\scriptsize 60c}}$, C.
Wang${}^{\textrm{\scriptsize 60b}}$, F. Wang${}^{\textrm{\scriptsize 181}}$,
H. Wang${}^{\textrm{\scriptsize 18}}$, H. Wang${}^{\textrm{\scriptsize 3}}$,
J. Wang${}^{\textrm{\scriptsize 157}}$, J. Wang${}^{\textrm{\scriptsize
61b}}$, P. Wang${}^{\textrm{\scriptsize 42}}$, Q. Wang${}^{\textrm{\scriptsize
128}}$, R.-J. Wang${}^{\textrm{\scriptsize 99}}$, R.
Wang${}^{\textrm{\scriptsize 60a}}$, R. Wang${}^{\textrm{\scriptsize 6}}$,
S.M. Wang${}^{\textrm{\scriptsize 158}}$, W.T. Wang${}^{\textrm{\scriptsize
60a}}$, W. Wang${}^{\textrm{\scriptsize 15c,af}}$, W.X.
Wang${}^{\textrm{\scriptsize 60a,af}}$, Y. Wang${}^{\textrm{\scriptsize
60a,an}}$, Z. Wang${}^{\textrm{\scriptsize 60c}}$, C.
Wanotayaroj${}^{\textrm{\scriptsize 46}}$, A.
Warburton${}^{\textrm{\scriptsize 103}}$, C.P. Ward${}^{\textrm{\scriptsize
32}}$, D.R. Wardrope${}^{\textrm{\scriptsize 94}}$, N.
Warrack${}^{\textrm{\scriptsize 57}}$, A. Washbrook${}^{\textrm{\scriptsize
50}}$, A.T. Watson${}^{\textrm{\scriptsize 21}}$, M.F.
Watson${}^{\textrm{\scriptsize 21}}$, G. Watts${}^{\textrm{\scriptsize 148}}$,
B.M. Waugh${}^{\textrm{\scriptsize 94}}$, A.F. Webb${}^{\textrm{\scriptsize
11}}$, S. Webb${}^{\textrm{\scriptsize 99}}$, C. Weber${}^{\textrm{\scriptsize
183}}$, M.S. Weber${}^{\textrm{\scriptsize 20}}$, S.A.
Weber${}^{\textrm{\scriptsize 34}}$, S.M. Weber${}^{\textrm{\scriptsize
61a}}$, A.R. Weidberg${}^{\textrm{\scriptsize 135}}$, J.
Weingarten${}^{\textrm{\scriptsize 47}}$, M. Weirich${}^{\textrm{\scriptsize
99}}$, C. Weiser${}^{\textrm{\scriptsize 52}}$, P.S.
Wells${}^{\textrm{\scriptsize 36}}$, T. Wenaus${}^{\textrm{\scriptsize 29}}$,
T. Wengler${}^{\textrm{\scriptsize 36}}$, S. Wenig${}^{\textrm{\scriptsize
36}}$, N. Wermes${}^{\textrm{\scriptsize 24}}$, M.D.
Werner${}^{\textrm{\scriptsize 78}}$, M. Wessels${}^{\textrm{\scriptsize
61a}}$, T.D. Weston${}^{\textrm{\scriptsize 20}}$, K.
Whalen${}^{\textrm{\scriptsize 131}}$, N.L. Whallon${}^{\textrm{\scriptsize
148}}$, A.M. Wharton${}^{\textrm{\scriptsize 89}}$, A.S.
White${}^{\textrm{\scriptsize 105}}$, A. White${}^{\textrm{\scriptsize 8}}$,
M.J. White${}^{\textrm{\scriptsize 1}}$, D. Whiteson${}^{\textrm{\scriptsize
171}}$, B.W. Whitmore${}^{\textrm{\scriptsize 89}}$, W.
Wiedenmann${}^{\textrm{\scriptsize 181}}$, M. Wielers${}^{\textrm{\scriptsize
144}}$, N. Wieseotte${}^{\textrm{\scriptsize 99}}$, C.
Wiglesworth${}^{\textrm{\scriptsize 40}}$, L.A.M. Wiik-
Fuchs${}^{\textrm{\scriptsize 52}}$, F. Wilk${}^{\textrm{\scriptsize 100}}$,
H.G. Wilkens${}^{\textrm{\scriptsize 36}}$, L.J.
Wilkins${}^{\textrm{\scriptsize 93}}$, H.H. Williams${}^{\textrm{\scriptsize
137}}$, S. Williams${}^{\textrm{\scriptsize 32}}$, C.
Willis${}^{\textrm{\scriptsize 106}}$, S. Willocq${}^{\textrm{\scriptsize
102}}$, J.A. Wilson${}^{\textrm{\scriptsize 21}}$, I. Wingerter-
Seez${}^{\textrm{\scriptsize 5}}$, E. Winkels${}^{\textrm{\scriptsize 156}}$,
F. Winklmeier${}^{\textrm{\scriptsize 131}}$, O.J.
Winston${}^{\textrm{\scriptsize 156}}$, B.T. Winter${}^{\textrm{\scriptsize
52}}$, M. Wittgen${}^{\textrm{\scriptsize 153}}$, M.
Wobisch${}^{\textrm{\scriptsize 95}}$, A. Wolf${}^{\textrm{\scriptsize 99}}$,
T.M.H. Wolf${}^{\textrm{\scriptsize 120}}$, R. Wolff${}^{\textrm{\scriptsize
101}}$, R.W. Wölker${}^{\textrm{\scriptsize 135}}$, J.
Wollrath${}^{\textrm{\scriptsize 52}}$, M.W. Wolter${}^{\textrm{\scriptsize
84}}$, H. Wolters${}^{\textrm{\scriptsize 140a,140c}}$, V.W.S.
Wong${}^{\textrm{\scriptsize 175}}$, N.L. Woods${}^{\textrm{\scriptsize
146}}$, S.D. Worm${}^{\textrm{\scriptsize 21}}$, B.K.
Wosiek${}^{\textrm{\scriptsize 84}}$, K.W. Woźniak${}^{\textrm{\scriptsize
84}}$, K. Wraight${}^{\textrm{\scriptsize 57}}$, S.L.
Wu${}^{\textrm{\scriptsize 181}}$, X. Wu${}^{\textrm{\scriptsize 54}}$, Y.
Wu${}^{\textrm{\scriptsize 60a}}$, T.R. Wyatt${}^{\textrm{\scriptsize 100}}$,
B.M. Wynne${}^{\textrm{\scriptsize 50}}$, S. Xella${}^{\textrm{\scriptsize
40}}$, Z. Xi${}^{\textrm{\scriptsize 105}}$, L. Xia${}^{\textrm{\scriptsize
178}}$, X. Xiao${}^{\textrm{\scriptsize 105}}$, I.
Xiotidis${}^{\textrm{\scriptsize 156}}$, D. Xu${}^{\textrm{\scriptsize 15a}}$,
H. Xu${}^{\textrm{\scriptsize 60a,c}}$, L. Xu${}^{\textrm{\scriptsize 29}}$,
T. Xu${}^{\textrm{\scriptsize 145}}$, W. Xu${}^{\textrm{\scriptsize 105}}$, Z.
Xu${}^{\textrm{\scriptsize 60b}}$, Z. Xu${}^{\textrm{\scriptsize 153}}$, B.
Yabsley${}^{\textrm{\scriptsize 157}}$, S. Yacoob${}^{\textrm{\scriptsize
33a}}$, K. Yajima${}^{\textrm{\scriptsize 133}}$, D.P.
Yallup${}^{\textrm{\scriptsize 94}}$, D. Yamaguchi${}^{\textrm{\scriptsize
165}}$, Y. Yamaguchi${}^{\textrm{\scriptsize 165}}$, A.
Yamamoto${}^{\textrm{\scriptsize 81}}$, M. Yamatani${}^{\textrm{\scriptsize
163}}$, T. Yamazaki${}^{\textrm{\scriptsize 163}}$, Y.
Yamazaki${}^{\textrm{\scriptsize 82}}$, Z. Yan${}^{\textrm{\scriptsize 25}}$,
H.J. Yang${}^{\textrm{\scriptsize 60c,60d}}$, H.T.
Yang${}^{\textrm{\scriptsize 18}}$, S. Yang${}^{\textrm{\scriptsize 77}}$, X.
Yang${}^{\textrm{\scriptsize 60b,58}}$, Y. Yang${}^{\textrm{\scriptsize
163}}$, W-M. Yao${}^{\textrm{\scriptsize 18}}$, Y.C.
Yap${}^{\textrm{\scriptsize 46}}$, Y. Yasu${}^{\textrm{\scriptsize 81}}$, E.
Yatsenko${}^{\textrm{\scriptsize 60c,60d}}$, J. Ye${}^{\textrm{\scriptsize
42}}$, S. Ye${}^{\textrm{\scriptsize 29}}$, I.
Yeletskikh${}^{\textrm{\scriptsize 79}}$, M.R. Yexley${}^{\textrm{\scriptsize
89}}$, E. Yigitbasi${}^{\textrm{\scriptsize 25}}$, K.
Yorita${}^{\textrm{\scriptsize 179}}$, K. Yoshihara${}^{\textrm{\scriptsize
137}}$, C.J.S. Young${}^{\textrm{\scriptsize 36}}$, C.
Young${}^{\textrm{\scriptsize 153}}$, J. Yu${}^{\textrm{\scriptsize 78}}$, R.
Yuan${}^{\textrm{\scriptsize 60b,i}}$, X. Yue${}^{\textrm{\scriptsize 61a}}$,
S.P.Y. Yuen${}^{\textrm{\scriptsize 24}}$, M. Zaazoua${}^{\textrm{\scriptsize
35e}}$, B. Zabinski${}^{\textrm{\scriptsize 84}}$, G.
Zacharis${}^{\textrm{\scriptsize 10}}$, E. Zaffaroni${}^{\textrm{\scriptsize
54}}$, J. Zahreddine${}^{\textrm{\scriptsize 136}}$, A.M.
Zaitsev${}^{\textrm{\scriptsize 123,ap}}$, T.
Zakareishvili${}^{\textrm{\scriptsize 159b}}$, N.
Zakharchuk${}^{\textrm{\scriptsize 34}}$, S. Zambito${}^{\textrm{\scriptsize
59}}$, D. Zanzi${}^{\textrm{\scriptsize 36}}$, D.R.
Zaripovas${}^{\textrm{\scriptsize 57}}$, S.V. Zeißner${}^{\textrm{\scriptsize
47}}$, C. Zeitnitz${}^{\textrm{\scriptsize 182}}$, G.
Zemaityte${}^{\textrm{\scriptsize 135}}$, J.C. Zeng${}^{\textrm{\scriptsize
173}}$, O. Zenin${}^{\textrm{\scriptsize 123}}$, T.
Ženiš${}^{\textrm{\scriptsize 28a}}$, D. Zerwas${}^{\textrm{\scriptsize
132}}$, M. Zgubič${}^{\textrm{\scriptsize 135}}$, D.F.
Zhang${}^{\textrm{\scriptsize 15b}}$, G. Zhang${}^{\textrm{\scriptsize 15b}}$,
H. Zhang${}^{\textrm{\scriptsize 15c}}$, J. Zhang${}^{\textrm{\scriptsize
6}}$, L. Zhang${}^{\textrm{\scriptsize 15c}}$, L.
Zhang${}^{\textrm{\scriptsize 60a}}$, M. Zhang${}^{\textrm{\scriptsize 173}}$,
R. Zhang${}^{\textrm{\scriptsize 24}}$, X. Zhang${}^{\textrm{\scriptsize
60b}}$, Y. Zhang${}^{\textrm{\scriptsize 15a,15d}}$, Z.
Zhang${}^{\textrm{\scriptsize 63a}}$, Z. Zhang${}^{\textrm{\scriptsize 132}}$,
P. Zhao${}^{\textrm{\scriptsize 49}}$, Y. Zhao${}^{\textrm{\scriptsize 60b}}$,
Z. Zhao${}^{\textrm{\scriptsize 60a}}$, A. Zhemchugov${}^{\textrm{\scriptsize
79}}$, Z. Zheng${}^{\textrm{\scriptsize 105}}$, D.
Zhong${}^{\textrm{\scriptsize 173}}$, B. Zhou${}^{\textrm{\scriptsize 105}}$,
C. Zhou${}^{\textrm{\scriptsize 181}}$, M.S. Zhou${}^{\textrm{\scriptsize
15a,15d}}$, M. Zhou${}^{\textrm{\scriptsize 155}}$, N.
Zhou${}^{\textrm{\scriptsize 60c}}$, Y. Zhou${}^{\textrm{\scriptsize 7}}$,
C.G. Zhu${}^{\textrm{\scriptsize 60b}}$, H.L. Zhu${}^{\textrm{\scriptsize
60a}}$, H. Zhu${}^{\textrm{\scriptsize 15a}}$, J. Zhu${}^{\textrm{\scriptsize
105}}$, Y. Zhu${}^{\textrm{\scriptsize 60a}}$, X.
Zhuang${}^{\textrm{\scriptsize 15a}}$, K. Zhukov${}^{\textrm{\scriptsize
110}}$, V. Zhulanov${}^{\textrm{\scriptsize 122b,122a}}$, D.
Zieminska${}^{\textrm{\scriptsize 65}}$, N.I. Zimine${}^{\textrm{\scriptsize
79}}$, S. Zimmermann${}^{\textrm{\scriptsize 52}}$, Z.
Zinonos${}^{\textrm{\scriptsize 115}}$, M. Ziolkowski${}^{\textrm{\scriptsize
151}}$, L. Živković${}^{\textrm{\scriptsize 16}}$, G.
Zobernig${}^{\textrm{\scriptsize 181}}$, A. Zoccoli${}^{\textrm{\scriptsize
23b,23a}}$, K. Zoch${}^{\textrm{\scriptsize 53}}$, T.G.
Zorbas${}^{\textrm{\scriptsize 149}}$, R. Zou${}^{\textrm{\scriptsize 37}}$,
L. Zwalinski${}^{\textrm{\scriptsize 36}}$.

1Department of Physics, University of Adelaide, Adelaide; Australia.

2Physics Department, SUNY Albany, Albany NY; United States of America.

3Department of Physics, University of Alberta, Edmonton AB; Canada.

4(a)Department of Physics, Ankara University, Ankara;(b)Istanbul Aydin
University, Istanbul;(c)Division of Physics, TOBB University of Economics and
Technology, Ankara; Turkey.

5LAPP, Université Grenoble Alpes, Université Savoie Mont Blanc, CNRS/IN2P3,
Annecy; France.

6High Energy Physics Division, Argonne National Laboratory, Argonne IL; United
States of America.

7Department of Physics, University of Arizona, Tucson AZ; United States of
America.

8Department of Physics, University of Texas at Arlington, Arlington TX; United
States of America.

9Physics Department, National and Kapodistrian University of Athens, Athens;
Greece.

10Physics Department, National Technical University of Athens, Zografou;
Greece.

11Department of Physics, University of Texas at Austin, Austin TX; United
States of America.

12(a)Bahcesehir University, Faculty of Engineering and Natural Sciences,
Istanbul;(b)Istanbul Bilgi University, Faculty of Engineering and Natural
Sciences, Istanbul;(c)Department of Physics, Bogazici University,
Istanbul;(d)Department of Physics Engineering, Gaziantep University,
Gaziantep; Turkey.

13Institute of Physics, Azerbaijan Academy of Sciences, Baku; Azerbaijan.

14Institut de Física d’Altes Energies (IFAE), Barcelona Institute of Science
and Technology, Barcelona; Spain.

15(a)Institute of High Energy Physics, Chinese Academy of Sciences,
Beijing;(b)Physics Department, Tsinghua University, Beijing;(c)Department of
Physics, Nanjing University, Nanjing;(d)University of Chinese Academy of
Science (UCAS), Beijing; China.

16Institute of Physics, University of Belgrade, Belgrade; Serbia.

17Department for Physics and Technology, University of Bergen, Bergen; Norway.

18Physics Division, Lawrence Berkeley National Laboratory and University of
California, Berkeley CA; United States of America.

19Institut für Physik, Humboldt Universität zu Berlin, Berlin; Germany.

20Albert Einstein Center for Fundamental Physics and Laboratory for High
Energy Physics, University of Bern, Bern; Switzerland.

21School of Physics and Astronomy, University of Birmingham, Birmingham;
United Kingdom.

22Facultad de Ciencias y Centro de Investigaciónes, Universidad Antonio
Nariño, Bogota; Colombia.

23(a)INFN Bologna and Universita’ di Bologna, Dipartimento di Fisica;(b)INFN
Sezione di Bologna; Italy.

24Physikalisches Institut, Universität Bonn, Bonn; Germany.

25Department of Physics, Boston University, Boston MA; United States of
America.

26Department of Physics, Brandeis University, Waltham MA; United States of
America.

27(a)Transilvania University of Brasov, Brasov;(b)Horia Hulubei National
Institute of Physics and Nuclear Engineering, Bucharest;(c)Department of
Physics, Alexandru Ioan Cuza University of Iasi, Iasi;(d)National Institute
for Research and Development of Isotopic and Molecular Technologies, Physics
Department, Cluj-Napoca;(e)University Politehnica Bucharest, Bucharest;(f)West
University in Timisoara, Timisoara; Romania.

28(a)Faculty of Mathematics, Physics and Informatics, Comenius University,
Bratislava;(b)Department of Subnuclear Physics, Institute of Experimental
Physics of the Slovak Academy of Sciences, Kosice; Slovak Republic.

29Physics Department, Brookhaven National Laboratory, Upton NY; United States
of America.

30Departamento de Física, Universidad de Buenos Aires, Buenos Aires;
Argentina.

31California State University, CA; United States of America.

32Cavendish Laboratory, University of Cambridge, Cambridge; United Kingdom.

33(a)Department of Physics, University of Cape Town, Cape Town;(b)Department
of Mechanical Engineering Science, University of Johannesburg,
Johannesburg;(c)School of Physics, University of the Witwatersrand,
Johannesburg; South Africa.

34Department of Physics, Carleton University, Ottawa ON; Canada.

35(a)Faculté des Sciences Ain Chock, Réseau Universitaire de Physique des
Hautes Energies - Université Hassan II, Casablanca;(b)Faculté des Sciences,
Université Ibn-Tofail, Kénitra;(c)Faculté des Sciences Semlalia, Université
Cadi Ayyad, LPHEA-Marrakech;(d)Faculté des Sciences, Université Mohamed
Premier and LPTPM, Oujda;(e)Faculté des sciences, Université Mohammed V,
Rabat; Morocco.

36CERN, Geneva; Switzerland.

37Enrico Fermi Institute, University of Chicago, Chicago IL; United States of
America.

38LPC, Université Clermont Auvergne, CNRS/IN2P3, Clermont-Ferrand; France.

39Nevis Laboratory, Columbia University, Irvington NY; United States of
America.

40Niels Bohr Institute, University of Copenhagen, Copenhagen; Denmark.

41(a)Dipartimento di Fisica, Università della Calabria, Rende;(b)INFN Gruppo
Collegato di Cosenza, Laboratori Nazionali di Frascati; Italy.

42Physics Department, Southern Methodist University, Dallas TX; United States
of America.

43Physics Department, University of Texas at Dallas, Richardson TX; United
States of America.

44National Centre for Scientific Research "Demokritos", Agia Paraskevi;
Greece.

45(a)Department of Physics, Stockholm University;(b)Oskar Klein Centre,
Stockholm; Sweden.

46Deutsches Elektronen-Synchrotron DESY, Hamburg and Zeuthen; Germany.

47Lehrstuhl für Experimentelle Physik IV, Technische Universität Dortmund,
Dortmund; Germany.

48Institut für Kern- und Teilchenphysik, Technische Universität Dresden,
Dresden; Germany.

49Department of Physics, Duke University, Durham NC; United States of America.

50SUPA - School of Physics and Astronomy, University of Edinburgh, Edinburgh;
United Kingdom.

51INFN e Laboratori Nazionali di Frascati, Frascati; Italy.

52Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Freiburg;
Germany.

53II. Physikalisches Institut, Georg-August-Universität Göttingen, Göttingen;
Germany.

54Département de Physique Nucléaire et Corpusculaire, Université de Genève,
Genève; Switzerland.

55(a)Dipartimento di Fisica, Università di Genova, Genova;(b)INFN Sezione di
Genova; Italy.

56II. Physikalisches Institut, Justus-Liebig-Universität Giessen, Giessen;
Germany.

57SUPA - School of Physics and Astronomy, University of Glasgow, Glasgow;
United Kingdom.

58LPSC, Université Grenoble Alpes, CNRS/IN2P3, Grenoble INP, Grenoble; France.

59Laboratory for Particle Physics and Cosmology, Harvard University, Cambridge
MA; United States of America.

60(a)Department of Modern Physics and State Key Laboratory of Particle
Detection and Electronics, University of Science and Technology of China,
Hefei;(b)Institute of Frontier and Interdisciplinary Science and Key
Laboratory of Particle Physics and Particle Irradiation (MOE), Shandong
University, Qingdao;(c)School of Physics and Astronomy, Shanghai Jiao Tong
University, KLPPAC-MoE, SKLPPC, Shanghai;(d)Tsung-Dao Lee Institute, Shanghai;
China.

61(a)Kirchhoff-Institut für Physik, Ruprecht-Karls-Universität Heidelberg,
Heidelberg;(b)Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg,
Heidelberg; Germany.

62Faculty of Applied Information Science, Hiroshima Institute of Technology,
Hiroshima; Japan.

63(a)Department of Physics, Chinese University of Hong Kong, Shatin, N.T.,
Hong Kong;(b)Department of Physics, University of Hong Kong, Hong
Kong;(c)Department of Physics and Institute for Advanced Study, Hong Kong
University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong;
China.

64Department of Physics, National Tsing Hua University, Hsinchu; Taiwan.

65Department of Physics, Indiana University, Bloomington IN; United States of
America.

66(a)INFN Gruppo Collegato di Udine, Sezione di Trieste, Udine;(b)ICTP,
Trieste;(c)Dipartimento Politecnico di Ingegneria e Architettura, Università
di Udine, Udine; Italy.

67(a)INFN Sezione di Lecce;(b)Dipartimento di Matematica e Fisica, Università
del Salento, Lecce; Italy.

68(a)INFN Sezione di Milano;(b)Dipartimento di Fisica, Università di Milano,
Milano; Italy.

69(a)INFN Sezione di Napoli;(b)Dipartimento di Fisica, Università di Napoli,
Napoli; Italy.

70(a)INFN Sezione di Pavia;(b)Dipartimento di Fisica, Università di Pavia,
Pavia; Italy.

71(a)INFN Sezione di Pisa;(b)Dipartimento di Fisica E. Fermi, Università di
Pisa, Pisa; Italy.

72(a)INFN Sezione di Roma;(b)Dipartimento di Fisica, Sapienza Università di
Roma, Roma; Italy.

73(a)INFN Sezione di Roma Tor Vergata;(b)Dipartimento di Fisica, Università di
Roma Tor Vergata, Roma; Italy.

74(a)INFN Sezione di Roma Tre;(b)Dipartimento di Matematica e Fisica,
Università Roma Tre, Roma; Italy.

75(a)INFN-TIFPA;(b)Università degli Studi di Trento, Trento; Italy.

76Institut für Astro- und Teilchenphysik, Leopold-Franzens-Universität,
Innsbruck; Austria.

77University of Iowa, Iowa City IA; United States of America.

78Department of Physics and Astronomy, Iowa State University, Ames IA; United
States of America.

79Joint Institute for Nuclear Research, Dubna; Russia.

80(a)Departamento de Engenharia Elétrica, Universidade Federal de Juiz de Fora
(UFJF), Juiz de Fora;(b)Universidade Federal do Rio De Janeiro COPPE/EE/IF,
Rio de Janeiro;(c)Universidade Federal de São João del Rei (UFSJ), São João
del Rei;(d)Instituto de Física, Universidade de São Paulo, São Paulo; Brazil.

81KEK, High Energy Accelerator Research Organization, Tsukuba; Japan.

82Graduate School of Science, Kobe University, Kobe; Japan.

83(a)AGH University of Science and Technology, Faculty of Physics and Applied
Computer Science, Krakow;(b)Marian Smoluchowski Institute of Physics,
Jagiellonian University, Krakow; Poland.

84Institute of Nuclear Physics Polish Academy of Sciences, Krakow; Poland.

85Faculty of Science, Kyoto University, Kyoto; Japan.

86Kyoto University of Education, Kyoto; Japan.

87Research Center for Advanced Particle Physics and Department of Physics,
Kyushu University, Fukuoka ; Japan.

88Instituto de Física La Plata, Universidad Nacional de La Plata and CONICET,
La Plata; Argentina.

89Physics Department, Lancaster University, Lancaster; United Kingdom.

90Oliver Lodge Laboratory, University of Liverpool, Liverpool; United Kingdom.

91Department of Experimental Particle Physics, Jožef Stefan Institute and
Department of Physics, University of Ljubljana, Ljubljana; Slovenia.

92School of Physics and Astronomy, Queen Mary University of London, London;
United Kingdom.

93Department of Physics, Royal Holloway University of London, Egham; United
Kingdom.

94Department of Physics and Astronomy, University College London, London;
United Kingdom.

95Louisiana Tech University, Ruston LA; United States of America.

96Fysiska institutionen, Lunds universitet, Lund; Sweden.

97Centre de Calcul de l’Institut National de Physique Nucléaire et de Physique
des Particules (IN2P3), Villeurbanne; France.

98Departamento de Física Teorica C-15 and CIAFF, Universidad Autónoma de
Madrid, Madrid; Spain.

99Institut für Physik, Universität Mainz, Mainz; Germany.

100School of Physics and Astronomy, University of Manchester, Manchester;
United Kingdom.

101CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille; France.

102Department of Physics, University of Massachusetts, Amherst MA; United
States of America.

103Department of Physics, McGill University, Montreal QC; Canada.

104School of Physics, University of Melbourne, Victoria; Australia.

105Department of Physics, University of Michigan, Ann Arbor MI; United States
of America.

106Department of Physics and Astronomy, Michigan State University, East
Lansing MI; United States of America.

107B.I. Stepanov Institute of Physics, National Academy of Sciences of
Belarus, Minsk; Belarus.

108Research Institute for Nuclear Problems of Byelorussian State University,
Minsk; Belarus.

109Group of Particle Physics, University of Montreal, Montreal QC; Canada.

110P.N. Lebedev Physical Institute of the Russian Academy of Sciences, Moscow;
Russia.

111Institute for Theoretical and Experimental Physics of the National Research
Centre Kurchatov Institute, Moscow; Russia.

112National Research Nuclear University MEPhI, Moscow; Russia.

113D.V. Skobeltsyn Institute of Nuclear Physics, M.V. Lomonosov Moscow State
University, Moscow; Russia.

114Fakultät für Physik, Ludwig-Maximilians-Universität München, München;
Germany.

115Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), München;
Germany.

116Nagasaki Institute of Applied Science, Nagasaki; Japan.

117Graduate School of Science and Kobayashi-Maskawa Institute, Nagoya
University, Nagoya; Japan.

118Department of Physics and Astronomy, University of New Mexico, Albuquerque
NM; United States of America.

119Institute for Mathematics, Astrophysics and Particle Physics, Radboud
University Nijmegen/Nikhef, Nijmegen; Netherlands.

120Nikhef National Institute for Subatomic Physics and University of
Amsterdam, Amsterdam; Netherlands.

121Department of Physics, Northern Illinois University, DeKalb IL; United
States of America.

122(a)Budker Institute of Nuclear Physics and NSU, SB RAS,
Novosibirsk;(b)Novosibirsk State University Novosibirsk; Russia.

123Institute for High Energy Physics of the National Research Centre Kurchatov
Institute, Protvino; Russia.

124Department of Physics, New York University, New York NY; United States of
America.

125Ochanomizu University, Otsuka, Bunkyo-ku, Tokyo; Japan.

126Ohio State University, Columbus OH; United States of America.

127Faculty of Science, Okayama University, Okayama; Japan.

128Homer L. Dodge Department of Physics and Astronomy, University of Oklahoma,
Norman OK; United States of America.

129Department of Physics, Oklahoma State University, Stillwater OK; United
States of America.

130Palacký University, RCPTM, Joint Laboratory of Optics, Olomouc; Czech
Republic.

131Center for High Energy Physics, University of Oregon, Eugene OR; United
States of America.

132LAL, Université Paris-Sud, CNRS/IN2P3, Université Paris-Saclay, Orsay;
France.

133Graduate School of Science, Osaka University, Osaka; Japan.

134Department of Physics, University of Oslo, Oslo; Norway.

135Department of Physics, Oxford University, Oxford; United Kingdom.

136LPNHE, Sorbonne Université, Université de Paris, CNRS/IN2P3, Paris; France.

137Department of Physics, University of Pennsylvania, Philadelphia PA; United
States of America.

138Konstantinov Nuclear Physics Institute of National Research Centre
"Kurchatov Institute", PNPI, St. Petersburg; Russia.

139Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh
PA; United States of America.

140(a)Laboratório de Instrumentação e Física Experimental de Partículas - LIP,
Lisbon;(b)Departamento de Física, Faculdade de Ciências, Universidade de
Lisboa, Lisbon;(c)Departamento de Física, Universidade de Coimbra,
Coimbra;(d)Centro de Física Nuclear da Universidade de Lisboa,
Lisbon;(e)Departamento de Física, Universidade do Minho, Braga;(f)Universidad
de Granada, Granada (Spain);(g)Dep Física and CEFITEC of Faculdade de Ciências
e Tecnologia, Universidade Nova de Lisboa, Caparica;(h)Av. Rovisco Pais, 1
1049-001 Lisbon, Portugal; Portugal.

141Institute of Physics of the Czech Academy of Sciences, Prague; Czech
Republic.

142Czech Technical University in Prague, Prague; Czech Republic.

143Charles University, Faculty of Mathematics and Physics, Prague; Czech
Republic.

144Particle Physics Department, Rutherford Appleton Laboratory, Didcot; United
Kingdom.

145IRFU, CEA, Université Paris-Saclay, Gif-sur-Yvette; France.

146Santa Cruz Institute for Particle Physics, University of California Santa
Cruz, Santa Cruz CA; United States of America.

147(a)Departamento de Física, Pontificia Universidad Católica de Chile,
Santiago;(b)Departamento de Física, Universidad Técnica Federico Santa María,
Valparaíso; Chile.

148Department of Physics, University of Washington, Seattle WA; United States
of America.

149Department of Physics and Astronomy, University of Sheffield, Sheffield;
United Kingdom.

150Department of Physics, Shinshu University, Nagano; Japan.

151Department Physik, Universität Siegen, Siegen; Germany.

152Department of Physics, Simon Fraser University, Burnaby BC; Canada.

153SLAC National Accelerator Laboratory, Stanford CA; United States of
America.

154Physics Department, Royal Institute of Technology, Stockholm; Sweden.

155Departments of Physics and Astronomy, Stony Brook University, Stony Brook
NY; United States of America.

156Department of Physics and Astronomy, University of Sussex, Brighton; United
Kingdom.

157School of Physics, University of Sydney, Sydney; Australia.

158Institute of Physics, Academia Sinica, Taipei; Taiwan.

159(a)E. Andronikashvili Institute of Physics, Iv. Javakhishvili Tbilisi State
University, Tbilisi;(b)High Energy Physics Institute, Tbilisi State
University, Tbilisi; Georgia.

160Department of Physics, Technion, Israel Institute of Technology, Haifa;
Israel.

161Raymond and Beverly Sackler School of Physics and Astronomy, Tel Aviv
University, Tel Aviv; Israel.

162Department of Physics, Aristotle University of Thessaloniki, Thessaloniki;
Greece.

163International Center for Elementary Particle Physics and Department of
Physics, University of Tokyo, Tokyo; Japan.

164Graduate School of Science and Technology, Tokyo Metropolitan University,
Tokyo; Japan.

165Department of Physics, Tokyo Institute of Technology, Tokyo; Japan.

166Tomsk State University, Tomsk; Russia.

167Department of Physics, University of Toronto, Toronto ON; Canada.

168(a)TRIUMF, Vancouver BC;(b)Department of Physics and Astronomy, York
University, Toronto ON; Canada.

169Division of Physics and Tomonaga Center for the History of the Universe,
Faculty of Pure and Applied Sciences, University of Tsukuba, Tsukuba; Japan.

170Department of Physics and Astronomy, Tufts University, Medford MA; United
States of America.

171Department of Physics and Astronomy, University of California Irvine,
Irvine CA; United States of America.

172Department of Physics and Astronomy, University of Uppsala, Uppsala;
Sweden.

173Department of Physics, University of Illinois, Urbana IL; United States of
America.

174Instituto de Física Corpuscular (IFIC), Centro Mixto Universidad de
Valencia - CSIC, Valencia; Spain.

175Department of Physics, University of British Columbia, Vancouver BC;
Canada.

176Department of Physics and Astronomy, University of Victoria, Victoria BC;
Canada.

177Fakultät für Physik und Astronomie, Julius-Maximilians-Universität
Würzburg, Würzburg; Germany.

178Department of Physics, University of Warwick, Coventry; United Kingdom.

179Waseda University, Tokyo; Japan.

180Department of Particle Physics, Weizmann Institute of Science, Rehovot;
Israel.

181Department of Physics, University of Wisconsin, Madison WI; United States
of America.

182Fakultät für Mathematik und Naturwissenschaften, Fachgruppe Physik,
Bergische Universität Wuppertal, Wuppertal; Germany.

183Department of Physics, Yale University, New Haven CT; United States of
America.

184Yerevan Physics Institute, Yerevan; Armenia.

a Also at Borough of Manhattan Community College, City University of New York,
New York NY; United States of America.

b Also at CERN, Geneva; Switzerland.

c Also at CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille; France.

d Also at Département de Physique Nucléaire et Corpusculaire, Université de
Genève, Genève; Switzerland.

e Also at Departament de Fisica de la Universitat Autonoma de Barcelona,
Barcelona; Spain.

f Also at Departamento de Física, Instituto Superior Técnico, Universidade de
Lisboa, Lisboa; Portugal.

g Also at Department of Applied Physics and Astronomy, University of Sharjah,
Sharjah; United Arab Emirates.

h Also at Department of Financial and Management Engineering, University of
the Aegean, Chios; Greece.

i Also at Department of Physics and Astronomy, Michigan State University, East
Lansing MI; United States of America.

j Also at Department of Physics and Astronomy, University of Louisville,
Louisville, KY; United States of America.

k Also at Department of Physics, Ben Gurion University of the Negev, Beer
Sheva; Israel.

l Also at Department of Physics, California State University, East Bay; United
States of America.

m Also at Department of Physics, California State University, Fresno; United
States of America.

n Also at Department of Physics, California State University, Sacramento;
United States of America.

o Also at Department of Physics, King’s College London, London; United
Kingdom.

p Also at Department of Physics, St. Petersburg State Polytechnical
University, St. Petersburg; Russia.

q Also at Department of Physics, Stanford University, Stanford CA; United
States of America.

r Also at Department of Physics, University of Adelaide, Adelaide; Australia.

s Also at Department of Physics, University of Fribourg, Fribourg;
Switzerland.

t Also at Department of Physics, University of Michigan, Ann Arbor MI; United
States of America.

u Also at Dipartimento di Matematica, Informatica e Fisica, Università di
Udine, Udine; Italy.

v Also at Faculty of Physics, M.V. Lomonosov Moscow State University, Moscow;
Russia.

w Also at Giresun University, Faculty of Engineering, Giresun; Turkey.

x Also at Graduate School of Science, Osaka University, Osaka; Japan.

y Also at Hellenic Open University, Patras; Greece.

z Also at Institucio Catalana de Recerca i Estudis Avancats, ICREA, Barcelona;
Spain.

aa Also at Institut für Experimentalphysik, Universität Hamburg, Hamburg;
Germany.

ab Also at Institute for Mathematics, Astrophysics and Particle Physics,
Radboud University Nijmegen/Nikhef, Nijmegen; Netherlands.

ac Also at Institute for Nuclear Research and Nuclear Energy (INRNE) of the
Bulgarian Academy of Sciences, Sofia; Bulgaria.

ad Also at Institute for Particle and Nuclear Physics, Wigner Research Centre
for Physics, Budapest; Hungary.

ae Also at Institute of Particle Physics (IPP), Vancouver; Canada.

af Also at Institute of Physics, Academia Sinica, Taipei; Taiwan.

ag Also at Institute of Physics, Azerbaijan Academy of Sciences, Baku;
Azerbaijan.

ah Also at Institute of Theoretical Physics, Ilia State University, Tbilisi;
Georgia.

ai Also at Instituto de Fisica Teorica, IFT-UAM/CSIC, Madrid; Spain.

aj Also at Istanbul University, Dept. of Physics, Istanbul; Turkey.

ak Also at Joint Institute for Nuclear Research, Dubna; Russia.

al Also at LAL, Université Paris-Sud, CNRS/IN2P3, Université Paris-Saclay,
Orsay; France.

am Also at Louisiana Tech University, Ruston LA; United States of America.

an Also at LPNHE, Sorbonne Université, Université de Paris, CNRS/IN2P3, Paris;
France.

ao Also at Manhattan College, New York NY; United States of America.

ap Also at Moscow Institute of Physics and Technology State University,
Dolgoprudny; Russia.

aq Also at National Research Nuclear University MEPhI, Moscow; Russia.

ar Also at Physics Department, An-Najah National University, Nablus;
Palestine.

as Also at Physics Dept, University of South Africa, Pretoria; South Africa.

at Also at Physikalisches Institut, Albert-Ludwigs-Universität Freiburg,
Freiburg; Germany.

au Also at School of Physics, Sun Yat-sen University, Guangzhou; China.

av Also at The City College of New York, New York NY; United States of
America.

aw Also at The Collaborative Innovation Center of Quantum Matter (CICQM),
Beijing; China.

ax Also at Tomsk State University, Tomsk, and Moscow Institute of Physics and
Technology State University, Dolgoprudny; Russia.

ay Also at TRIUMF, Vancouver BC; Canada.

az Also at Universita di Napoli Parthenope, Napoli; Italy.

∗ Deceased


disposition

# An algorithm to approximate the real trilogarithm for a real argument

Alexander Voigt  Flensburg University of Applied Sciences,
Kanzleistraße 91–93, 24943 Flensburg, Germany

## Abstract

We present an algorithm to approximate the real trilogarithm for a real
argument with IEEE 754/1985 double precision accuracy. The approximation is
structured such that it can make use of instruction/level parallelism when
executed on appropriate CPUs.

##  1 Introduction

The trilogarithm $\operatorname{Li}_{3}$ is a special function that appears in
the calculation of loop integrals in high/energy physics, for example in
two/loop self/energy integrals in the calculation of the Higgs boson pole mass
or in two/loop three/point integrals in the calculation of Higgs boson decays.
For this reason the trilogarithm is implemented in Feynman integral libraries
such as TSIL [1] and 3vil [2] or in the CHAPLIN [3], HPOLY.f [4] and handyG
[5] libraries as one basis function to numerically evaluate harmonic or
generalized polylogarithms, respectively, in terms of which certain classes of
Feynman integrals can be expressed. Since such loop integrals must be
numerically evaluated many times in parameter studies of models beyond the
Standard Model of particle physics, a time/efficient implementation of the
trilogarithm is advantageous.

An often used strategy to implement a special function is to map its argument
to one or more small regions on which a suitable approximant for the function
can be given. This strategy is, for example, usually used to implement the
real dilogarithm of a real argument: Using the known functional relations for
the real dilogarithm, it is possible to map its argument to the interval
$[0,1/2]$, where an expansion in terms of Chebyshev polynomials [6] or a
rational minimax approximant [7, 8] can be used for the numerical evaluation.
For a particularly time/efficient evaluation of a special function on some
interval one should try to minimize the number of costly arithmetic
floating/point operations, such as division or multiplication. In addition one
can try to make use of so/called instruction/level parallelism (ILP), which is
the ability of modern CPUs to execute multiple independent operations at the
same time. The combination of all these strategies was for example used in
Ref. [8] to obtain a time/efficient implementation of the real dilogarithm for
a real argument with IEEE 754/1985 double precision accuracy. In this
publication we will use the strategy from Ref. [8] to construct a
time/efficient algorithm for the numerical evaluation of the real trilogarithm
for a real argument with IEEE 754/1985 double precision accuracy.

##  2 The trilogarithm

For all $z\in\mathbb{C}$ the trilogarithm
$\operatorname{Li}_{3}:\mathbb{C}\to\mathbb{C}$ is defined as (see e.g. [9])

$\operatorname{Li}_{3}(z)=\int_{0}^{z}\frac{\operatorname{Li}_{2}(t)}{t}\textnormal{d}t,$
(1)

where $\operatorname{Li}_{2}:\mathbb{C}\to\mathbb{C}$ is the dilogarithm,
defined as

$\operatorname{Li}_{2}(z)=-\int_{0}^{z}\frac{\ln(1-t)}{t}\textnormal{d}t.$ (2)

For $|z|<1$ the trilogarithm has the series expansion

$\operatorname{Li}_{3}(z)=\sum_{k=1}^{\infty}\frac{z^{k}}{k^{3}}.$ (3)

For $z\neq 0$ the following relations hold:

$\displaystyle\operatorname{Li}_{3}(z)$
$\displaystyle=\operatorname{Li}_{3}(1/z)-\ln(-z)\zeta(2)-\frac{1}{6}\ln^{3}(-z),$
(4) $\displaystyle\operatorname{Li}_{3}(z)$
$\displaystyle=-\operatorname{Li}_{3}(1-1/z)-\operatorname{Li}_{3}(1-z)+\zeta(3)$
$\displaystyle+\ln(z)\zeta(2)-\frac{1}{2}\ln^{2}(z)\ln(1-z)+\frac{1}{6}\ln^{3}(z),$
(5)

where $\zeta$ is the Riemann zeta function with $\zeta(2)=\pi^{2}/6$. For the
following considerations we define the real trilogarithm for a real argument,
$\operatorname{li}_{3}:\mathbb{R}\to\mathbb{R}$, as

$\operatorname{li}_{3}(x)=\Re[\operatorname{Li}_{3}(x)].$ (6)

##  3 Algorithm to approximate the real trilogarithm

To obtain a time/efficient approximation of the real trilogarithm we proceed
similarly to Ref. [5]: We use Eqs. (4)–(5) to transform the argument of the
trilogarithm to the intervals $[-1,0]$ and/or $[0,1/2]$. On each of these
intervals Ref. [5] approximates $\operatorname{li}_{3}$ in terms of Chebyshev
polynomials, which are evaluated using Clenshaw’s algorithm [10]. Clenshaw’s
algorithm, however, is purely sequential and thus cannot make use of ILP. For
this reason we use a different technique to approximate
$\operatorname{li}_{3}$ on these intervals to allow for ILP: We use a rational
minimax approximant of the form

$\operatorname{li}_{3}(x)=x\sum_{k=0}^{\infty}\frac{x^{k}}{(k+1)^{3}}\approx
x\frac{\sum_{k=0}^{5}p_{k}x^{k}}{\sum_{k=0}^{6}q_{k}x^{k}},$ (7)

where we evaluate the numerator and denominator polynomials using Estrin’s
scheme [11]. The coefficients $p_{k}$ and $q_{k}$ in Eq. (7) are calculated
using the MiniMaxApproximation function from Wolfram/Mathematica [12] and are
listed in Table 1 and 2. For arguments $x\in[-1,0]$ the error of the
approximant in Eq. (7) is less than $2.050\cdot 10^{-17}$, while for
$x\in[0,1/2]$ it is less than $1.066\cdot 10^{-17}$. These maximum errors are
small enough to achieve IEEE 754/1985 double precision accuracy in the numeric
evaluation of $\operatorname{li}_{3}$.

In detail, our algorithm to numerically evaluate $\operatorname{li}_{3}$ is as
follows: We split the domain of $\operatorname{li}_{3}$ into the sub/domains
$\mathbb{R}=(-\infty,-1)\cup\\{-1\\}\cup(-1,0)\cup\\{0\\}\cup(0,1/2)\cup\\{1/2\\}\cup(1/2,1)\cup\\{1\\}\cup(1,2)\cup\\{2\\}\cup(2,\infty)$.
For arguments $x\in(-\infty,-1)$ we use Eq. (4) to transform the argument to
the interval $(-1,0)$, where we use the rational minimax approximant from Eq.
(7) with the coefficients listed in Table 1. For $x=-1$ we implement the known
value $\operatorname{li}_{3}(-1)=-3\zeta(3)/4$. For $x\in(-1,0)$ we directly
use the appropriate approximant from Eq. (7). For $x=0$ we use the known value
$\operatorname{li}_{3}(0)=0$. For $x\in(0,1/2)$ we directly use the
approximant in Eq. (7) with the coefficients listed in Table 2. For $x=1/2$ we
use the known value
$\operatorname{li}_{3}(1/2)=[21\zeta(3)+4\ln^{3}(2)-2\pi^{2}\ln(2)]/24$. For
$x\in(1/2,1)$ we use Eq. (5) to transform the argument to the intervals
$(-1,0)$ and $(0,1/2)$, where we use the appropriate approximant from Eq. (7)
on each interval. For $x=1$ we use the known value
$\operatorname{li}_{3}(1)=\zeta(3)$. For $x\in(1,2)$ we use Eq. (5) to
transform the argument to the intervals $(-1,0)$ and $(0,1/2)$, where we use
the appropriate approximant from Eq. (7) on each interval. For $x\geq 2$ we
use Eq. (4) to transform the argument to the interval $(0,1/2]$, where we use
the appropriate approximant from Eq. (7).

Table 1: Coefficients of the numerator and denominator polynomials for the minimax approximant in Eq. (7) for $x\in[-1,0]$. $p_{0}$ | $0.9999999999999999795\cdot 10^{+0}$
---|---
$p_{1}$ | $-2.0281801754117129576\cdot 10^{+0}$
$p_{2}$ | $1.4364029887561718540\cdot 10^{+0}$
$p_{3}$ | $-4.2240680435713030268\cdot 10^{-1}$
$p_{4}$ | $4.7296746450884096877\cdot 10^{-2}$
$p_{5}$ | $-1.3453536579918419568\cdot 10^{-3}$
$q_{0}$ | $1.0000000000000000000\cdot 10^{+0}$
$q_{1}$ | $-2.1531801754117049035\cdot 10^{+0}$
$q_{2}$ | $1.6685134736461140517\cdot 10^{+0}$
$q_{3}$ | $-5.6684857464584544310\cdot 10^{-1}$
$q_{4}$ | $8.1999463370623961084\cdot 10^{-2}$
$q_{5}$ | $-4.0756048502924149389\cdot 10^{-3}$
$q_{6}$ | $3.4316398489103212699\cdot 10^{-5}$
Table 2: Coefficients of the numerator and denominator polynomials for the minimax approximant in Eq. (7) for $x\in[0,1/2]$. $p_{0}$ | $0.9999999999999999893\cdot 10^{+0}$
---|---
$p_{1}$ | $-2.5224717303769789628\cdot 10^{+0}$
$p_{2}$ | $2.3204919140887894133\cdot 10^{+0}$
$p_{3}$ | $-9.3980973288965037869\cdot 10^{-1}$
$p_{4}$ | $1.5728950200990509052\cdot 10^{-1}$
$p_{5}$ | $-7.5485193983677071129\cdot 10^{-3}$
$q_{0}$ | $1.0000000000000000000\cdot 10^{+0}$
$q_{1}$ | $-2.6474717303769836244\cdot 10^{+0}$
$q_{2}$ | $2.6143888433492184741\cdot 10^{+0}$
$q_{3}$ | $-1.1841788297857667038\cdot 10^{+0}$
$q_{4}$ | $2.4184938524793651120\cdot 10^{-1}$
$q_{5}$ | $-1.8220900115898156346\cdot 10^{-2}$
$q_{6}$ | $2.4927971540017376759\cdot 10^{-4}$

An implementation of the described algorithm in C is given in Appendix A. This
C implementation is also provided in an ancillary file in the arXiv submission
of this publication under the CC-BY-4.0 license.

##  4 Benchmark

In the following we investigate the run/time of the C implementation given in
Appendix A. Table 3 shows the average run/time of a single call of
$\operatorname{li}_{3}$ in nano seconds for arguments on different intervals
and on different 64-bit CPU architectures (compiled with gcc 10.2.1 with -O2
optimization level). The run/times shown in the table have been obtained by
measuring the average run/time of $\operatorname{li}_{3}$ on $10^{6}$ random
values on each interval using the Google benchmark library version 1.5.2 [13].
For comparison we also show in Table 3 the average run/time for the real
natural logarithm $\ln$, the real cosine function $\cos$ and the real identity
function $\operatorname{id}(x)=x$.

Table 3: Average run/time in nano seconds for one invocation of $\operatorname{li}_{3}$, $\ln$, $\cos$ and $\operatorname{id}$ for arguments on different intervals on different CPU architectures (compiled with gcc 10.2.1 with -O2 optimization level). Function | Interval | i5-8265U | i7-5600U
---|---|---|---
$\operatorname{li}_{3}$ | $[-2,-1]$ | $16.7$ | $20.3$
$\operatorname{li}_{3}$ | $[-1,0]$ | $4.31$ | $5.81$
$\operatorname{li}_{3}$ | $[0,1/2]$ | $5.09$ | $6.42$
$\operatorname{li}_{3}$ | $[1/2,1]$ | $24.3$ | $29.4$
$\operatorname{li}_{3}$ | $[1,2]$ | $24.2$ | $31.0$
$\operatorname{li}_{3}$ | $[2,3]$ | $13.2$ | $15.9$
$\ln$ | $[1,2]$ | $5.20$ | $6.34$
$\cos$ | $[0,\pi/2]$ | $11.0$ | $13.3$
$\operatorname{id}$ | $[-2,2]$ | $0.259$ | $0.315$

We find that the run/time of $\operatorname{li}_{3}$ is similar to the
run/time of $\ln$ for arguments on the intervals $[-1,0]$ and $[0,1/2]$, where
no transformation is performed and the rational minimax approximants from Eq.
(7) are used directly. For arguments $x\in[-2,-1]$ or $x\in[2,3]$ the
transformation onto $[-1,0]$ and $[0,1/2]$, respectively, involves extra
arithmetic floating/point operations and one additional call of $\ln$, which
leads to an increased run/time of $\operatorname{li}_{3}$ by approximately a
factor $3$ in total. For arguments $x\in[1/2,1]$ or $x\in[1,2]$ a
transformation onto both $[-1,0]$ and $[0,1/2]$ intervals is performed, where
the corresponding approximants from Eq. (7) are used, respectively. This
transformation, which requires extra arithmetic floating/point operations and
two additional calls of $\ln$, and the necessity to use two approximants leads
to an increased run/time of $\operatorname{li}_{3}$ by approximately a factor
$5$ on these intervals.111A potential performance optimization for arguments
$x\in[1/2,1]$ or $x\in[1,2]$ could be to not perform a transformation onto
other intervals, but instead directly use dedicated rational minimax
approximants on these intervals, at the cost of an increased number of
coefficients to be stored in the source code.

##  5 Summary

We have presented an algorithm to approximate the real trilogarithm of a real
argument with IEEE 754/1985 double precision. The approximation is structured
to allow for the use of instruction/level parallelism on appropriate CPUs. A C
implementation of the real trilogarithm, using the algorithm presented in this
publication, can be found in the appendix as well as in an ancillary file in
the arXiv submission of this publication under the CC-BY-4.0 license.
Implementations in C, C++, Fortran, Julia and Rust can be found in Refs. [14,
15, 16].

##  Appendix A Implementation of the real trilogarithm

⬇

@ifdisplaystyle

#include <math.h>

// Re[Li_3(x)] for x in [-1, 0]

static double li3_neg(double x)

{

const double P[] = {

0.9999999999999999795e+0,

-2.0281801754117129576e+0,

1.4364029887561718540e+0,

-4.2240680435713030268e-1,

4.7296746450884096877e-2,

-1.3453536579918419568e-3

};

const double Q[] = {

1.0000000000000000000e+0,

-2.1531801754117049035e+0,

1.6685134736461140517e+0,

-5.6684857464584544310e-1,

8.1999463370623961084e-2,

-4.0756048502924149389e-3,

3.4316398489103212699e-5

};

const double x2 = x*x;

const double x4 = x2*x2;

const double p = P[0] + x*P[1]

+ x2*(P[2] + x*P[3])

+ x4*(P[4] + x*P[5]);

const double q = Q[0] + x*Q[1]

+ x2*(Q[2] + x*Q[3])

+ x4*(Q[4] + x*Q[5] + x2*Q[6]);

return x*p/q;

}

// Re[Li_3(x)] for x in [0, 1/2]

static double li3_pos(double x)

{

const double P[] = {

0.9999999999999999893e+0,

-2.5224717303769789628e+0,

2.3204919140887894133e+0,

-9.3980973288965037869e-1,

1.5728950200990509052e-1,

-7.5485193983677071129e-3

};

const double Q[] = {

1.0000000000000000000e+0,

-2.6474717303769836244e+0,

2.6143888433492184741e+0,

-1.1841788297857667038e+0,

2.4184938524793651120e-1,

-1.8220900115898156346e-2,

2.4927971540017376759e-4

};

const double x2 = x*x;

const double x4 = x2*x2;

const double p = P[0] + x*P[1]

+ x2*(P[2] + x*P[3])

+ x4*(P[4] + x*P[5]);

const double q = Q[0] + x*Q[1]

+ x2*(Q[2] + x*Q[3])

+ x4*(Q[4] + x*Q[5] + x2*Q[6]);

return x*p/q;

}

// Re[Li_3(x)] for x in (-inf, +inf)

double li3(double x)

{

const double zeta2 = 1.6449340668482264;

const double zeta3 = 1.2020569031595943;

if (x < -1) {

const double l = log(-x);

return li3_neg(1/x)

- l*(zeta2 + 1.0/6*l*l);

} else if (x == -1) {

return -0.75*zeta3;

} else if (x < 0) {

return li3_neg(x);

} else if (x == 0) {

return 0;

} else if (x < 0.5) {

return li3_pos(x);

} else if (x == 0.5) {

return 0.53721319360804020;

} else if (x < 1) {

const double l = log(x);

return -li3_neg(1 - 1/x)

- li3_pos(1 - x) + zeta3

+ l*(zeta2

+ l*(-0.5*log(1 - x)

+ 1.0/6*l));

} else if (x == 1) {

return zeta3;

} else if (x < 2) {

const double l = log(x);

return -li3_neg(1 - x)

- li3_pos(1 - 1/x) + zeta3

+ l*(zeta2

+ l*(-0.5*log(x - 1)

+ 1.0/6*l));

} else {

const double l = log(x);

return li3_pos(1/x)

+ l*(2*zeta2 - 1.0/6*l*l);

}

}

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