Multijet production in neutral current deep inelastic scattering at HERA and determination of $\alpha_s$
Multijet production rates in neutral current deep inelastic scattering have been measured in the range of exchanged boson virtualities $10 < Q^2 < 5000\,\hbox{GeV}^2$. The data were taken at the $ep$ collider HERA with centre-of-mass energy $\sqrt{s} = 318\,\hbox{GeV}$ using the ZEUS detector and correspond to an integrated luminosity of $82.2\,\hbox{pb}^{-1}$. Jets were identified in the Breit frame using the $k_T$ cluster algorithm in the longitudinally invariant inclusive mode. Measurements of differential dijet and trijet cross sections are presented as functions of jet transverse energy ($E_{T,B}^{jet}$), pseudorapidity ($\eta_{LAB}^{jet}$) and $Q^2$ with $E_{T,B}^{jet} > 5\,\hbox{GeV}$ and $-1 < \eta_{LAB}^{jet} < 2.5$. Next-to-leading-order QCD calculations describe the data well. The value of the strong coupling constant $\alpha_s(M_Z)$, determined from the ratio of the trijet to dijet cross sections, is $\alpha_s(M_Z) = 0.1179 \pm 0.0013~(stat.)^{+0.0028}_{-0.0046}~(exp.)^{+0.0064}_{-0.0046}~(th.)$.
DESY-05-019-aut.tex
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\begin{center}                                                                                     
{                      \Large  The ZEUS Collaboration              }                               
\end{center}                                                                                       
                                                   %                                               
  S.~Chekanov,                                                                                     
  M.~Derrick,                                                                                      
  S.~Magill,                                                                                       
  S.~Miglioranzi$^{   1}$,                                                                         
  B.~Musgrave,                                                                                     
  \mbox{J.~Repond},                                                                                
  R.~Yoshida\\                                                                                     
 {\it Argonne National Laboratory, Argonne, Illinois 60439-4815}, USA~$^{n}$                       
\par \filbreak                                                                                     
                                                  %                                                
  M.C.K.~Mattingly \\                                                                              
 {\it Andrews University, Berrien Springs, Michigan 49104-0380}, USA                               
\par \filbreak                                                                                     
                          %                                                                        
  N.~Pavel, A.G.~Yag\"ues Molina \\                                                                
  {\it Institut f\"ur Physik der Humboldt-Universit\"at zu Berlin,                                 
           Berlin, Germany}                                                                        
\par \filbreak                                                                                     
                                                  %                                                
  P.~Antonioli,                                                                                    
  G.~Bari,                                                                                         
  M.~Basile,                                                                                       
  L.~Bellagamba,                                                                                   
  D.~Boscherini,                                                                                   
  A.~Bruni,                                                                                        
  G.~Bruni,                                                                                        
  G.~Cara~Romeo,                                                                                   
\mbox{L.~Cifarelli},                                                                               
  F.~Cindolo,                                                                                      
  A.~Contin,                                                                                       
  M.~Corradi,                                                                                      
  S.~De~Pasquale,                                                                                  
  P.~Giusti,                                                                                       
  G.~Iacobucci,                                                                                    
\mbox{A.~Margotti},                                                                                
  A.~Montanari,                                                                                    
  R.~Nania,                                                                                        
  F.~Palmonari,                                                                                    
  A.~Pesci,                                                                                        
  A.~Polini,                                                                                       
  L.~Rinaldi,                                                                                      
  G.~Sartorelli,                                                                                   
  A.~Zichichi  \\                                                                                  
  {\it University and INFN Bologna, Bologna, Italy}~$^{e}$                                         
\par \filbreak                                                                                     
                                                    %                                              
  G.~Aghuzumtsyan,                                                                                 
  D.~Bartsch,                                                                                      
  I.~Brock,                                                                                        
  S.~Goers,                                                                                        
  H.~Hartmann,                                                                                     
  E.~Hilger,                                                                                       
  P.~Irrgang,                                                                                      
  H.-P.~Jakob,                                                                                     
  O.~Kind,                                                                                         
  U.~Meyer,                                                                                        
  E.~Paul$^{   2}$,                                                                                
  J.~Rautenberg,                                                                                   
  R.~Renner,                                                                                       
  K.C.~Voss$^{   3}$,                                                                              
  M.~Wang,                                                                                         
  M.~Wlasenko\\                                                                                    
  {\it Physikalisches Institut der Universit\"at Bonn,                                             
           Bonn, Germany}~$^{b}$                                                                   
\par \filbreak                                                                                     
                                                  %                                                
  D.S.~Bailey$^{   4}$,                                                                            
  N.H.~Brook,                                                                                      
  J.E.~Cole,                                                                                       
  G.P.~Heath,                                                                                      
  T.~Namsoo,                                                                                       
  S.~Robins\\                                                                                      
   {\it H.H.~Wills Physics Laboratory, University of Bristol,                                      
           Bristol, United Kingdom}~$^{m}$                                                         
\par \filbreak                                                                                     
  %                                                                                                
  M.~Capua,                                                                                        
  A. Mastroberardino,                                                                              
  M.~Schioppa,                                                                                     
  G.~Susinno,                                                                                      
  E.~Tassi  \\                                                                                     
  {\it Calabria University,                                                                        
           Physics Department and INFN, Cosenza, Italy}~$^{e}$                                     
\par \filbreak                                                                                     
                                      %                                                            
  J.Y.~Kim,                                                                                        
  K.J.~Ma$^{   5}$\\                                                                               
  {\it Chonnam National University, Kwangju, South Korea}~$^{g}$                                   
 \par \filbreak                                                                                    
                                                   %                                               
  M.~Helbich,                                                                                      
  Y.~Ning,                                                                                         
  Z.~Ren,                                                                                          
  W.B.~Schmidke,                                                                                   
  F.~Sciulli\\                                                                                     
  {\it Nevis Laboratories, Columbia University, Irvington on Hudson,                               
New York 10027}~$^{o}$                                                                             
\par \filbreak                                                                                     
                                                    %                                              
  J.~Chwastowski,                                                                                  
  A.~Eskreys,                                                                                      
  J.~Figiel,                                                                                       
  A.~Galas,                                                                                        
  K.~Olkiewicz,                                                                                    
  P.~Stopa,                                                                                        
  D.~Szuba,                                                                                        
  L.~Zawiejski  \\                                                                                 
  {\it Institute of Nuclear Physics, Cracow, Poland}~$^{i}$                                        
\par \filbreak                                                                                     
                                                           %                                       
  L.~Adamczyk,                                                                                     
  T.~Bo\l d,                                                                                       
  I.~Grabowska-Bo\l d,                                                                             
  D.~Kisielewska,                                                                                  
  A.M.~Kowal,                                                                                      
  J. \L ukasik,                                                                                    
  \mbox{M.~Przybycie\'{n}},                                                                        
  L.~Suszycki,                                                                                     
  J.~Szuba$^{   6}$\\                                                                              
{\it Faculty of Physics and Applied Computer Science,                                              
           AGH-University of Science and Technology, Cracow, Poland}~$^{p}$                        
\par \filbreak                                                                                     
                                                           %                                       
  A.~Kota\'{n}ski$^{   7}$,                                                                        
  W.~S{\l}omi\'nski\\                                                                              
  {\it Department of Physics, Jagellonian University, Cracow, Poland}                              
\par \filbreak                                                                                     
                                                           %                                       
  V.~Adler,                                                                                        
  U.~Behrens,                                                                                      
  I.~Bloch,                                                                                        
  K.~Borras,                                                                                       
  G.~Drews,                                                                                        
  J.~Fourletova,                                                                                   
  A.~Geiser,                                                                                       
  D.~Gladkov,                                                                                      
  P.~G\"ottlicher$^{   8}$,                                                                        
  O.~Gutsche,                                                                                      
  T.~Haas,                                                                                         
  W.~Hain,                                                                                         
  C.~Horn,                                                                                         
  B.~Kahle,                                                                                        
  U.~K\"otz,                                                                                       
  H.~Kowalski,                                                                                     
  G.~Kramberger,                                                                                   
  D.~Lelas$^{   9}$,                                                                               
  H.~Lim,                                                                                          
  B.~L\"ohr,                                                                                       
  R.~Mankel,                                                                                       
  I.-A.~Melzer-Pellmann,                                                                           
  C.N.~Nguyen,                                                                                     
  D.~Notz,                                                                                         
  A.E.~Nuncio-Quiroz,                                                                              
  A.~Raval,                                                                                        
  R.~Santamarta,                                                                                   
  \mbox{U.~Schneekloth},                                                                           
  U.~St\"osslein,                                                                                  
  G.~Wolf,                                                                                         
  C.~Youngman,                                                                                     
  \mbox{W.~Zeuner} \\                                                                              
  {\it Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany}                                    
\par \filbreak                                                                                     
                                                           %                                       
  \mbox{S.~Schlenstedt}\\                                                                          
   {\it Deutsches Elektronen-Synchrotron DESY, Zeuthen, Germany}                                   
\par \filbreak                                                                                     
                                                           %                                       
  G.~Barbagli,                                                                                     
  E.~Gallo,                                                                                        
  C.~Genta,                                                                                        
  P.~G.~Pelfer  \\                                                                                 
  {\it University and INFN, Florence, Italy}~$^{e}$                                                
\par \filbreak                                                                                     
                                                           %                                       
                                                           %                                       
  A.~Bamberger,                                                                                    
  A.~Benen,                                                                                        
  F.~Karstens,                                                                                     
  D.~Dobur,                                                                                        
  N.N.~Vlasov$^{  10}$\\                                                                           
  {\it Fakult\"at f\"ur Physik der Universit\"at Freiburg i.Br.,                                   
           Freiburg i.Br., Germany}~$^{b}$                                                         
\par \filbreak                                                                                     
  P.J.~Bussey,                                                                                     
  A.T.~Doyle,                                                                                      
  J.~Ferrando,                                                                                     
  J.~Hamilton,                                                                                     
  S.~Hanlon,                                                                                       
  D.H.~Saxon,                                                                                      
  I.O.~Skillicorn\\                                                                                
  {\it Department of Physics and Astronomy, University of Glasgow,                                 
           Glasgow, United Kingdom}~$^{m}$                                                         
\par \filbreak                                                                                     
  I.~Gialas$^{  11}$\\                                                                             
  {\it Department of Engineering in Management and Finance, Univ. of                               
            Aegean, Greece}                                                                        
\par \filbreak                                                                                     
  T.~Carli,                                                                                        
  T.~Gosau,                                                                                        
  U.~Holm,                                                                                         
  N.~Krumnack$^{  12}$,                                                                            
  E.~Lohrmann,                                                                                     
  M.~Milite,                                                                                       
  H.~Salehi,                                                                                       
  P.~Schleper,                                                                                     
  \mbox{T.~Sch\"orner-Sadenius},                                                                   
  S.~Stonjek$^{  13}$,                                                                             
  K.~Wichmann,                                                                                     
  K.~Wick,                                                                                         
  A.~Ziegler,                                                                                      
  Ar.~Ziegler\\                                                                                    
  {\it Hamburg University, Institute of Exp. Physics, Hamburg,                                     
           Germany}~$^{b}$                                                                         
\par \filbreak                                                                                     
  C.~Collins-Tooth$^{  14}$,                                                                       
  C.~Foudas,                                                                                       
  C.~Fry,                                                                                          
  R.~Gon\c{c}alo$^{  15}$,                                                                         
  K.R.~Long,                                                                                       
  A.D.~Tapper\\                                                                                    
   {\it Imperial College London, High Energy Nuclear Physics Group,                                
           London, United Kingdom}~$^{m}$                                                          
\par \filbreak                                                                                     
  M.~Kataoka$^{  16}$,                                                                             
  K.~Nagano,                                                                                       
  K.~Tokushuku$^{  17}$,                                                                           
  S.~Yamada,                                                                                       
  Y.~Yamazaki\\                                                                                    
  {\it Institute of Particle and Nuclear Studies, KEK,                                             
       Tsukuba, Japan}~$^{f}$                                                                      
\par \filbreak                                                                                     
                                                    %                                              
  A.N. Barakbaev,                                                                                  
  E.G.~Boos,                                                                                       
  N.S.~Pokrovskiy,                                                                                 
  B.O.~Zhautykov \\                                                                                
  {\it Institute of Physics and Technology of Ministry of Education and                            
  Science of Kazakhstan, Almaty, \mbox{Kazakhstan}}                                                
  \par \filbreak                                                                                   
                                                      %                                            
  D.~Son \\                                                                                        
  {\it Kyungpook National University, Center for High Energy Physics, Daegu,                       
  South Korea}~$^{g}$                                                                              
  \par \filbreak                                                                                   
  J.~de~Favereau,                                                                                  
  K.~Piotrzkowski\\                                                                                
  {\it Institut de Physique Nucl\'{e}aire, Universit\'{e} Catholique de                            
  Louvain, Louvain-la-Neuve, Belgium}~$^{q}$                                                       
  \par \filbreak                                                                                   
                                                         %                                         
  F.~Barreiro,                                                                                     
  C.~Glasman$^{  18}$,                                                                             
  O.~Gonz\'alez,                                                                                   
  M.~Jimenez,                                                                                      
  L.~Labarga,                                                                                      
  J.~del~Peso,                                                                                     
  J.~Terr\'on,                                                                                     
  M.~Zambrana\\                                                                                    
  {\it Departamento de F\'{\i}sica Te\'orica, Universidad Aut\'onoma                               
  de Madrid, Madrid, Spain}~$^{l}$                                                                 
  \par \filbreak                                                                                   
  M.~Barbi,                                                    %                                   
  F.~Corriveau,                                                                                    
  C.~Liu,                                                                                          
  S.~Padhi,                                                                                        
  M.~Plamondon,                                                                                    
  D.G.~Stairs,                                                                                     
  R.~Walsh,                                                                                        
  C.~Zhou\\                                                                                        
  {\it Department of Physics, McGill University,                                                   
           Montr\'eal, Qu\'ebec, Canada H3A 2T8}~$^{a}$                                            
\par \filbreak                                                                                     
                                             %                                                     
  T.~Tsurugai \\                                                                                   
  {\it Meiji Gakuin University, Faculty of General Education,                                      
           Yokohama, Japan}~$^{f}$                                                                 
\par \filbreak                                                                                     
                                                           %                                       
  A.~Antonov,                                                                                      
  P.~Danilov,                                                                                      
  B.A.~Dolgoshein,                                                                                 
  V.~Sosnovtsev,                                                                                   
  A.~Stifutkin,                                                                                    
  S.~Suchkov \\                                                                                    
  {\it Moscow Engineering Physics Institute, Moscow, Russia}~$^{j}$                                
\par \filbreak                                                                                     
                                                           %                                       
  R.K.~Dementiev,                                                                                  
  P.F.~Ermolov,                                                                                    
  L.K.~Gladilin,                                                                                   
  I.I.~Katkov,                                                                                     
  L.A.~Khein,                                                                                      
  I.A.~Korzhavina,                                                                                 
  V.A.~Kuzmin,                                                                                     
  B.B.~Levchenko,                                                                                  
  O.Yu.~Lukina,                                                                                    
  A.S.~Proskuryakov,                                                                               
  L.M.~Shcheglova,                                                                                 
  D.S.~Zotkin,                                                                                     
  S.A.~Zotkin \\                                                                                   
  {\it Moscow State University, Institute of Nuclear Physics,                                      
           Moscow, Russia}~$^{k}$                                                                  
\par \filbreak                                                                                     
  I.~Abt,                                                                                          
  C.~B\"uttner,                                                                                    
  A.~Caldwell,                                                                                     
  X.~Liu,                                                                                          
  J.~Sutiak\\                                                                                      
{\it Max-Planck-Institut f\"ur Physik, M\"unchen, Germany}                                         
\par \filbreak                                                                                     
  N.~Coppola,                                                                                      
  G.~Grigorescu,                                                                                   
  S.~Grijpink,                                                                                     
  A.~Keramidas,                                                                                    
  E.~Koffeman,                                                                                     
  P.~Kooijman,                                                                                     
  E.~Maddox,                                                                                       
\mbox{A.~Pellegrino},                                                                              
  S.~Schagen,                                                                                      
  H.~Tiecke,                                                                                       
  M.~V\'azquez,                                                                                    
  L.~Wiggers,                                                                                      
  E.~de~Wolf \\                                                                                    
  {\it NIKHEF and University of Amsterdam, Amsterdam, Netherlands}~$^{h}$                          
\par \filbreak                                                                                     
  N.~Br\"ummer,                                                                                    
  B.~Bylsma,                                                                                       
  L.S.~Durkin,                                                                                     
  T.Y.~Ling\\                                                                                      
  {\it Physics Department, Ohio State University,                                                  
           Columbus, Ohio 43210}~$^{n}$                                                            
\par \filbreak                                                                                     
  P.D.~Allfrey,                                                                                    
  M.A.~Bell,                                                         %                             
  A.M.~Cooper-Sarkar,                                                                              
  A.~Cottrell,                                                                                     
  R.C.E.~Devenish,                                                                                 
  B.~Foster,                                                                                       
  G.~Grzelak,                                                                                      
  C.~Gwenlan$^{  19}$,                                                                             
  T.~Kohno,                                                                                        
  S.~Patel,                                                                                        
  P.B.~Straub,                                                                                     
  R.~Walczak \\                                                                                    
  {\it Department of Physics, University of Oxford,                                                
           Oxford United Kingdom}~$^{m}$                                                           
\par \filbreak                                                                                     
  P.~Bellan,                                                                                       
  A.~Bertolin,                                                         %                           
  R.~Brugnera,                                                                                     
  R.~Carlin,                                                                                       
  R.~Ciesielski,                                                                                   
  F.~Dal~Corso,                                                                                    
  S.~Dusini,                                                                                       
  A.~Garfagnini,                                                                                   
  S.~Limentani,                                                                                    
  A.~Longhin,                                                                                      
  L.~Stanco,                                                                                       
  M.~Turcato\\                                                                                     
  {\it Dipartimento di Fisica dell' Universit\`a and INFN,                                         
           Padova, Italy}~$^{e}$                                                                   
\par \filbreak                                                                                     
                                                           %                                       
  E.A.~Heaphy,                                                                                     
  F.~Metlica,                                                                                      
  B.Y.~Oh,                                                                                         
  J.J.~Whitmore$^{  20}$\\                                                                         
  {\it Department of Physics, Pennsylvania State University,                                       
           University Park, Pennsylvania 16802}~$^{o}$                                             
\par \filbreak                                                                                     
                                                           %                                       
  Y.~Iga \\                                                                                        
{\it Polytechnic University, Sagamihara, Japan}~$^{f}$                                             
\par \filbreak                                                                                     
                                                           %                                       
  G.~D'Agostini,                                                                                   
  G.~Marini,                                                                                       
  A.~Nigro \\                                                                                      
  {\it Dipartimento di Fisica, Universit\`a 'La Sapienza' and INFN,                                
           Rome, Italy}~$^{e}~$                                                                    
\par \filbreak                                                                                     
                                                           %                                       
  J.C.~Hart\\                                                                                      
  {\it Rutherford Appleton Laboratory, Chilton, Didcot, Oxon,                                      
           United Kingdom}~$^{m}$                                                                  
\par \filbreak                                                                                     
                          %                                                           %            
  H.~Abramowicz$^{  21}$,                                                                          
  A.~Gabareen,                                                                                     
  S.~Kananov,                                                                                      
  A.~Kreisel,                                                                                      
  A.~Levy\\                                                                                        
  {\it Raymond and Beverly Sackler Faculty of Exact Sciences,                                      
School of Physics, Tel-Aviv University, Tel-Aviv, Israel}~$^{d}$                                   
\par \filbreak                                                                                     
                                                           %                                       
  M.~Kuze \\                                                                                       
  {\it Department of Physics, Tokyo Institute of Technology,                                       
           Tokyo, Japan}~$^{f}$                                                                    
\par \filbreak                                                                                     
  S.~Kagawa,                                                                                       
  T.~Tawara\\                                                                                      
  {\it Department of Physics, University of Tokyo,                                                 
           Tokyo, Japan}~$^{f}$                                                                    
\par \filbreak                                                                                     
                                                           %                                       
  R.~Hamatsu,                                                                                      
  H.~Kaji,                                                                                         
  S.~Kitamura$^{  22}$,                                                                            
  K.~Matsuzawa,                                                                                    
  O.~Ota,                                                                                          
  Y.D.~Ri\\                                                                                        
  {\it Tokyo Metropolitan University, Department of Physics,                                       
           Tokyo, Japan}~$^{f}$                                                                    
\par \filbreak                                                                                     
                                                           %                                       
  M.~Costa,                                                                                        
  M.I.~Ferrero,                                                                                    
  V.~Monaco,                                                                                       
  R.~Sacchi,                                                                                       
  A.~Solano\\                                                                                      
  {\it Universit\`a di Torino and INFN, Torino, Italy}~$^{e}$                                      
\par \filbreak                                                                                     
  M.~Arneodo,                                                                                      
  M.~Ruspa\\                                                                                       
 {\it Universit\`a del Piemonte Orientale, Novara, and INFN, Torino,                               
Italy}~$^{e}$                                                                                      
                                                           %                                       
\par \filbreak                                                                                     
                                    %                                                              
  S.~Fourletov,                                                                                    
  T.~Koop,                                                                                         
  J.F.~Martin,                                                                                     
  A.~Mirea\\                                                                                       
   {\it Department of Physics, University of Toronto, Toronto, Ontario,                            
Canada M5S 1A7}~$^{a}$                                                                             
\par \filbreak                                                                                     
  J.M.~Butterworth$^{  23}$,                                                                       
  R.~Hall-Wilton,                                                                                  
  T.W.~Jones,                                                                                      
  J.H.~Loizides$^{  24}$,                                                                          
  M.R.~Sutton$^{   4}$,                                                                            
  C.~Targett-Adams,                                                                                
  M.~Wing  \\                                                                                      
  {\it Physics and Astronomy Department, University College London,                                
           London, United Kingdom}~$^{m}$                                                          
\par \filbreak                                                                                     
                                                           %                                       
  J.~Ciborowski$^{  25}$,                                                                          
  P.~Kulinski,                                                                                     
  P.~{\L}u\.zniak$^{  26}$,                                                                        
  J.~Malka$^{  26}$,                                                                               
  R.J.~Nowak,                                                                                      
  J.M.~Pawlak,                                                                                     
  J.~Sztuk$^{  27}$,                                                                               
  T.~Tymieniecka,                                                                                  
  A.~Tyszkiewicz$^{  26}$,                                                                         
  A.~Ukleja,                                                                                       
  J.~Ukleja$^{  28}$,                                                                              
  A.F.~\.Zarnecki \\                                                                               
   {\it Warsaw University, Institute of Experimental Physics,                                      
           Warsaw, Poland}                                                                         
\par \filbreak                                                                                     
                                                           %                                       
  M.~Adamus,                                                                                       
  P.~Plucinski\\                                                                                   
  {\it Institute for Nuclear Studies, Warsaw, Poland}                                              
\par \filbreak                                                                                     
                                                           %                                       
  Y.~Eisenberg,                                                                                    
  D.~Hochman,                                                                                      
  U.~Karshon,                                                                                      
  M.S.~Lightwood\\                                                                                 
    {\it Department of Particle Physics, Weizmann Institute, Rehovot,                              
           Israel}~$^{c}$                                                                          
\par \filbreak                                                                                     
  A.~Everett,                                                                                      
  D.~K\c{c}ira,                                                                                    
  S.~Lammers,                                                                                      
  L.~Li,                                                                                           
  D.D.~Reeder,                                                                                     
  M.~Rosin,                                                                                        
  P.~Ryan,                                                                                         
  A.A.~Savin,                                                                                      
  W.H.~Smith\\                                                                                     
  {\it Department of Physics, University of Wisconsin, Madison,                                    
Wisconsin 53706}, USA~$^{n}$                                                                       
\par \filbreak                                                                                     
                                                           %                                       
  S.~Dhawan\\                                                                                      
  {\it Department of Physics, Yale University, New Haven, Connecticut                              
06520-8121}, USA~$^{n}$                                                                            
 \par \filbreak                                                                                    
                                                           %                                       
  S.~Bhadra,                                                                                       
  C.D.~Catterall,                                                                                  
  Y.~Cui,                                                                                          
  G.~Hartner,                                                                                      
  S.~Menary,                                                                                       
  U.~Noor,                                                                                         
  M.~Soares,                                                                                       
  J.~Standage,                                                                                     
  J.~Whyte\\                                                                                       
  {\it Department of Physics, York University, Ontario, Canada M3J                                 
1P3}~$^{a}$                                                                                        
                                                           %                                       
\newpage                                                                                           
                                                           %                                       
$^{\    1}$ also affiliated with University College London, UK \\                                  
$^{\    2}$ retired \\                                                                             
$^{\    3}$ now at the University of Victoria, British Columbia, Canada \\                         
$^{\    4}$ PPARC Advanced fellow \\                                                               
$^{\    5}$ supported by a scholarship of the World Laboratory                                     
Bj\"orn Wiik Research Project\\                                                                    
$^{\    6}$ partly supported by Polish Ministry of Scientific Research and Information             
Technology, grant no.2P03B 12625\\                                                                 
$^{\    7}$ supported by the Polish State Committee for Scientific Research, grant no.             
2 P03B 09322\\                                                                                     
$^{\    8}$ now at DESY group FEB, Hamburg, Germany \\                                             
$^{\    9}$ now at LAL, Universit\'e de Paris-Sud, IN2P3-CNRS, Orsay, France \\                    
$^{  10}$ partly supported by Moscow State University, Russia \\                                   
$^{  11}$ also affiliated with DESY \\                                                             
$^{  12}$ now at Baylor University, USA \\                                                         
$^{  13}$ now at University of Oxford, UK \\                                                       
$^{  14}$ now at the Department of Physics and Astronomy, University of Glasgow, UK \\             
$^{  15}$ now at Royal Holloway University of London, UK \\                                        
$^{  16}$ also at Nara Women's University, Nara, Japan \\                                          
$^{  17}$ also at University of Tokyo, Japan \\                                                    
$^{  18}$ Ram{\'o}n y Cajal Fellow \\                                                              
$^{  19}$ PPARC Postdoctoral Research Fellow \\                                                    
$^{  20}$ on leave of absence at The National Science Foundation, Arlington, VA, USA \\            
$^{  21}$ also at Max Planck Institute, Munich, Germany, Alexander von Humboldt                    
Research Award\\                                                                                   
$^{  22}$ present address: Tokyo Metropolitan University of Health                                 
Sciences, Tokyo 116-8551, Japan\\                                                                  
$^{  23}$ also at University of Hamburg, Germany, Alexander von Humboldt Fellow \\                 
$^{  24}$ partially funded by DESY \\                                                              
$^{  25}$ also at \L\'{o}d\'{z} University, Poland \\                                              
$^{  26}$ \L\'{o}d\'{z} University, Poland \\                                                      
$^{  27}$ \L\'{o}d\'{z} University, Poland, supported by the KBN grant 2P03B12925 \\               
$^{  28}$ supported by the KBN grant 2P03B12725 \\                                                 
                                                           %                                       
                                                           %                                       
% \par         % if index listing & table fit to 1 page, put gap here                              
\newpage   % alternatively: go to newpage, if page is too small                                    
                                                           %                                       
% \institute_references_start    % do not touch or move this line !                                
                                                           %                                       
\begin{tabular}[h]{rp{14cm}}                                                                       
                                                           %                                       
$^{a}$ &  supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) \\  
$^{b}$ &  supported by the German Federal Ministry for Education and Research (BMBF), under        
          contract numbers HZ1GUA 2, HZ1GUB 0, HZ1PDA 5, HZ1VFA 5\\                                
$^{c}$ &  supported in part by the MINERVA Gesellschaft f\"ur Forschung GmbH, the Israel Science   
          Foundation (grant no. 293/02-11.2), the U.S.-Israel Binational Science Foundation and    
          the Benozyio Center for High Energy Physics\\                                            
$^{d}$ &  supported by the German-Israeli Foundation and the Israel Science Foundation\\           
$^{e}$ &  supported by the Italian National Institute for Nuclear Physics (INFN) \\                
$^{f}$ &  supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology 
          (MEXT) and its grants for Scientific Research\\                                          
$^{g}$ &  supported by the Korean Ministry of Education and Korea Science and Engineering          
          Foundation\\                                                                             
$^{h}$ &  supported by the Netherlands Foundation for Research on Matter (FOM)\\                   
$^{i}$ &  supported by the Polish State Committee for Scientific Research, grant no.               
          620/E-77/SPB/DESY/P-03/DZ 117/2003-2005 and grant no. 1P03B07427/2004-2006\\             
$^{j}$ &  partially supported by the German Federal Ministry for Education and Research (BMBF)\\   
$^{k}$ &  supported by RF Presidential grant N 1685.2003.2 for the leading scientific schools and  
          by the Russian Ministry of Education and Science through its grant for Scientific        
          Research on High Energy Physics\\                                                        
$^{l}$ &  supported by the Spanish Ministry of Education and Science through funds provided by     
          CICYT\\                                                                                  
$^{m}$ &  supported by the Particle Physics and Astronomy Research Council, UK\\                   
$^{n}$ &  supported by the US Department of Energy\\                                               
$^{o}$ &  supported by the US National Science Foundation\\                                        
$^{p}$ &  supported by the Polish Ministry of Scientific Research and Information Technology,      
          grant no. 112/E-356/SPUB/DESY/P-03/DZ 116/2003-2005 and 1 P03B 065 27\\                  
$^{q}$ &  supported by FNRS and its associated funds (IISN and FRIA) and by an Inter-University    
          Attraction Poles Programme subsidised by the Belgian Federal Science Policy Office\\     
                                                       %                                           
\end{tabular}                                                                                      
                                                           %                                       
% \institute_references_end     % do not touch or move this line !                                 
                                                           %                                       
%\end{document}                                                                                     

DESY-05-019-cit.tex
%==============================================================================
%	Some definitions of citations
%==============================================================================
%------------------------------------------------------------------------------
%	-> CTD
%------------------------------------------------------------------------------
\def\citeCTD{{\cite{%
nim:a279:290,*npps:b32:181,*nim:a338:254%
}}\xspace}
%------------------------------------------------------------------------------
%	-> CAL
%------------------------------------------------------------------------------
\def\citeCAL{{\cite{%
nim:a309:77,*nim:a309:101,*nim:a321:356,*nim:a336:23%
}}\xspace}

DESY-05-019-fig.tex
%------------------------------------------------------------------------------
%       Results
%------------------------------------------------------------------------------
\begin{figure}[p]
\vfill
\begin{center}
%\framebox[13.cm]{\rule{0.pt}{10.cm}}
\centerline{\epsfig{figure=DESY-05-019_1.eps,width=\linewidth}}
\end{center}
\caption{(a) The inclusive trijet cross sections as functions of
$E_{T,B}^{\rm jet}$ with the jets ordered in $E_{T,B}^{\rm jet}$. The cross sections of the
second and third jet were scaled by the factors shown for readability. 
The inner error bars
represent the statistical uncertainties. The outer error bars
represent the quadratic sum of statistical and systematic uncertainties
not associated with the calorimeter energy scale. The shaded band
indicates the calorimeter energy scale uncertainty. The predictions
of perturbative QCD in next-to-leading order, corrected for hadronisation
effects and using the CTEQ6 parameterisations of the proton PDFs, are compared to the data.
(b), (c) and (d) show the ratio of the data to the predictions. The hatched band
represents the renormalisation scale uncertainty of the QCD calculation.
}
\label{fig-e3}
\vfill
\end{figure}

\begin{figure}[p]
\vfill
\begin{center}
\centerline{\epsfig{figure=DESY-05-019_2.eps,width=\linewidth}}
\end{center}
\caption{(a) The inclusive trijet cross sections as functions of
$\eta_{\rm LAB}^{\rm jet}$ with the jets ordered in $\eta_{\rm LAB}^{\rm jet}$. The cross sections of
the second and third jet were scaled up for readability only. 
The predictions of perturbative QCD in next-to-leading order are 
compared to the data. (b), (c) and (d) show the ratio of the data to the 
predictions. Other details are as in the caption to Fig.~\ref{fig-e3}.
}
\label{fig-h3}
\vfill
\end{figure}

\begin{figure}[p]
\vfill
\begin{center}
\centerline{\epsfig{figure=DESY-05-019_3.eps,width=\linewidth}}
\end{center}
\caption{(a) The inclusive dijet and trijet cross sections as functions
of $Q^2$. The predictions of perturbative QCD in
next-to-leading order are compared to the data. (b) and (c) show the
ratio of the data to the predictions. Other details are as 
in the caption to Fig.~\ref{fig-e3}.
}
\label{fig-q1}
\vfill
\end{figure}

\begin{figure}[p]
\vfill
\begin{center}
\centerline{\epsfig{figure=DESY-05-019_4.eps,width=\linewidth}}
\end{center}
\caption{(a) The ratio of inclusive trijet to dijet cross sections as a
function of $Q^2$. The predictions of perturbative QCD in
next-to-leading order are compared to the data. (b) shows the ratio of the 
data to the predictions. 
Other details are as in the caption to Fig.~\ref{fig-e3}.
}
\label{fig-q2}
\vfill
\end{figure}

\begin{figure}[p]
\begin{center}
\centerline{\epsfig{figure=DESY-05-019_5.eps,width=\linewidth}}
\end{center}
\caption{(a) The ratio of inclusive trijet to dijet cross sections as a
function of $Q^2$. The predictions of perturbative QCD in
next-to-leading order using five sets of CTEQ4 PDF are compared to the data.
(b) shows the $\alpha_s(M_Z)$ values determined from the 
ratio of inclusive trijet to dijet cross
sections in different regions of $Q^2$.
The shaded band indicates the current world average value of $\alpha_s(M_Z)$.
The inner error bars represent the statistical uncertainty of the data. 
The outer error bars show the statistical and systematic uncertainties added in quadrature.
The dashed error bars display the theoretical uncertainties. 
}
\label{fig-alphas}
\end{figure}

DESY-05-019-ref.tex
{
\def\bibname{\Large\bf References}
\def\refname{\Large\bf References}
\pagestyle{plain}
\ifzeusbst
  \bibliographystyle{./BiBTeX/bst/l4z_default}
\fi
\ifzdrftbst
  \bibliographystyle{./BiBTeX/bst/l4z_draft}
\fi
\ifzbstepj
  \bibliographystyle{./BiBTeX/bst/l4z_epj}
\fi
\ifzbstnp
  \bibliographystyle{./BiBTeX/bst/l4z_np}
\fi
\ifzbstpl
  \bibliographystyle{./BiBTeX/bst/l4z_pl}
\fi
{\raggedright
\bibliography{./BiBTeX/user/syn.bib,%
              ./BiBTeX/bib/l4z_articles.bib,%
              ./BiBTeX/bib/l4z_books.bib,%
              ./BiBTeX/bib/l4z_conferences.bib,%
              ./BiBTeX/bib/l4z_h1.bib,%
              ./BiBTeX/bib/l4z_misc.bib,%
              ./BiBTeX/bib/l4z_old.bib,%
              ./BiBTeX/bib/l4z_preprints.bib,%
              ./BiBTeX/bib/l4z_replaced.bib,%
              ./BiBTeX/bib/l4z_temporary.bib,%
              ./BiBTeX/bib/l4z_zeus.bib}}
}
\vfill\eject

DESY-05-019-tab.tex
%-------------------------------------------------------------------------------
%       An example table
%-------------------------------------------------------------------------------
\begin{table}[p]
\begin{center}
\begin{tabular}{||c|cccc||c||c||}
\hline
$E_{T,B}^{\rm jet,1}$ & $d\sigma/dE_{T,B}^{\rm jet,1}$ & 
$\delta_{\rm stat}$ & $\delta_{\rm syst}$ & $\delta_{\rm ES}$ & $C_{\rm QED}$ &
$C_{\rm had}$ \\
($\hbox{GeV}$) & ($\hbox{pb}/\hbox{GeV}$) & & & &  &  \\
\hline\hline
5 - 8 &  5.22 & $\pm$ 0.29 & $^{+0.75}_{-0.89}$ & $^{+0.63}_{-0.50}$ & 0.97 &
2.11 \\
8 - 12 & 20.2 & $\pm$ 0.5 & $^{+3.0}_{-3.5}$ & $^{+3.0}_{-2.9}$ & 0.96 & 1.33 \\ 
12 - 16 & 14.4 & $\pm $ 0.4 & $^{+1.6}_{-1.4}$ & $^{+1.6}_{-1.3}$ & 0.96 &
1.21 \\
16 - 20 & 6.05 & $\pm$ 0.22 & $^{+0.68}_{-0.54}$ & $^{+0.42}_{-0.44}$ & 0.96 &
1.19 \\
20 - 25 & 2.19 & $\pm$ 0.11 & $^{+0.32}_{-0.21}$ & $^{+0.20}_{-0.16}$ & 0.96 &
1.19 \\
25 - 30 & 0.828 & $\pm$ 0.068 & $^{+0.12}_{-0.10}$ & $^{+0.063}_{-0.065}$ & 0.97
& 1.22 \\
30 - 40 & 0.222 & $\pm$ 0.027 & $^{+0.036}_{-0.026}$ & $^{+0.021}_{-0.017}$ &
0.97 & 1.20 \\
40 - 60 & 0.047 & $\pm$ 0.012 & $^{+0.004}_{-0.012}$ & $^{+0.001}_{-0.004}$ &
1.07 & 1.31 \\
\hline
\end{tabular}
\caption{ The inclusive trijet cross section $d\sigma/dE_{T,B}^{\rm jet,1}$ for jets of
hadrons in the Breit frame, selected using the $k_T$ cluster algorithm in the 
longitudinally invariant inclusive mode. The statistical ($\delta_{\rm stat}$), 
systematic ($\delta_{\rm syst}$) and 
the absolute energy scale uncertainties ($\delta_{\rm ES}$) are shown separately. 
The multiplicative
correction factors for QED radiative effects ($C_{\rm QED}$), 
applied to the data, 
and for hadronisation
effects ($C_{\rm had}$), applied to the NLO predictions, are shown in the last two columns.}
\label{et1}
\end{center}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{||c|cccc||c||c||}
\hline
$E_{T,B}^{\rm jet,2}$ & $d\sigma/dE_{T,B}^{\rm jet,2}$ & 
$\delta_{\rm stat}$ & $\delta_{\rm syst}$ & $\delta_{\rm ES}$ & $C_{\rm QED}$ &
$C_{\rm had}$ \\
($\hbox{GeV}$) & ($\hbox{pb}/\hbox{GeV}$) & & & &  &  \\
\hline\hline
5 - 8 &  24.0 & $\pm$ 0.6 & $^{+2.6}_{-2.6}$ & $^{+2.5}_{-2.4}$ & 0.96 & 1.48 \\
8 - 12 & 20.7 & $\pm$ 0.4 & $^{+3.0}_{-2.8}$ & $^{+3.0}_{-2.6}$ & 0.97 & 1.22 \\ 
12 - 16 & 6.70 & $\pm $ 0.24 & $^{+0.65}_{-0.66}$ & $^{+0.65}_{-0.56}$ & 0.96 &
1.28 \\
16 - 20 & 2.16 & $\pm$ 0.13 & $^{+0.14}_{-0.22}$ & $^{+0.13}_{-0.14}$ & 0.95 &
1.32 \\
20 - 25 & 0.780 & $\pm$ 0.067 & $^{+0.087}_{-0.10}$ & $^{+0.086}_{-0.070}$ &
0.97 & 1.34 \\
25 - 30 & 0.225 & $\pm$ 0.038 & $^{+0.034}_{-0.005}$ & $^{+0.005}_{-0.005}$ &
0.97 & 1.40 \\
30 - 40 & 0.054 & $\pm$ 0.014 & $^{+0.013}_{-0.006}$ & $^{+0.004}_{-0.001}$ &
1.02 & 1.51 \\
\hline
\end{tabular}
\caption{ The inclusive trijet cross section $d\sigma/dE_{T,B}^{\rm jet,2}$ for jets of
hadrons in the Breit frame, selected using the $k_T$ cluster algorithm in the 
longitudinally invariant inclusive mode. Other details are as in the caption to 
Table~\ref{et1}.}
\label{et2}
\end{center}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{||c|cccc||c||c||}
\hline
$E_{T,B}^{\rm jet,3}$ & $d\sigma/dE_{T,B}^{\rm jet,3}$ & 
$\delta_{\rm stat}$ & $\delta_{\rm syst}$ & $\delta_{\rm ES}$ & $C_{\rm QED}$ &
$C_{\rm had}$ \\
($\hbox{GeV}$) & ($\hbox{pb}/\hbox{GeV}$) & & & &  &  \\
\hline\hline
5 - 8 &  52.8 & $\pm$ 0.8 & $^{+6.3}_{-6.2}$ & $^{+6.1}_{-5.3}$ & 0.96 & 1.28 \\
8 - 12 & 8.06 & $\pm$ 0.25 & $^{+1.1}_{-1.1}$ & $^{+1.1}_{-1.1}$ & 0.96 & 1.49 \\ 
12 - 16 & 1.11 & $\pm $ 0.09 & $^{+0.17}_{-0.08}$ & $^{+0.07}_{-0.08}$ & 0.97 &
1.66 \\
16 - 20 & 0.208 & $\pm$ 0.039 & $^{+0.025}_{-0.035}$ & $^{+0.025}_{-0.018}$ &
0.98 & 1.77 \\
20 - 25 & 0.052 & $\pm$ 0.020 & $^{+0.005}_{-0.019}$ & $^{+0.000}_{-0.003}$ &
0.94 & 1.77 \\
\hline
\end{tabular}
\caption{ The inclusive trijet cross section $d\sigma/dE_{T,B}^{\rm jet,3}$ for jets of
hadrons in the Breit frame, selected using the $k_T$ cluster algorithm in the 
longitudinally invariant inclusive mode. Other details are as in the caption to 
Table~\ref{et1}.}
\label{et3}
\end{center}
\end{table}


\begin{table}[p]
\begin{center}
\begin{tabular}{||c|cccc||c||c||}
\hline
$\eta_{\rm LAB}^{\rm jet,1}$ & $d\sigma/d\eta_{\rm LAB}^{\rm jet,1}$ & 
$\delta_{\rm stat}$ & $\delta_{\rm syst}$ & $\delta_{\rm ES}$ & $C_{\rm QED}$ &
$C_{\rm had}$ \\
 & ($\hbox{pb}$) & & & & &  \\
\hline\hline
0.5 - 1.0 & 19.1 & $\pm$ 1.2 & $^{+2.9}_{-2.7}$ & $^{+2.9}_{-2.6}$ & 0.95 &
2.18 \\
1.0 - 1.5 & 61.2 & $\pm$ 2.0 & $^{+7.3}_{-8.7}$ & $^{+7.3}_{-7.3}$ & 0.96 &
1.51 \\
1.5 - 2.0 & 125 & $\pm$ 3 & $^{+15}_{-20}$ & $^{+15}_{-13}$ & 0.96 & 1.31 \\
2.0 - 2.5 & 186 & $\pm$ 4 & $^{+23}_{-20}$ & $^{+20}_{-18}$ & 0.96 & 1.17 \\
\hline
\end{tabular}
\caption{ The inclusive trijet cross section $d\sigma/d\eta_{\rm LAB}^{\rm jet,1}$ for jets of
hadrons in the Breit frame, selected using the $k_T$ cluster algorithm in the 
longitudinally invariant inclusive mode. Other details are as in the caption to 
Table~\ref{et1}.}
\label{eta1}
\end{center}
\end{table}

\begin{table}[p]
\begin{center}
\begin{tabular}{||c|cccc||c||c||}
\hline
$\eta_{\rm LAB}^{\rm jet,2}$ & $d\sigma/d\eta_{\rm LAB}^{\rm jet,2}$ & 
$\delta_{\rm stat}$ & $\delta_{\rm syst}$ & $\delta_{\rm ES}$ & $C_{\rm QED}$ &
$C_{\rm had}$ \\
 & ($\hbox{pb}$) & & & & &  \\
\hline\hline
-1.0 - -0.5 & 0.34 & $\pm$ 0.14 & $^{+0.04}_{-0.18}$ & $^{+0.03}_{-0.09}$ &
0.91 & 8.72 \\
-0.5 - 0.0 & 13.6 & $\pm$ 1.0 & $^{+3.0}_{-2.4}$ & $^{+2.9}_{-2.1}$ & 0.98 &
2.30 \\
0.0 - 0.5 & 62.1 & $\pm$ 2.1 & $^{+9.3}_{-11.0}$ & $^{+9.1}_{-8.4}$ & 0.96 & 1.50 \\
0.5 - 1.0 & 115 & $\pm$ 3 & $^{+14}_{-14}$ & $^{+14}_{-13}$ & 0.97 & 1.29 \\
1.0 - 1.5 & 108 & $\pm$ 3 & $^{+12}_{-14}$ & $^{+11}_{-10}$ & 0.96 & 1.22 \\
1.5 - 2.0 & 73.3 & $\pm$ 2.2 & $^{+7.3}_{-7.8}$ & $^{+7.3}_{-6.3}$ & 0.96 &
1.19 \\
2.0 - 2.5 & 20.3 & $\pm$ 1.4 & $^{+5.6}_{-2.1}$ & $^{+1.7}_{-1.6}$ & 0.97 &
1.16 \\
\hline
\end{tabular}
\caption{ The inclusive trijet cross section $d\sigma/d\eta_{\rm LAB}^{\rm jet,2}$ for jets of
hadrons in the Breit frame, selected using the $k_T$ cluster algorithm in the 
longitudinally invariant inclusive mode. Other details are as in the caption to 
Table~\ref{et1}.}
\label{eta2}
\end{center}
\end{table}

\begin{table}[p]
\begin{center}
\begin{tabular}{||c|cccc||c||c||}
\hline
$\eta_{\rm LAB}^{\rm jet,3}$ & $d\sigma/d\eta_{\rm LAB}^{\rm jet,3}$ & 
$\delta_{\rm stat}$ & $\delta_{\rm syst}$ & $\delta_{\rm ES}$ & $C_{\rm QED}$ &
$C_{\rm had}$ \\
 & ($\hbox{pb}$) & & & & &  \\
\hline\hline
-1.0 - -0.5 & 63.7 & $\pm$ 2.3 & $^{+9.2}_{-11.0}$ & $^{+9.0}_{-9.3}$ & 0.96 &
1.32 \\
-0.5 - 0.0 & 112 & $\pm$ 3 & $^{+15}_{-14}$ & $^{+15}_{-12}$ & 0.96 & 1.24 \\
0.0 - 0.5 & 108 & $\pm$ 3 & $^{+12}_{-15}$ & $^{+12}_{-11}$ & 0.96 & 1.25 \\
0.5 - 1.0 & 68.6 & $\pm$ 2.1 & $^{+8.7}_{-7.6}$ & $^{+7.4}_{-6.5}$ & 0.97 & 1.32 \\
1.0 - 1.5 & 31.4 & $\pm$ 1.5 & $^{+2.3}_{-4.6}$ & $^{+2.1}_{-2.7}$ & 0.94 & 1.44 \\
1.5 - 2.0 & 6.21 & $\pm$ 0.67 & $^{+1.50}_{-0.67}$ & $^{+0.61}_{-0.61}$ & 0.93 &
1.50 \\
\hline
\end{tabular}
\caption{ The inclusive trijet cross section $d\sigma/d\eta_{\rm LAB}^{\rm jet,3}$ for jets of
hadrons in the Breit frame, selected using the $k_T$ cluster algorithm in the 
longitudinally invariant inclusive mode. Other details are as in the caption to 
Table~\ref{et1}.}
\label{eta3}
\end{center}
\end{table}


\begin{table}[p]
\begin{center}
\begin{tabular}{||c|cccc||c||c||}
\hline
$Q^2$  & $(d\sigma/dQ^2)_{\rm dijet}$ & 
$\delta_{\rm stat}$ & $\delta_{\rm syst}$ & $\delta_{\rm ES}$ & $C_{\rm QED}$ &
$C_{\rm had}$ \\
($\hbox{GeV}^2$) & ($\hbox{pb}/\hbox{GeV}^2$) & & & & &  \\
\hline\hline
10 - 35 &  9.70 & $\pm$ 0.10 & $^{+1.10}_{-0.94}$ & $^{+0.87}_{-0.76}$ & 0.97 &
1.18 \\
35 - 85 &  2.50 & $\pm$ 0.04 & $^{+0.21}_{-0.20}$ & $^{+0.20}_{-0.18}$ & 0.95
& 1.16 \\ 
85 - 220 & 0.649 & $\pm $ 0.011 & $^{+0.049}_{-0.044}$ & $^{+0.044}_{-0.041}$ &
0.96 & 1.15 \\
220 - 700 & 0.104 & $\pm$ 0.002 & $^{+0.006}_{-0.006}$ &
$^{+0.006}_{-0.006}$ & 0.94 & 1.12 \\
700 - 5000 & 0.00403 & $\pm$ 0.00014 & $^{+0.00015}_{-0.00024}$ &
$^{+0.00014}_{-0.00014}$ & 0.92 & 1.09 \\
\hline
\end{tabular}
\caption{ The inclusive dijet cross section $(d\sigma/dQ^2)_{\rm dijet}$ for jets of
hadrons in the Breit frame, selected using the $k_T$ cluster algorithm in the 
longitudinally invariant inclusive mode. Other details are as in the caption to 
Table~\ref{et1}.}
\label{dq2}
\end{center}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{||c|cccc||c||c||}
\hline
$Q^2$  & $(d\sigma/dQ^2)_{\rm trijet}$ & 
$\delta_{\rm stat}$ & $\delta_{\rm syst}$ & $\delta_{\rm ES}$ & $C_{\rm QED}$ &
$C_{\rm had}$ \\
($\hbox{GeV}^2$) & ($\hbox{pb}/\hbox{GeV}^2$) & & & & &  \\
\hline\hline
10 - 35 &  3.94 & $\pm$ 0.08 & $^{+0.54}_{-0.55}$ & $^{+0.49}_{-0.46}$ & 0.98 &
1.35 \\
35 - 85 &  0.94 & $\pm$ 0.02 & $^{+0.12}_{-0.11}$ & $^{+0.11}_{-0.09}$ & 0.95
& 1.31 \\ 
85 - 220 & 0.227 & $\pm $ 0.007 & $^{+0.024}_{-0.027}$ & $^{+0.024}_{-0.023}$
& 0.96 & 1.32 \\
220 - 700 & 0.0320 & $\pm$ 0.0013 & $^{+0.0036}_{-0.0031}$ &
$^{+0.0027}_{-0.0026}$ & 0.94 & 1.35 \\
700 - 5000 & 0.00112 & $\pm$ 0.00007 & $^{+0.00008}_{-0.00014}$ &
$^{+0.00008}_{-0.00009}$ & 0.92 & 1.33 \\
\hline
\end{tabular}
\caption{ The inclusive trijet cross section $(d\sigma/dQ^2)_{\rm trijet}$ for jets of
hadrons in the Breit frame, selected using the $k_T$ cluster algorithm in the 
longitudinally invariant inclusive mode. Other details are as in the caption to 
Table~\ref{et1}.}
\label{tq2}
\end{center}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{||c|cccc||c||c||}
\hline
$Q^2$  & $R_{3/2}$ & 
$\delta_{\rm stat}$ & $\delta_{\rm syst}$ & $\delta_{\rm ES}$ & $C_{\rm QED}$ &
$C_{\rm had}$ \\
($\hbox{GeV}^2$) & & & & & & \\
\hline\hline
10 - 35 &  0.406 & $\pm$ 0.008 & $^{+0.014}_{-0.027}$ & $^{+0.013}_{-0.017}$ &
1.01 & 1.14 \\
35 - 85 &  0.375 & $\pm$ 0.010 & $^{+0.014}_{-0.019}$ & $^{+0.014}_{-0.010}$ &
0.99 & 1.02 \\ 
85 - 220 & 0.350 & $\pm $ 0.011 & $^{+0.014}_{-0.031}$ & $^{+0.013}_{-0.015}$ &
1.00 & 1.15 \\
220 - 700 & 0.306 & $\pm$ 0.013 & $^{+0.024}_{-0.022}$ & $^{+0.008}_{-0.009}$
& 1.01 & 1.21 \\
700 - 5000 & 0.279 & $\pm$ 0.018 & $^{+0.016}_{-0.029}$ & $^{+0.009}_{-0.014}$ &
1.01 & 1.21 \\
\hline
\end{tabular}
\caption{ The ratio of inclusive trijet to dijet cross sections for jets of
hadrons in the Breit frame, selected using the $k_T$ cluster algorithm in the 
longitudinally invariant inclusive mode. Other details are as in the caption to 
Table~\ref{et1}.}
\label{rq2}
\end{center}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{||c|cccc||}
\hline
$Q^2$  & $\alpha_s(M_Z)$ & 
$\delta_{\rm stat}$ & $\delta_{\rm exp}$ & $\delta_{\rm theo}$ \\
($\hbox{GeV}^2$) & & & & \\
\hline\hline
10 - 35 &  0.1210 & $\pm$ 0.0022 & $^{+0.0031}_{-0.0058}$ & $^{+0.0074}_{-0.0080}$ \\
35 - 85 &  0.1148 & $\pm$ 0.0024 & $^{+0.0028}_{-0.0033}$ & $^{+0.0056}_{-0.0039}$ \\ 
85 - 220 & 0.1178 & $\pm$ 0.0027 & $^{+0.0027}_{-0.0063}$ & $^{+0.0064}_{-0.0016}$ \\
220 - 700 & 0.1171 & $\pm$ 0.0039 & $^{+0.0043}_{-0.0056}$ & $^{+0.0068}_{-0.0023}$ \\
700 - 5000 & 0.1170 & $\pm$ 0.0064 & $^{+0.0045}_{-0.0064}$ & $^{+0.011}_{-0.0037}$ \\
\hline
10 - 5000 & 0.1179 & $\pm$ 0.0013 & $^{+0.0028}_{-0.0046}$ & $^{+0.0064}_{-0.0046}$ \\
\hline
\end{tabular}
\caption{The $\alpha_s(M_Z)$ values as determined in this analysis.
The statistical ($\delta_{\rm stat}$), experimental ($\delta_{\rm exp}$) and 
theoretical ($\delta_{\rm theo}$) systematic uncertainties are shown separately.
}
\label{alphas}
\end{center}
\end{table}


DESY-05-019-tit.tex

\prepnum{DESY-05-019}

\title{
Multijet production in neutral current deep inelastic scattering at HERA and 
determination of $\alpha_s$ 
}  

\author{ZEUS Collaboration}
 
\date{January 2005}

\abstract{
Multijet production rates in neutral current deep inelastic 
scattering have been measured in the range of exchanged boson
virtualities $10 < Q^2 < 5000\,\hbox{GeV}^2$. The data were taken at
the $ep$ collider HERA with centre-of-mass energy  $\sqrt{s} =
318\,\hbox{GeV}$ using the ZEUS detector and correspond to an
integrated luminosity of $82.2\,\hbox{pb}^{-1}$. Jets were identified in the Breit frame
using the $k_T$ cluster algorithm in the longitudinally invariant inclusive mode. 
Measurements of differential dijet and trijet cross sections are presented as
functions of jet transverse energy ($E_{T,B}^{\rm jet}$), pseudorapidity
($\eta_{\rm LAB}^{\rm jet}$) and $Q^2$ with $E_{T,B}^{\rm jet} > 5\,\hbox{GeV}$ and $-1 <
\eta_{\rm LAB}^{\rm jet} < 2.5$. Next-to-leading-order QCD calculations describe the data well.
The value of the strong coupling constant $\alpha_s(M_Z)$, determined from
the ratio of the trijet to dijet cross sections, is
$\alpha_s(M_Z) = 0.1179 \pm 0.0013~\rm(stat.)^{+0.0028}_{-0.0046}~\rm(exp.)^{+0.0064}_{-0.0046}~\rm(th.)$. 
}

\makezeustitle

DESY-05-019-txt.tex
\pagenumbering{arabic} 
\pagestyle{plain}
\raggedbottom
% ----------------------------------------------------------------------------
%       Introduction
% ----------------------------------------------------------------------------
\section{Introduction}
\label{intro}

Measurements of multijet production from initial-state hadrons and leptons have been carried
out previously in collisions at the SPS \cite{pl:b158:494,*zfp:c30:341}, the ISR 
\cite{np:b303:569,*zfp:c32:317}, the TEVATRON \cite{prl:86:1955,*pr:d53:6000}
and at LEP \cite{proc:moriond:1996:223,*npps:74:44} as well as in photoproduction \cite{pl:b443:394} and deep inelastic 
scattering (DIS) \cite{epj:c6:575} at HERA.
Multijet production in DIS at HERA has been used to test the predictions of perturbative QCD (pQCD) 
calculations 
over a large range of four-momentum transfer squared, $Q^2$ \cite{pl:b515:17}.
Recently, the ZEUS and H1 collaborations have determined the strong coupling
constant $\alpha_s$ from a variety of measurements of jet production and jet properties
in both DIS \cite{epj:c6:575,pl:b363:201,pl:b507:70,pl:b547:164,pl:b558:41,epj:c19:289,epj:c21:33,np:b700:3}
and photoproduction \cite{pl:b560:7}.

At leading order (LO) in $\alpha_s$, dijet production in neutral current DIS
proceeds via the boson-gluon-fusion (BGF, $V^{*} g \rightarrow q\bar{q}$ with $V=\gamma$, $Z^0$) 
and QCD-Compton (QCDC, $V^{*} q \rightarrow qg$) processes.
Events with three jets can be seen as dijet processes with 
an additional gluon radiation or splitting of a gluon
into a quark-antiquark pair and are directly sensitive to
$\mathcal{O}(\alpha_s^{2})$ QCD effects.
The higher sensitivity to $\alpha_s$ and the large number of degrees of freedom of the trijet final state 
allow detailed testing of QCD predictions.

In the present analysis, the differential cross sections for the trijet production have been measured with high statistical precision. 
Measurements of the inclusive trijet cross section as a function 
of $Q^2$ and the jet transverse energy, $E_{T,B}^{\rm jet}$, in the Breit frame and the jet 
pseudorapidity, $\eta_{\rm LAB}^{\rm jet}$, in the laboratory frame are presented.
Predictions of pQCD at next-to-leading order (NLO) are compared to the measurements. 
In addition, the analysis includes the first $\alpha_s$ determination using the cross-section ratio 
of trijet to dijet production, $R_{3/2}$, at HERA. In this ratio, correlated
experimental and 
theoretical uncertainties cancel, allowing for an extension 
of the measurement to low $Q^2$. 


\section{Experimental set-up}
\label{exp}
The data used in this analysis were collected during the 1998-2000 running
period, when HERA operated with protons of energy
$E_p=920$~GeV and electrons or positrons\footnote{In the
following, the term ``electron'' denotes generically both the
electron ($e^-$) and the positron ($e^+$).}
of energy $E_e=27.5$~GeV, and correspond to an integrated luminosity
of $82.2\pm 1.9$~pb$^{-1}$. A detailed description of the ZEUS detector
can be found elsewhere~\cite{pl:b293:465,zeus:1993:bluebook}. A brief
outline of the components that are most relevant for this analysis is
given below.

Charged particles are measured in the central tracking detector
(CTD)~\cite{nim:a279:290,*npps:b32:181,*nim:a338:254}, which operates in a
magnetic field of $1.43\Tesla$ provided by a thin superconducting
solenoid. The CTD consists of 72~cylindrical drift chamber
layers, organised in nine superlayers covering the
polar-angle\footnote{The 
ZEUS coordinate system is a right-handed Cartesian system, with the $Z$ axis 
pointing in the proton beam direction, referred to as the 
``forward direction'', and the $X$ axis pointing left towards the centre of 
HERA. The coordinate origin is at the nominal interaction point.} 
region \mbox{$15^\circ<\theta<164^\circ$}. The transverse momentum
resolution for full-length tracks can be parameterised as
$\sigma(p_T)/p_T=0.0058p_T\oplus0.0065\oplus0.0014/p_T$, with $p_T$ in
$\Gev$. The tracking system was used to measure the interaction vertex
with a typical resolution along (transverse to) the beam direction of
0.4~(0.1)~cm and also to cross-check the energy scale of the calorimeter.

The high-resolution uranium-scintillator calorimeter
(CAL)~\cite{nim:a309:77,*nim:a309:101,*nim:a321:356,*nim:a336:23} covers
$99.7\%$ of the total solid angle and consists of three parts: the
forward (FCAL), the barrel (BCAL) and the rear (RCAL) calorimeters. Each
part is subdivided transversely into towers and longitudinally into one
electromagnetic section (EMC) and either one (in RCAL) or two (in BCAL and
FCAL) hadronic sections (HAC). The smallest subdivision of the calorimeter
is called a cell. Under test-beam conditions, the CAL single-particle
relative energy resolutions were $\sigma(E)/E=0.18/\sqrt{E}$ for
electrons and $\sigma(E)/E=0.35/\sqrt{E}$ for hadrons, with $E$ in GeV.

The luminosity was measured from the rate of the bremsstrahlung process
$ep\rightarrow e\gamma p$. The resulting small angle energetic photons
were measured by the luminosity
monitor~\cite{desy-92-066,*zfp:c63:391,*acpp:b32:2025}, a
lead-scintillator calorimeter placed in the HERA tunnel at $Z=-107$ m.


\section{Kinematics and event selection}
\label{ks}
A three-level trigger system was used to select events online \cite{zeus:1993:bluebook,zfp:c74:207,thesis:krumnack:2004,thesis:li:2004}.
Neutral current DIS events were selected by requiring that the scattered electron with energy more than 4 $\gev$ was measured in the 
CAL \cite{epj:c11:427}.

The offline kinematic variables $Q^2$ (four-momentum transfer squared), $x_{\rm Bj}$ (Bjorken scaling variable) and $y=Q^2/(sx_{\rm Bj})$ 
($s$ is the centre-of-mass energy squared)  were reconstructed by the electron ($e$), 
double angle (DA) \cite{proc:hera:1991:23}
and Jacquet-Blondel (JB) \cite{proc:epfacility:1979:391} methods. 
The angle of the hadronic system, $\gamma_{\rm had}$, corresponds, in the quark-parton model, to the direction of the scattered quark 
and was reconstructed from the CAL measurements of the hadronic final state.

%The angle of the hadronic system is defined as:
% $\cos\gamma_{had} \equiv \frac{(\sum p_x)^2+(\sum p_y)^2-(\sum E-p_z)^2}
%                              {(\sum p_x)^2+(\sum p_y)^2+(\sum E-p_z)^2}$, where the sum runs over all hadronic energy deposits.
If $\gamma_{\rm had}$ was less than 90$^\circ$ and the scattered-electron track could be 
well reconstructed by the CTD, the DA method was used; otherwise, the electron method was used.
The offline selection of DIS events was similar to that used in a previous ZEUS measurement \cite{epj:c23:13}
and was based on the following requirements:

\begin{itemize}
\item $E_e^\prime \gre 10\gev$, where $E_e^\prime$ is the scattered-electron energy after correction for energy loss in inactive material in front 
of the CAL, to achieve a high-purity sample of DIS events;

\item $y_e \les 0.6$, where $y_e$ is $y$ reconstructed by the electron method, to reduce the photoproduction background;

\item $y_{\rm JB} \gre 0.04$, where $y_{\rm JB}$ is $y$ reconstructed by the JB
method, to ensure sufficient accuracy for the DA reconstruction of $Q^2$;

\item $\cos\gamma_{\rm had} \les 0.7$, to ensure good reconstruction of jets in the Breit frame;

\item $40 \les \sum_i(E-P_Z)_i \les 60\gev$, where the sum runs over all CAL energy deposits. The lower cut removed background from
photoproduction and events with large initial-state QED radiation. The higher cut removed cosmic-ray background;

\item $|Z_{\rm vertex}| \les 50$~cm, where $Z_{\rm vertex}$ is the
reconstructed primary vertex $Z$-position, to select events consistent with $ep$
collisions; 

\item $|X| \gre 13$ or $|Y| \gre 7~{\rm cm}$, where $X$ and $Y$
are the impact positions of the scattered electron on the RCAL, to 
avoid the low-acceptance region adjacent to the rear beampipe.

\end{itemize}

The kinematic range of the analysis is defined as:

\begin{center}
$10<Q^2<5000\gev^2$ and $0.04<y<0.6$.
\end{center}
	
Jets were reconstructed using the $k_T$ cluster algorithm \cite{np:b406:187} in the 
longitudinally invariant inclusive mode \cite{pr:d48:3160}. 
The jet search was conducted in the Breit frame \cite{feynman:1972:photon,*zfp:c2:237}. 
For each event, the jet search was performed using a combination of track and CAL information,
excluding the cells and the track associated with the scattered electron. The selected tracks and CAL
clusters were treated as massless Energy Flow Objects (EFOs) \cite{desy-04-053}. The clustering
of objects was done according to the Snowmass convention\cite{proc:snowmass:1990:134}. 

The jet phase space is defined by selection cuts on the jet pseudorapidity
$\eta_{\rm LAB}^{\rm jet}$ in the laboratory frame and on the jet transverse energy
$E_{T,B}^{\rm jet}$ in the Breit frame:
\begin{center}
$-1 \les \eta_{\rm LAB}^{\rm jet} \les 2.5$ and $E_{T,B}^{\rm jet} \gre 5\gev$.
\end{center}
Events with two (three) or more jets were selected by requiring the invariant
mass of the two (three) highest $E_{T,B}^{\rm jet}$ jets to be:
\begin{center}
$M_{\rm 2jets (3jets)} \gre\ 25\gev$. 
\end{center}
%Events with three or more jets were selected by requiring the invariant mass of the three highest $E_{T,B}^{\rm jet}$ jets to be:
%\begin{center}
%$M_{\rm 3J} \gre\ 25\gev$. 
%\end{center}
These requirements were necessary to ensure a reliable prediction of the 
cross sections at NLO (see Section~\ref{nlo}). 

After all cuts, 37089 events with two or more jets (dijets) and
13665 events with three or more jets (trijets) remained.

\section{Monte Carlo simulation}
\label{mc}
Monte Carlo (MC) simulations were used to correct the data for detector 
effects, inefficiencies of the event selection and that of the jet reconstruction, as well as for QED effects. 
Neutral current DIS events were generated using the {\sc Ariadne}~4.08 program \cite{cpc:71:15} 
and the {\sc Lepto}~6.5 program \cite{cpc:101:108} interfaced to {\sc Heracles}~4.5.2 \cite{cpc:69:155} 
via {\sc Django}~6.2.4 \cite{cpc:81:381}. The {\sc Heracles} program includes
QED effects up to $\mathcal{O}(\alpha_{\rm EM}^{2})$.
In case of {\sc Ariadne}, the QCD cascade is 
simulated using the colour-dipole model \cite{np:b306:746}, whereas for {\sc Lepto}, the matrix 
elements plus parton shower model is used. Both models use the Lund string model \cite{prep:97:31}, 
as implemented in {\sc Jetset} 7.4 \cite{cpc:46:43,cpc:82:74}, for hadronisation. 

The ZEUS detector response was simulated with a program based on {\sc Geant} 3.13
\cite{tech:cern-dd-ee-84-1}. The generated events were passed through the 
detector simulation, subjected to the same trigger requirements as the data,
and processed by the same reconstruction and offline programs.

Measured distributions of kinematic variables are well described by both the {\sc Ariadne} 
and {\sc Lepto} MC models after reweighting in $Q^2$.
The {\sc Lepto} simulation gives a better overall description of the $E_{T,B}^{\rm jet}$
and invariant mass distributions. Therefore, the events generated with the {\sc Lepto}
program were used to determine the acceptance corrections. The events generated with
{\sc Ariadne} were used to estimate the systematic uncertainty associated with the treatment of
the parton shower.


\section{NLO QCD calculations}
\label{nlo}
The NLO calculations were carried out in the \MSbar scheme for
five massless quark flavors with the program {\sc Nlojet} \cite{prl:87:082001} using 
CTEQ6 \cite{hep-ph-0201195}, CTEQ4 \cite{pr:d55:1280}, MRST99 \cite{epj:c14:133} and \mbox{ZEUS-S \cite{pr:d67:012007}}
for the proton parton density functions (PDFs). {\sc Nlojet} allows a computation of
the trijet production cross sections to next-to-leading order, i.e. including all
terms up to $\mathcal{O}(\alpha_s^{3})$.
It was checked that the LO and NLO calculations from {\sc Nlojet} agree with those 
of {\sc Disent} \cite{np:b485:291} at the 1-2\% level for the dijet cross sections \cite{thesis:krumnack:2004,thesis:li:2004}.


For comparison with the data, the CTEQ6 
parameterisations of the proton PDFs were used
 and the renormalisation and factorisation scales 
were both chosen to be $(\bar{E}_T^2+Q^2)/4$, where 
for dijets (trijets) $\bar{E}_T$ is the average $E_T$ of the two (three) highest $E_T$ jets in a given event.
The strong coupling constant 
was set to the value used in the CTEQ6 analysis,
$\alpha_s(M_Z)=0.1179$, and evolved according to the two-loop
solution of the renormalisation group equation. 

The NLO QCD predictions were corrected for hadronisation effects using a
bin-by-bin procedure. Hadronisation correction factors
were defined for each bin as the ratio of the parton- to 
hadron-level cross sections and were calculated using the {\sc Lepto} MC program.
The correction factors $C_{\rm had}$ were typically in the range $1.15-1.35$ for most of the phase space.
 
\section{Corrections and systematic uncertainties}
\label{correction}

The jet transverse energy was corrected for energy losses in the 
inactive material in front of the CAL using the samples of MC simulated events \cite{np:b700:3}.
The cross sections for jets of hadrons in bins of $Q^2$, $E_{T,B}^{\rm jet}$ and $\eta_{\rm LAB}^{\rm jet}$ were 
obtained by applying a bin-by-bin correction to the measured jet distributions using 
the {\sc Lepto} program. The corrections take into account the efficiency of the trigger, the selection criteria and 
the purity and efficiency of the jet reconstruction. Additional 
corrections for QED effects, $C_{\rm QED}$, 
calculated using {\sc Heracles}, were applied 
to the measured cross sections, $\sigma_{\rm Born}=\sigma_{\rm meas.} \cdot
C_{\rm QED}$.  

A detailed study of the sources contributing to the systematic uncertainties of the measurements was 
performed \cite{thesis:krumnack:2004, thesis:li:2004}. The main sources contributing to
the systematic uncertainties are listed below (typical values of the systematic uncertainties in the 
dijet cross section and cross-section ratio $R_{3/2}$ are indicated in parentheses):

\begin{itemize}

\item {\it jet-pseudorapidity cut} - a change of $\pm$0.1 (corresponding to the resolution) 
      in the $\eta_{\rm LAB}^{\rm jet}$ cuts imposed on the jets in the laboratory frame for 
      both data and MC simulated events (1\%,1\%);

\item {\it jet transverse energy and invariant mass cuts} -  $E_{T,B}^{\rm jet}$
and $M_{\rm 2jets} (M_{\rm 3jets})$ 
      were simultaneously varied by the corresponding resolution near the cuts for both data and MC simulated events. 
      Along with the previous systematic check, this takes into account the 
      effect of the remaining differences 
      between the data and the MC simulation (3\%,3\%);
   
\item {\it use of different parton shower model} - using {\sc Ariadne} instead of {\sc Lepto} 
      to evaluate the acceptance corrections (2\%,4\%);
   
\item {\it the absolute energy scale of the CAL} - varying $E_{T,B}^{\rm jet}$ by its uncertainty of 
      $\pm1\% (\gre 10\gev)$ and $\pm3\% (\les 10\gev)$ for MC events \cite{pl:b547:164} (6\%,3.5\%).
\end{itemize}

The systematic uncertainties not associated with the absolute energy scale of the CAL were
added in quadrature to the statistical uncertainties and are shown on the figures as error bars.
The uncertainty due to the absolute energy scale of the CAL is highly correlated from bin-to-bin 
and is shown separately as a shaded band. The total systematic uncertainty and the uncertainty 
due to the absolute energy scale are also shown in Tables~\ref{et1}$-$\ref{rq2}. 


The main contributions to the theoretical uncertainties of the NLO QCD predictions are:
\begin{itemize}

 \item {\it uncertainties in the proton PDFs}, which were estimated by repeating the calculations
 using 40 additional sets obtained under different theoretical assumptions as part of the CTEQ6 release (2.5\%,2\%);
 
 \item {\it uncertainties in the correction factors, $C_{had}$}, which were estimated by using the 
      {\sc Ariadne} program instead of {\sc Lepto} (6\%,4\%);
      
 \item {\it uncertainties due to terms beyond NLO}, which were estimated by varying
      both $\mu_R$ and $\mu_F$ between $(\bar{E}_T^2+Q^2)$ and $(\bar{E}_T^2+Q^2)/16$ (10\%,7\%).
 
\end{itemize}
The total theoretical uncertainty was obtained by adding in quadrature
the individual uncertainties listed above.


\section{Results}
\label{results}

\subsection{Differential cross sections}
\label{cross}

The differential trijet cross sections as functions of $E_{T,B}^{\rm jet}$ are presented in Fig.~\ref{fig-e3} 
and in Tables~\ref{et1}$-$\ref{et3}.
The three highest $E_{T,B}^{\rm jet}$ jets were ordered in 
$E_{T,B}^{\rm jet}$ ($E_{T,B}^{\rm jet,1} \gre E_{T,B}^{\rm jet,2} \gre E_{T,B}^{\rm jet,3}$).
The observed decrease of the cross section for the first jet towards small values of $E_{T,B}^{\rm jet}$ 
is caused by the $E_{T,B}^{\rm jet}$ ordering combined with the requirement that the second and third jet have $E_{T,B}^{\rm jet} \gre 5\gev$.
For the second jet, a similar but less pronounced effect is observed. The NLO predictions using {\sc Nlojet}, 
corrected for hadronisation effects, are compared to the data in Fig.~~\ref{fig-e3}. The QCD predictions provide a good 
description of both the shape and magnitude of the measured cross sections, even at low $E_{T,B}^{\rm jet}$.

Figure~\ref{fig-h3} and Tables~\ref{eta1}$-$\ref{eta3} show the differential trijet cross sections 
as functions of $\eta_{\rm LAB}^{\rm jet}$. The three highest 
$E_{T,B}^{\rm jet}$ jets were ordered in $\eta_{\rm LAB}^{\rm jet}$ 
($\eta_{\rm LAB}^{\rm jet,1} \gre \eta_{\rm LAB}^{\rm jet,2} \gre \eta_{\rm LAB}^{\rm jet,3}$). 
Figure~\ref{fig-q1} and Tables~\ref{dq2} and ~\ref{tq2} show both the differential dijet and trijet cross section as functions of
$Q^2$. In Figs.~\ref{fig-h3} and \ref{fig-q1}, the data are generally well described by the NLO QCD predictions.
The largest difference is a slightly different slope of the $\eta_{\rm LAB}^{\rm jet}$ dependence of the third jet.


\subsection{Cross-section ratio and determination of $\alpha_s$}
\label{sec-alp}
Figure~\ref{fig-q2} and Table~\ref{rq2} show the cross-section ratio $R_{3/2}$
of the trijet cross section to the dijet cross section,
as a function of $Q^2$. The correlated systematic and the renormalisation scale uncertainties largely cancel
in the ratio. The agreement between the data and NLO predictions is good. The
total experimental and theoretical
uncertainties are about $5\%$ and $7\%$, respectively. These uncertainties are substantially reduced with respect to those of
the di- and trijet cross sections. In particular, at low $Q^2$ ($Q^2 < 100\gev^2$), the theoretical uncertainties are reduced by as
much as a factor of four.
This reduction allows the determination of $\alpha_s(M_Z)$ at a much lower $Q^2$
than in
previous analyses ~\cite{np:b700:3,pl:b547:164}.

The measurement of $R_{3/2}$ as a function of $Q^2$ was used to determine $\alpha_s(M_Z)$ with
a method similar to that of a previous ZEUS publication \cite{pl:b507:70}:
\begin{itemize}

\item the NLO QCD calculation of $R_{3/2}$ was 
    performed using the five sets of proton PDFs of the CTEQ4
    A-series
    ~\cite{pr:d55:1280} {\footnote {The CTEQ4 PDF was chosen 
    because the 
    CTEQ6 does not provide PDF sets obtained with different 
    $\alpha_s(M_Z)$ values and therefore cannot be used for the 
    determination of $\alpha_s$.}}.
    The value of $\alpha_s(M_{Z})$ used in each partonic
    cross-section calculation was that associated with the
    corresponding set of PDFs: 0.110, 0.113, 0.116, 0.119, 0.122; 

\item for each bin, $i$, in $Q^2$, the NLO QCD calculations, corrected for 
hadronisation effects, were used to parameterise the $\alpha_s(M_Z)$ 
dependence of $R_{3/2}$ according to the functional form:

\begin{equation}
[R_{3/2}(\alpha_s(M_Z))]^i = C^i_1 \cdot \alpha_s(M_Z) + C^i_2 \cdot \alpha_s^2(M_Z),
\label{eqr}
\end{equation}

where $C^i_1$ and $C^i_2$ are fitting parameters. This simple parameterisation gives a good description of the $\alpha_s(M_Z)$ 
dependence of $R_{3/2}(Q^2)$ over the entire $\alpha_s$ range spanned by the PDF sets;

\item a value of $\alpha_s(M_Z)$ was then determined in each bin of $Q^2$, 
as well as in the entire $Q^2$ region,
by a $\chi^2$-fit of the measured $R_{3/2}(Q^2)$ values using the 
parameterisation in Eq.~(\ref{eqr}). 
\end{itemize}

This procedure correctly handles the complete $\alpha_s$-dependence of
the NLO differential cross sections (the explicit dependence coming from the
partonic cross sections and the implicit dependence coming from the PDFs) in the
fit, while preserving the correlation between $\alpha_s$ and the PDFs.
Taking into account only the statistical uncertainties on the measured cross-section ratio,
$\alpha_s(M_{Z})$ is determined to be $\alpha_s(M_Z)=0.1179\plm0.0013(\rm stat.)$.

Figure~\ref{fig-alphas}a shows the sensitivity of the cross-section 
ratio $R_{3/2}$ to the value of $\alpha_s$. Figure~\ref{fig-alphas}b and 
Table~\ref{alphas} show $\alpha_s(M_Z)$
determined in the five bins of $Q^2$.

As a cross-check of the extracted value of $\alpha_s(M_Z)$, the fit procedure was
repeated by using the three sets of the MRST99 PDF corresponding to 
$\alpha_s (M_Z)$ equal to 0.1125, 0.1175 and 0.1225. 
The result is $\alpha_s(M_Z)=0.1178\plm0.0010(\rm stat.)$ 

In addition, the NLO QCD analysis used to obtain the ZEUS-S PDF ~\cite{pr:d67:012007}
was repeated to obtain a set
of five PDFs corresponding to the values of $\alpha_s(M_{Z})$: 0.115, 0.117,
0.119, 0.121, 0.123. 
These sets
were used in the current analysis yielding $\alpha_s(M_Z)=0.1191\plm0.0010(\rm stat.)$, in good agreement 
with the other determinations. 
 
The experimental and theoretical uncertainties of the extracted value of 
$\alpha_s(M_Z)$ were evaluated by repeating the analysis above for each systematic 
check, as described in Section~\ref{correction}. The main contributions to the
experimental systematic uncertainty were:
\begin{itemize}

\item {\it jet pseudorapidity cut} ($^{+1\%}_{-1.5\%}$);

\item {\it jet transverse energy and invariant mass cuts}  ($^{+0.5\%}_{-2\%}$);
   
\item {\it use of different parton shower model}  ($-2\%$) ;
   
\item {\it the absolute energy scale of the CAL}  ($^{+2\%}_{-2.5\%}$).
\end{itemize}

The main contributions to the theoretical uncertainty are: 
\begin{itemize}

 \item {\it uncertainties in the proton PDFs}  ($^{+1.5\%}_{-2\%}$);
 
 \item {\it uncertainties in the correction factor, $C_{\rm had}$}  ($+2\%$);
      
 \item {\it uncertainties due to terms beyond NLO} ($^{+5\%}_{-3.5\%}$).
 
\end{itemize}

The value of $\alpha_s(M_Z)$ as determined from the measurements of 
$R_{3/2}$ is therefore:

\begin{center}
$\alpha_s(M_Z) = 0.1179 \pm 0.0013~\rm(stat.)^{+0.0028}_{-0.0046}~\rm(exp.)^{+0.0064}_{-0.0046}~\rm(th.)$. 
\end{center}

The result is in good agreement with recent determinations at HERA 
\cite{pl:b363:201,pl:b507:70,pl:b547:164,pl:b558:41,pl:b560:7,np:b700:3,epj:c6:575,epj:c19:289,epj:c21:33} 
and the current world average of
$\alpha_s(M_Z)=0.1182\plm0.0027$~\cite{jp:g26:r27,*hep-ex-0407021}.

\section{Summary}
\label{sec-sum}
Differential dijet and trijet cross sections have been measured with high precision 
in neutral current deep inelastic scattering for $10<Q^2<5000\gev^2$ at HERA using the ZEUS detector. 
The inclusive trijet cross section has been measured as a
function of $E_{T,B}^{\rm jet}$, $\eta_{\rm LAB}^{\rm jet}$ and $Q^2$. The ratio $R_{3/2}$ of the
trijet and dijet cross sections has been measured
as a function of $Q^2$. The predictions of perturbative QCD calculations in
next-to-leading order give a good description of the dijet and trijet cross
sections and the cross-section ratio $R_{3/2}$ over the whole range of $Q^2$.
The cancellation of uncertainties in the ratio, in particular 
those from theory, allow the extraction of $\alpha_s$ with good
precision down to $Q^2$ of 10 GeV$^2$.
The value of the strong coupling constant $\alpha_s$ was measured to be 
$\alpha_s(M_Z) = 0.1179 \pm 0.0013~\rm(stat.)^{+0.0028}_{-0.0046}~\rm(exp.)^{+0.0064}_{-0.0046}~\rm(th.)$, 
in good agreement with the current world average value 
and previous determinations of $\alpha_s(M_Z)$ at HERA.


\section{Acknowlegements}
\label{sec-ack}
  
  We thank the DESY Directorate for their strong support and encouragement. The
  remarkable achievements of the HERA machine group were essential for the
  successful completion of this work and are greatly appreciated. 
  The design, construction and installation of the
  ZEUS detector has been made possible by the effort of many
  people who are not listed as authors.
  We would like
  to thank Z.~Nagy for useful discussions.


\vfill\eject

DESY-05-019.tex
%==============================================================================
%
%       <Insert ZEUS note title here>
%       _____________________________
%
%      Please consult the WWW LaTeX4zeus documentation for explanations
%      and usage information related to this LaTeX setup:
%
%      http://www-zeus.desy.de/ZEUS_ONLY/LaTeX4zeus/WWW/html/l4z.html
%
%==============================================================================
%
%       Use 'zeus_paper' class
%       ----------------------
%
%       Options specific to zeus_paper.cls:
%
%       Paper type (mutually exclusive):
%       -> zpreprint    - DESY preprint (always 12pt)
%       -> zdraft       - draft paper (always 12pt)
%       -> zconf        - paper contributed to a conference (always 12pt)
%       -> znote        - ZEUS note (default=12pt, 10pt or 11pt can be selected)
%
%       Bibliography style (mutually exclusive):
%       ->zbstdefault   - standard ZEUS BiBTeX style file (default)    
%       ->zbstdraft     - standard ZEUS BiBTeX style file for drafts
%                         (default for drafts)   
%       ->zbstepj       - ZEUS BiBTeX style file for Eur.J.Phys papers 
%       ->zbstnp        - ZEUS BiBTeX style file for Nucl.Phys. papers 
%       ->zbstpl        - ZEUS BiBTeX style file for Phys.Lett. papers 
%
\documentclass[zpreprint,zbstepj]{zeus_paper}
%
%       Use English language 
%       --------------------
%       (change "english" -> "american" to conform to American standards)
%
\usepackage[english]{babel}
%
%       Default auxiliary packages
%       --------------------------
%
%       The following packages are automatically loaded by zeus_paper.cls:
%       -> rotating     - to rotate boxes, pictures etc.
%       -> epsfig       - to include (encapsulated) postscript files
%       -> amsmath      - to use the AMS extended math package
%                         [centertags] -> equation numbers vertically centered
%                                         in split enevironment
%                         [intlimits]  -> integral limits on bottom and top
%                                         of integral sign
%       -> amssymb      - to obtain additional math symbols in AMS fonts
%       -> amscd        - to obtain AMS flow chart utilities
%       -> array        - to obtain additional tabular functionality
%       -> multirow     - for multirow-entries in tables
%       -> supertabular - for multi-page tables
%       -> dcolumn      - decimal-point aligned colums in tables
%       -> xspace       - to add empty space after commands
%       -> axodraw      - to draw Feynman diagrams (by J.A.M.Vermaseren)
%       -> upgreek      - provide upright greek letters
%       -> color        - color support
%       -> colors       - some color definitions
%       -> calc         - enhanced calculus in LaTeX macros
%       -> ifthen       - enhance logical structures in LaTeX macros
%       -> cite         - for multiple citations like [1-4] instead of [1,2,3,4]
%       -> mcite        - for multiple citations in one reference
%          
%       <Here you can include additional auxiliary packages:>
%       -----------------------------------------------------
%
%       use \usepackage[options]{package name}
%
%       File './LaTeX/zeus/zeus_def.tex' contains the standard set 
%                                        of TeX definitions etc.
%
\input zeus_def.tex
%
%       File './LaTeX/user/def.tex' contains additional user definitions
%
%\input def.tex
%
%       File 'zdraft-cit.tex' contains citation definitions
%
\input  DESY-05-019-cit.tex
%==============================================================================
%       Here comes the document.
%==============================================================================
\includeonly{DESY-05-019-tit,DESY-05-019-aut,DESY-05-019-txt,DESY-05-019-ref,DESY-05-019-fig,DESY-05-019-tab}                  
\begin{document}
%------------------------------------------------------------------------------
%       Title sheet
%------------------------------------------------------------------------------
\include{DESY-05-019-tit}
%------------------------------------------------------------------------------
%       Author list
%------------------------------------------------------------------------------
\include{DESY-05-019-aut}                                                                                      
%------------------------------------------------------------------------------
%       Text
%------------------------------------------------------------------------------
\include{DESY-05-019-txt}
%------------------------------------------------------------------------------
%       Bibliography
%------------------------------------------------------------------------------
\include{DESY-05-019-ref}
%------------------------------------------------------------------------------
%       Tables
%------------------------------------------------------------------------------
\include{DESY-05-019-tab}
%------------------------------------------------------------------------------
%       Figures
%------------------------------------------------------------------------------
\include{DESY-05-019-fig}
%
%       ... that's it
%
\end{document}

zeus_def.tex
%------------------------------------------------------------------------------
%       Standard ZEUS text fragments
%       ----------------------------
%       -> references system footnote
%------------------------------------------------------------------------------
\newcommand{\ZcoosysA}{%
The ZEUS coordinate system is a right-handed Cartesian system, with the $Z$
axis pointing in the proton beam direction, referred to as the ``forward
direction'', and the $X$ axis pointing left towards the center of HERA.
The coordinate origin is at the nominal interaction point.\xspace}
\newcommand{\ZcoosysB}{%
The ZEUS coordinate system is a right-handed Cartesian system, with the $Z$
axis pointing in the proton beam direction, referred to as the ``forward
direction'', and the $X$ axis pointing left towards the centre of HERA.
The coordinate origin is at the nominal interaction point.\xspace}
\newcommand{\Zpsrap}{}
\newcommand{\ZcoosysfnA}{\footnote{\ZcoosysA}}
\newcommand{\ZcoosysfnB}{\footnote{\ZcoosysB}}
\newcommand{\ZcoosysfnAeta}{\footnote{\ZcoosysA\Zpsrap}}
\newcommand{\ZcoosysfnBeta}{\footnote{\ZcoosysB\Zpsrap}}
%------------------------------------------------------------------------------
%       -> introductory sentence for detector description
%------------------------------------------------------------------------------
\newcommand{\Zdetdesc}{%
A detailed description of the ZEUS detector can be found 
elsewhere~\cite{zeus:1993:bluebook}. A brief outline of the 
components that are most relevant for this analysis is given
below.\xspace}
%------------------------------------------------------------------------------
%       -> CTD description (coordinate system footnote is given as argument)
%------------------------------------------------------------------------------
\newcommand{\Zctddesc}[1]{%
Charged particles are tracked in the central tracking detector (CTD)~\citeCTD,
which operates in a magnetic field of $1.43\Tesla$ provided by a thin 
superconducting solenoid. The CTD consists of 72~cylindrical drift chamber 
layers, organised in nine superlayers covering the polar-angle#1 region 
\mbox{$15^\circ<\theta<164^\circ$}. The transverse-momentum resolution for
full-length tracks is $\sigma(p_T)/p_T=0.0058p_T\oplus0.0065\oplus0.0014/p_T$,
with $p_T$ in $\Gev$.}
%------------------------------------------------------------------------------
%       -> CAL description
%------------------------------------------------------------------------------
\newcommand{\Zcaldesc}{%
The high-resolution uranium--scintillator calorimeter (CAL)~\citeCAL consists 
of three parts: the forward, the barrel and the rear 
calorimeters. 
The smallest subdivision of
the calorimeter is called a cell.  The CAL energy resolutions, as measured under
test-beam conditions, are $\sigma(E)/E=0.18/\sqrt{E}$ for electrons and
$\sigma(E)/E=0.35/\sqrt{E}$ for hadrons, with $E$ in $\Gev$.}
%------------------------------------------------------------------------------
%       Handy macros, mainly for text mode:
%       -----------------------------------
%       -> unknown marker
%------------------------------------------------------------------------------
\newcommand{\xxx}{\hbox{$\qsm\qsm\qsm$}\xspace}
%------------------------------------------------------------------------------
%       -> D0 acronym
%------------------------------------------------------------------------------
\newcommand{\DO}{{D{\O}}\xspace}
%------------------------------------------------------------------------------
%       -> (anti-)something with correct hyphenation
%------------------------------------------------------------------------------
\newcommand{\Anti}[1]{\hbox{(anti-)}\discretionary{}{}{}#1}
\newcommand{\anti}[1]{(anti\discretionary{-)}{}{)}#1}
%------------------------------------------------------------------------------
%       -> characters for typewriter font (used for http addresses)
%------------------------------------------------------------------------------
\chardef\usc=95
\chardef\til=126
%------------------------------------------------------------------------------
%       -> xdotspace (like xspace, but adds period unless
%          another period follows)
%       -> "nn% C.L."
%------------------------------------------------------------------------------
\catcode`\@=11 % @ signs are now treated as letters
\DeclareRobustCommand\xdotspace{\futurelet\@let@token\@xdotspace}
\def\@xdotspace{%
  \ifx\@let@token.\else
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  \ifx\@let@token\/.\else
  \ifx\@let@token\ .\else
  \ifx\@let@token~.\else
  \ifx\@let@token!.\else
  \ifx\@let@token,.\else
  \ifx\@let@token:.\else
  \ifx\@let@token;.\else
  \ifx\@let@token?.\else
  \ifx\@let@token/.\else
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  \ifx\@let@token\@sptoken.\else
   .\space
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\catcode`\@=12 % @ signs are no longer letters
\newcommand{\CL}[1]{$#1\%$~C.L\xdotspace}
%------------------------------------------------------------------------------
%       -> vertical rule of width zero and variable height & depth
%------------------------------------------------------------------------------
\newcommand{\stru}[2]{%
   \relax\ifmmode\hbox{\vrule height#1 depth#2 width0pt}%
   \else\vrule height#1 depth#2 width0pt\fi}
%------------------------------------------------------------------------------
%       -> underline with resonable distance text - line
%------------------------------------------------------------------------------
\newcommand{\uline}[1]{$\underline{\hbox{#1\stru{0.pt}{.485ex}}}$}
%------------------------------------------------------------------------------
%       -> roman numbers (uppercase and lowercase)
%------------------------------------------------------------------------------
\newcommand{\Ronum}[1]{\uppercase\expandafter{\romannumeral#1}}
\newcommand{\ronum}[1]{\expandafter{\romannumeral#1}}
%------------------------------------------------------------------------------
%       -> definition for LaTeX4zeus
%------------------------------------------------------------------------------
\DeclareRobustCommand{\LaTeXZ}{%
  \LaTeX\kern-.05em4\kern-.1em
  {\raisebox{-0.2ex}{$\scriptstyle\text{ZEUS}$}}\xspace}
%------------------------------------------------------------------------------
%       -> umlaut definitions for references (needed since BiBTeX 
%          MACRO strings cannot contain quotes)
%------------------------------------------------------------------------------
\newcommand{\Auml}{\"A}
\newcommand{\Ouml}{\"O}
\newcommand{\Uuml}{\"U}
\newcommand{\auml}{\"a}
\newcommand{\ouml}{\"o}
\newcommand{\uuml}{\"u}
%------------------------------------------------------------------------------
%       -> shorthands for equation, figure, table, section references
%          (require standard use of labels: eq-... for equations, cha-...
%           for chapters, sec-... for (sub)sections, tab-... for tables
%           and fig-... for figures)
%------------------------------------------------------------------------------
\newcommand{\eq}[1]{(\ref{eq-#1})}
\newcommand{\eqsand}[2]{(\ref{eq-#1}) and~(\ref{eq-#2})}
\newcommand{\eqsto}[2]{(\ref{eq-#1})--(\ref{eq-#2})}
\newcommand{\eqstwo}[2]{(\ref{eq-#1},\ref{eq-#2})}
\newcommand{\eqsthr}[3]{(\ref{eq-#1},\ref{eq-#2},\ref{eq-#3})}
\newcommand{\eqsfou}[4]{(\ref{eq-#1},\ref{eq-#2},\ref{eq-#3},\ref{eq-#4})}
\newcommand{\Eq}[1]{Equation~(\ref{eq-#1})}
\newcommand{\Eqsto}[2]{Equations~(\ref{eq-#1})--(\ref{eq-#2})}
\newcommand{\fig}[1]{Fig.~\ref{fig-#1}}
\newcommand{\Fig}[1]{Figure~\ref{fig-#1}}
\newcommand{\figand}[2]{Figs.~\ref{fig-#1} and~\ref{fig-#2}}
\newcommand{\Figand}[2]{Figures~\ref{fig-#1} and~\ref{fig-#2}}
\newcommand{\tab}[1]{Table~\ref{tab-#1}}
\newcommand{\Tab}[1]{Table~\ref{tab-#1}}
\newcommand{\taband}[2]{Tables~\ref{tab-#1} and~\ref{tab-#2}}
\newcommand{\cha}[1]{Chap.~\ref{cha-#1}}
\newcommand{\Cha}[1]{Chapter~\ref{cha-#1}}
\newcommand{\sect}[1]{Sect.~\ref{sec-#1}}
\newcommand{\Sect}[1]{Section~\ref{sec-#1}}
\newcommand{\sectand}[2]{Sects.~\ref{sec-#1} and~\ref{sec-#2}}
\newcommand{\sectto}[2]{Sects.~\ref{sec-#1} to~\ref{sec-#2}}
\newcommand{\Sectand}[2]{Sections~\ref{sec-#1} and~\ref{sec-#2}}
\newcommand{\Sectto}[2]{Sections~\ref{sec-#1} to~\ref{sec-#2}}
%------------------------------------------------------------------------------
%       -> log file message
%------------------------------------------------------------------------------
\newcommand{\LogMess}[1]{\typeout{^^J==> #1^^J}}
%------------------------------------------------------------------------------
%       Some special symbols and settings for math mode:
%       ------------------------------------------------
%       -> redefine mathbf
%------------------------------------------------------------------------------
\DeclareMathAlphabet{\mathbf}{OT1}{cmr}{bx}{sl}
%------------------------------------------------------------------------------
%       -> physics units (eV and multiples)
%------------------------------------------------------------------------------
\newcommand{\eVdist}{\kern-0.06667em}
\newcommand{\Ev}{{\text{e}\eVdist\text{V\/}}}
\newcommand{\Kev}{{\text{ke}\eVdist\text{V\/}}}
\newcommand{\Mev}{{\text{Me}\eVdist\text{V\/}}}
\newcommand{\Gev}{{\text{Ge}\eVdist\text{V\/}}}
\newcommand{\Tev}{{\text{Te}\eVdist\text{V\/}}}
\newcommand{\ev}{{\,\text{e}\eVdist\text{V\/}}}
\newcommand{\kev}{{\,\text{ke}\eVdist\text{V\/}}}
\newcommand{\mev}{{\,\text{Me}\eVdist\text{V\/}}}
\newcommand{\gev}{{\,\text{Ge}\eVdist\text{V\/}}}
\newcommand{\tev}{{\,\text{Te}\eVdist\text{V\/}}}
%------------------------------------------------------------------------------
%       -> various other units
%------------------------------------------------------------------------------
\newcommand{\mb}{\,\text{mb}}
\newcommand{\mub}{\,\upmu\text{b}}
\newcommand{\nb}{\,\text{nb}}
\newcommand{\pb}{\,\text{pb}}
\newcommand{\fb}{\,\text{fb}}
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\newcommand{\mbi}{\,\text{mb}^{-1}}
\newcommand{\nbi}{\,\text{nb}^{-1}}
\newcommand{\pbi}{\,\text{pb}^{-1}}
\newcommand{\fbi}{\,\text{fb}^{-1}}
\newcommand{\met}{\,\text{m}}
\newcommand{\nm}{\,\text{nm}}
\newcommand{\mum}{\,\upmu\text{m}}
\newcommand{\mm}{\,\text{mm}}
\newcommand{\cm}{\,\text{cm}}
\newcommand{\km}{\,\text{km}}
\newcommand{\Amp}{\,\text{A}}
\newcommand{\nA}{\,\text{nA}}
\newcommand{\muA}{\,\upmu\text{A}}
\newcommand{\mA}{\,\text{mA}}
\newcommand{\kA}{\,\text{kA}}
\newcommand{\Hz}{\,\text{Hz}}
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\newcommand{\ppm}{\,\text{ppm}}
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\newcommand{\gra}{\,\text{g}}
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\newcommand{\nang}{\,\text{ng}}
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\newcommand{\mg}{\,\text{mg}}
\newcommand{\kg}{\,\text{kg}}
\newcommand{\Tesla}{\,\text{T}}
\newcommand{\Kelvin}{\,\text{K}}
%------------------------------------------------------------------------------
%       -> additional operators
%------------------------------------------------------------------------------
\newcommand{\odiv}{{\operatorname*{div}}}
\newcommand{\ograd}{{\operatorname*{grad}}}
\newcommand{\orot}{{\operatorname*{rot}}}
\newcommand{\odiag}{{\operatorname*{diag}}}
\newcommand{\ocov}{{\operatorname*{cov}}}
%------------------------------------------------------------------------------
%       -> fraction with slash instead of ratio bar
%------------------------------------------------------------------------------
\newcommand{\slashfrac}[2]{%
  \raisebox{0.5ex}{\ensuremath #1}\kern-0.12em/\kern-0.08em
  \raisebox{-.8ex}{\ensuremath #2}}
%------------------------------------------------------------------------------
%       -> shorthand for "integral limits below and above"
%------------------------------------------------------------------------------
\newcommand{\intl}{\int\limits}
\newcommand{\ointl}{\oint\limits}
%------------------------------------------------------------------------------
%       -> d'Alembert operator
%------------------------------------------------------------------------------
\newcommand{\sqr}[3]{%
    {\vcenter{\hrule height.#3ex\hbox{\vrule width.#2ex height#1ex
     \kern#1ex\vrule width.#3ex}\hrule height.#2ex}}}
\newcommand{\dalem}{\mathchoice\sqr{1.3}{08}{24}\sqr{1.4}{08}{24}
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%------------------------------------------------------------------------------
%       -> vectors and matrices
%------------------------------------------------------------------------------
\newcommand{\vect}[1]{\begin{matrix}#1\end{matrix}}
\newcommand{\pvect}[1]{\begin{pmatrix}#1\end{pmatrix}}
\newcommand{\bvect}[1]{\begin{bmatrix}#1\end{bmatrix}}
\newcommand{\vvect}[1]{\begin{vmatrix}#1\end{vmatrix}}
\newcommand{\Vvect}[1]{\begin{Vmatrix}#1\end{Vmatrix}}
%------------------------------------------------------------------------------
%       -> antiparticles and particles with a bar in parentheses
%------------------------------------------------------------------------------
\newcommand{\widebar}[1]{%
   \mkern1.5mu\overline{\mkern-1.5mu#1\mkern-1.mu}\mkern1.mu}
\catcode`\@=11 % @ signs are now treated as letters
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\catcode`\@=12 % @ signs are no longer letters
\newcommand{\nuan}{\parenbar{\nu}}
\newcommand{\nunubar}{\parenbar{\nu}}
\newcommand{\ppbar}{\parenbar{p}}
\newcommand{\qqbar}{\parenbar{q}}
\newcommand{\pbar}{\widebar{p}}
\newcommand{\nbar}{\widebar{n}}
\newcommand{\qbar}{\widebar{q}}
\newcommand{\dbar}{\widebar{d}}
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\newcommand{\cbar}{\widebar{c}}
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\newcommand{\Hebar}{\widebar{\text{He}}}
\newcommand{\Cbar}{\widebar{\text{C}}}
\newcommand{\Abar}{\widebar{\text{A}}}
\newcommand{\Kbar}{\widebar{K}}
\newcommand{\chibar}{\widebar{\chi}}
\newcommand{\LQbar}{\widebar{\rm LQ}}
%------------------------------------------------------------------------------
%       -> small numbers
%------------------------------------------------------------------------------
\newcommand{\zero}{{\scriptscriptstyle 0}}
\newcommand{\sone}{{\scriptscriptstyle 1}}
\newcommand{\stwo}{{\scriptscriptstyle 2}}
\newcommand{\sthr}{{\scriptscriptstyle 3}}
\newcommand{\sfou}{{\scriptscriptstyle 4}}
\newcommand{\sfiv}{{\scriptscriptstyle 5}}
\newcommand{\ssix}{{\scriptscriptstyle 6}}
\newcommand{\ssev}{{\scriptscriptstyle 7}}
\newcommand{\seig}{{\scriptscriptstyle 8}}
\newcommand{\snin}{{\scriptscriptstyle 9}}
\newcommand{\sten}{{\scriptscriptstyle 10}}
\newcommand{\half}{{\scriptscriptstyle 1/2}}
%------------------------------------------------------------------------------
%       -> abbreviations for common math mode symbols
%------------------------------------------------------------------------------
\newcommand{\MSbar}{\hbox{$\overline{\rm MS}$}\xspace}
\newcommand{\UO}{{\rm U}(1)_\sY}
\newcommand{\SUT}{{\rm SU}(2)_\sL}
\newcommand{\SUTH}{{\rm SU}(3)_c}
\newcommand{\SUTUO}{{\rm SU}(2)_\sL\times{\rm U}(1)_\sY}
\newcommand{\SUTSUTUO}{{\rm SU}(3)_c\times{\rm SU}(2)_\sL\times{\rm U}(1)_\sY}
\newcommand{\alsmu}[1]{{\alpha_s(\mu_{#1}^2)}}
\newcommand{\alsmz}{{\alpha_s(M_Z^2)}}
\newcommand{\alsqs}{{\alpha_s(Q^2)}}
\newcommand{\als}{\alpha_s}
\newcommand{\ctwb}{\cos\theta_W}
\newcommand{\ctws}{\cos^2\theta_W}
\newcommand{\diff}{{\rm d}}
\newcommand{\gh}{{\gamma_h}}
\newcommand{\lsq}[1]{{\lambda_{#1}}}
\newcommand{\lsqp}[1]{{\lambda'_{#1}}}
\newcommand{\lsqpp}[1]{{{\lambda''}_{#1}}}
\newcommand{\rnge}{\hbox{$\,\text{--}\,$}}
\newcommand{\sihat}{{\hat\sigma}}
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\newcommand{\stwss}{\sin^4\theta_W}
\newcommand{\stws}{\sin^2\theta_W}
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\newcommand{\tilR}{\tilde R}
\newcommand{\tilS}{\tilde S}
\newcommand{\tilU}{\tilde U}
\newcommand{\tilV}{\tilde V}
\newcommand{\tl}{{\theta_\ell}}
\newcommand{\tw}{\theta_W}
%------------------------------------------------------------------------------
%       -> roman character combinations to be used in math mode, mainly
%          sub- or superscripts
%------------------------------------------------------------------------------
\newcommand{\Born}{{\rm Born}}
\newcommand{\BR}{{\rm BR}}
\newcommand{\CC}{{\rm CC}}
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\newcommand{\tot}{{\rm tot}}
\newcommand{\true}{{\rm true}}
%------------------------------------------------------------------------------
%       -> some calligraphic symbols and applications
%------------------------------------------------------------------------------
\newcommand{\Cor}{{\cal C}}
\newcommand{\F}{{\cal F}}
\newcommand{\Lumi}{{\cal L}}
\newcommand{\Prob}{{\cal P}}
\newcommand{\ord}[1]{{\cal O}(#1)}
\newcommand{\ordlr}[1]{{\cal O}\left(#1\right)}
%------------------------------------------------------------------------------
%       -> some math symbols (+,-,...) for usage as mathchar's
%------------------------------------------------------------------------------
\mathchardef\qsm=63
\mathchardef\pls=43
\mathchardef\mns=512
\mathchardef\plm=518
\mathchardef\eql=61
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\mathchardef\smallright=301
\mathchardef\les=316
\mathchardef\gre=318
\mathchardef\leq=532
\mathchardef\grq=533
%------------------------------------------------------------------------------
%       Additional functionality for tables and figures
%       -----------------------------------------------
%       -> alignment tools for tables
%------------------------------------------------------------------------------
\newcommand{\cA}{{\phantom{0}}}
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\newcommand{\cD}{{\phantom{0000}}}
\newcommand{\cAp}{{\phantom{0.}}}
\newcommand{\cM}{{\phantom{\mns}}}
\newcommand{\cP}{{\phantom{\pls}}}
%------------------------------------------------------------------------------
%       -> comments on figures (allow to add axodraw and picture items)
%------------------------------------------------------------------------------
\catcode`\@=11 % @ signs are now treated as letters
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{#4}
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}
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DESY-05-019.bbl
\providecommand{\etal}{et al.\xspace}
\providecommand{\coll}{Coll.\xspace}
\catcode`\@=11
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