Omm$#2\text{\,}#\mathrm{3}$ Om$#\mathrm{2}$

11institutetext: ${}^{\ref{902889}}$INFN Sezione di Torino, Italy;
${}^{\ref{902953}}$Physics Division, Lawrence Berkeley National Laboratory,
United States; ${}^{\ref{943385}}$Università di Pavia, Italy;
${}^{\ref{902682}}$Department of Physics, Brandeis University, United States;
${}^{\ref{912409}}$Center for High Energy Physics (CHEP-FU), Fayoum
University, Egypt; ${}^{\ref{902884}}$INFN Sezione di Padova,Italy;
${}^{\ref{902885}}$INFN Sezione di Pavia, Italy; ${}^{\ref{902888}}$INFN
Sezione di Trieste, Italy; ${}^{\ref{903113}}$Dipartimento di Fisica e
Astronomia, Universit’a di Padova, Italy; ${}^{\ref{902877}}$INFN Sezione di
Bari, Italy; ${}^{\ref{906718}}$European Centre for Theoretical Studies in
Nuclear Physics and Related Areas (ECT*), Italy; ${}^{\ref{1241166}}$MPS
School, University of Sussex, United Kingdom; ${}^{\ref{905303}}$Laboratório
de Instrumentação e Física Experimental de Partículas (LIP), Portugal;
${}^{\ref{902660}}$Department of Physics, Università degli Studi di Bari,
Italy; ${}^{\ref{902882}}$INFN Sezione di Milano, Italy;
${}^{\ref{903349}}$University of Wisconsin, United States;
${}^{\ref{902689}}$Brookhaven National Laboratory, United States;
${}^{\ref{908452}}$Department of Physics, Bolu Abant Izzet Baysal University,
Turkey; ${}^{\ref{906528}}$Dipartimento di Matematica e Fisica, Università
Roma Tre, Italy; ${}^{\ref{1276460}}$Helmholtz-Zentrum Dresden-Rossendorf,
Germany; ${}^{\ref{1623978}}$University of Tennessee, United States;
${}^{\ref{902796}}$Fermi National Accelerator Laboratory, United States;
${}^{\ref{902912}}$Department of Physics and Astronomy, United States;
${}^{\ref{903603}}$Peking University, China; ${}^{\ref{902868}}$Imperial
College London, United Kingdom; ${}^{\ref{910783}}$Centre for Cosmology,
Particle Physics and Phenomenology (CP3), Université Catholique de Louvain,
Belgium; ${}^{\ref{902770}}$Deutsches Elektronen-Synchrotron DESY, Germany;
${}^{\ref{903112}}$Department of Physics, Oxford University, United Kingdom;
${}^{\ref{902725}}$CERN, Switzerland; ${}^{\ref{1210798}}$State Key Laboratory
of Nuclear Physics and Technology, Peking University, China;
${}^{\ref{903290}}$TRIUMF, Canada; ${}^{\ref{903702}}$Sun Yat-sen University,
China; ${}^{\ref{902858}}$Institut für Physik, Humboldt-Universität zu Berlin,
Germany; ${}^{\ref{912328}}$INFN-TIFPA Trento Institute of Fundamental Physics
and Applications, Italy; ${}^{\ref{907692}}$INFN Sezione di Roma Tre, Italy;
${}^{\ref{903174}}$STFC, United Kingdom; ${}^{\ref{903123}}$Institute of High-
Energy Physics, China; ${}^{\ref{903156}}$Physics & Astronomy Department, Rice
University, United States; ${}^{\ref{902990}}$Maryland Center for Fundamental
Physics, University of Maryland, United States; ${}^{\ref{907623}}$DAMTP,
University of Cambridge, United Kingdom; ${}^{\ref{902887}}$INFN Sezione di
Roma, Italy; ${}^{\ref{1237813}}$Center for Theoretical Physics, Massachusetts
Institute of Technology, United States; ${}^{\ref{910429}}$YITP, Stony Brook,
United States; ${}^{\ref{904416}}$SISSA International School for Advanced
Studies, Italy; ${}^{\ref{903734}}$Department of Physics, University of
Warwick, United Kingdom; ${}^{\ref{905268}}$INFN Sezione di Ferrara, Italy;
${}^{\ref{911708}}$The University of Huddersfield, United Kingdom;
${}^{\ref{902881}}$INFN Sezione di Genova, Italy;
${}^{\ref{903196}}$Department of Physics and Astronomy, University of
Sheffield, United Kingdom; ${}^{\ref{903128}}$Scuola Normale Superiore, Italy;
${}^{\ref{902886}}$INFN Sezione di Pisa, Italy; ${}^{\ref{926589}}$CNRS/IN2P3,
France; ${}^{\ref{908474}}$Perimeter Institute, Canada;
${}^{\ref{903168}}$Dipartimento di Fisica, Sapienza Univ. Roma, Italy;
${}^{\ref{902879}}$INFN Sezione di Catania, Italy;
${}^{\ref{902835}}$Department of Physics, Harvard University, United States;
${}^{\ref{912490}}$IRFU, CEA, Université Paris-Saclay, France;
${}^{\ref{1471035}}$Theoretical Particle Physics Laboratory (LPTP), Institute
of Physics, EPFL, Switzerland; ${}^{\ref{1275736}}$Physics and Astronomy
Department, Georgia State University, United States;
${}^{\ref{903307}}$University of California, Santa Barbara, United States;
${}^{\ref{903370}}$University of Zurich, Switzerland;
${}^{\ref{903282}}$Department of Physics, University of Toronto, Canada;
${}^{\ref{1087875}}$Universitè Paris Saclay, CNRS, CEA, Institut de Physique
Thèorique, France; ${}^{\ref{903836}}$CAFPE and Departamento de Física Teórica
y del Cosmos, Universidad de Granada, Spain; ${}^{\ref{907142}}$Laboratori
Acceleratori e Superconduttività Applicata (LASA), INFN, Italy;
${}^{\ref{1743848}}$IP2I, Université Lyon 1, CNRS/IN2P3, France;
${}^{\ref{903085}}$University of Notre Dame, United States;
${}^{\ref{902874}}$Physics Department, Indiana University, United States;
${}^{\ref{903305}}$University of California San Diego, United States;
${}^{\ref{902825}}$Chalmers University of Technology, Sweden;
${}^{\ref{903075}}$Radboud University and Nikhef, The Netherlands;
${}^{\ref{902948}}$University of Lancaster, Department of Physics, United
Kingdom; ${}^{\ref{1508424}}$Laboratoire Charles Coulomb, CNRS and University
of Montpellier, France; ${}^{\ref{903287}}$Physics Department, University of
Trieste, Italy; ${}^{\ref{903889}}$Kavli Institute for Theoretical Physics,
University of California, United States; ${}^{\ref{903237}}$SUNY at Stony
Brook, United States; ${}^{\ref{902999}}$School of Physics, University of
Melbourne, Australia; ${}^{\ref{903299}}$UC Berkeley, United States;
${}^{\ref{902668}}$Albert Einstein Center for Fundamental Physics, Institute
for Theoretical Physics, University of Bern, Switzerland;
${}^{\ref{903628}}$Department of Physics, Fudan University, China;
${}^{\ref{903094}}$Oklahoma State University, United States;
${}^{\ref{903130}}$University of Pittsburgh, United States;
${}^{\ref{902893}}$Iowa State University, United States;
${}^{\ref{902841}}$Max-Planck-Institut für Kernphysik, Germany;
${}^{\ref{902867}}$Department of Physics, University of Illinois at Urbana-
Champaign, United States; ${}^{\ref{902865}}$Illinois Institute of Technology,
United States; ${}^{\ref{903203}}$Department of Physics, University of Siegen,
Germany; ${}^{\ref{903832}}$Nikhef National Institute for Subatomic Physics,
The Netherlands; ${}^{\ref{903940}}$College of Engineering, Design and
Physical Sciences, Brunel University, United Kingdom;
${}^{\ref{903010}}$School of Physics and Astronomy, University of Minnesota,
United States; ${}^{\ref{903142}}$Purdue University, United States;
${}^{\ref{904187}}$Shandong University, China; ${}^{\ref{903009}}$Dipartimento
di Fisica Aldo Pontremoli, Universitá degli Studi di Milano, Italy;
${}^{\ref{902804}}$Physics Department, University of Florida, United States;
${}^{\ref{902711}}$California Institute of Technology, United States;
${}^{\ref{16750}}$Princeton University, United States; ${}^{\ref{903129}}$Pisa
University, Italy; ${}^{\ref{1513358}}$Dipartamento di Fisica e Astronomia “G.
Galilei", Università di Padova, Italy; ${}^{\ref{902748}}$Department of
Physics, University of Colorado, United States; ${}^{\ref{903326}}$University
of Vienna, Faculty of Physics, Austria; ${}^{\ref{902880}}$INFN Sezione di
Firenze, Italy; ${}^{\ref{902803}}$Florida State University, United States;
${}^{\ref{904961}}$JLab, United States; ${}^{\ref{902813}}$Départment de
Physique Théorique, Université de Genève, Switzerland;
${}^{\ref{902678}}$Rudjer Boskovic Institute, Croatia;
${}^{\ref{902756}}$Institute of Nuclear Physics – Polish Academy of Sciences
(IFJ PAN), Poland; ${}^{\ref{1189711}}$Lawrence Berkeley National Laboratory,
United States; ${}^{\ref{411233}}$International Institute of Physics,
Universidade Federal do Rio Grande do Norte, Brazil;
${}^{\ref{908583}}$Laboratoire de Physique Théorique et Hautes Énergies,
Sorbonne Université, CNRS, France; ${}^{\ref{909079}}$CAFPE, Spain;
${}^{\ref{903324}}$Institut für Hochenergiephysik der Österreichischen
Akademie der Wissenschaften, Austria; ${}^{\ref{903006}}$MIchigan State
University, United States; ${}^{\ref{903895}}$CAS Key Laboratory of
Theoretical Physics, Insitute of Theoretical Physics, Chinese Academy of
Sciences, P.R.China; ${}^{\ref{902823}}$School of Physics and Astronomy,
University of Glasgow, United Kingdom; ${}^{\ref{1218068}}$SCIPP, UC Santa
Cruz, United States; ${}^{\ref{1273509}}$University of Oklahoma, United
States; ${}^{\ref{902647}}$University of Arizona, United States;
${}^{\ref{912511}}$Maryland Center for Fundamental Physics, University of
Maryland, United States; ${}^{\ref{903302}}$University of California Irvine,
United States; ${}^{\ref{902883}}$Universitá di Napoli “Federico II" and INFN
Napoli, Italy; ${}^{\ref{902984}}$University of Manchester, United Kingdom;
${}^{\ref{902729}}$Department of Physics, University of Chicago, United
States; ${}^{\ref{904214}}$Physics, SUPA, United Kingdom;
${}^{\ref{902916}}$High Energy Accelerator Research Organization KEK, Japan;
${}^{\ref{903304}}$Department of Physics and Astronomy, University of
California, United States; ${}^{\ref{907907}}$Instituto de Física Corpuscular,
CSIC-Universitat de Valéncia, Spain; ${}^{\ref{1111512}}$Institut für
Allgemeine Elektrotechnik, Universität Rostock, Germany;
${}^{\ref{903092}}$Ohio State University, United States;
${}^{\ref{907960}}$Dipartimento di Fisica, Università Milano Bicocca, Italy;
${}^{\ref{903119}}$LPNHE, Sorbonne Université, France; ${}^{\ref{905405}}$Paul
Scherrer Institute, Switzerland; ${}^{\ref{903170}}$Royal Holloway University
of London, Department of Physics, United Kingdom;
${}^{\ref{902674}}$Dipartimento di Fisica e Astronomia, Università di Bologna,
Italy; ${}^{\ref{904536}}$Atominstitut, Technische Universität Wien, Austria

 https://muoncollider.web.cern.ch

#  Simulated Detector Performance at the Muon Collider Submitted to the
Proceedings of the US Community Study on the Future of Particle Physics
(Snowmass 2021)
This is one of the five reports submitted to Snowmass by the muon colliders
community at large. The reports’ preparation effort has been coordinated by
the International Muon Collider Collaboration. Authors and Signatories have
been collected with a subscription page, and are defined as follows:  • An
“Author” contributed to the results documented in the report in any form,
including e.g. by participating to the discussions of the community meetings
and sending comments on the draft, or plans to contribute to the future work.
• A “Signatory” expresses support to the efforts described in the report and
endorses the Collaboration plans.

Editors:
N. Bartosik${}^{\ref{902889}}$ K. Krizka${}^{\ref{902953}}$ S. Pagan
Griso${}^{\ref{902953}}$
Authors:
C. Aimè${}^{\ref{943385},\ref{902885}}$ A. Apyan${}^{\ref{902682}}$ M.A.
Mahmoud.${}^{\ref{912409}}$ A. Bertolin${}^{\ref{902884}}$ A.
Braghieri${}^{\ref{902885}}$ L. Buonincontri${}^{\ref{902884},\ref{903113}}$
S. Calzaferri${}^{\ref{902885}}$ M. Casarsa${}^{\ref{902888}}$ L.
Castelli${}^{\ref{903113}}$ M.G. Catanesi${}^{\ref{902877}}$ F.G.
Celiberto${}^{\ref{906718},\ref{912328}}$ A. Cerri${}^{\ref{1241166}}$ G.
Chachamis${}^{\ref{905303}}$ A. Colaleo${}^{\ref{902660},\ref{902877}}$ C.
Curatolo${}^{\ref{902882}}$ G. Da Molin${}^{\ref{903113}}$ S.
Dasu${}^{\ref{903349}}$ D. Denisov${}^{\ref{902689}}$ H.
Denizli${}^{\ref{908452}}$ B. Di Micco${}^{\ref{906528},\ref{907692}}$ T.
Dorigo${}^{\ref{902884}}$ F. Errico${}^{\ref{902660},\ref{902877}}$ A.
Ferrari${}^{\ref{1276460}}$ D. Fiorina${}^{\ref{902885}}$ M.
Gallinaro${}^{\ref{905303}}$ L. Giambastiani${}^{\ref{903113},\ref{902884}}$
A. Gianelle${}^{\ref{902884}}$ C. Giraldin${}^{\ref{903113}}$ M.
Herndon${}^{\ref{903349}}$ T.R. Holmes${}^{\ref{1623978}}$ S.
Jindariani${}^{\ref{902796}}$ G.K. Krintiras${}^{\ref{902912}}$ L.
Lee${}^{\ref{1623978}}$ Q. Li${}^{\ref{903603}}$ R. Lipton${}^{\ref{902796}}$
S. Lomte${}^{\ref{903349}}$ K.R. Long${}^{\ref{902868},\ref{903174}}$ D.
Lucchesi${}^{\ref{903113},\ref{902884}}$ P. Mastrapasqua${}^{\ref{910783}}$ F.
Meloni${}^{\ref{902770}}$ A. Montella${}^{\ref{902888}}$ F.
Nardi${}^{\ref{903113},\ref{902884}}$ N. Pastrone${}^{\ref{902889}}$ A.
Pellecchia${}^{\ref{902660}}$ K. Potamianos${}^{\ref{903112}}$ E.
Radicioni${}^{\ref{902877}}$ R. Radogna${}^{\ref{902660},\ref{902877}}$ C.
Riccardi${}^{\ref{943385},\ref{902885}}$ L. Ristori${}^{\ref{902796}}$ P.
Salvini${}^{\ref{902885}}$ I. Sarra D. Schulte${}^{\ref{902725}}$ A.
Senol${}^{\ref{908452}}$ L. Sestini${}^{\ref{902884}}$ F. M.
Simone${}^{\ref{902660},\ref{902877}}$ R. Simoniello${}^{\ref{902725}}$ A.
Stamerra${}^{\ref{902660},\ref{902877}}$ X. Sun${}^{\ref{1210798}}$ M.
Swiatlowski${}^{\ref{903290}}$ J. Tang${}^{\ref{903702},\ref{903123}}$ E. A.
Thompson${}^{\ref{902770}}$ I. Vai${}^{\ref{902885}}$ M.
Valente${}^{\ref{903290}}$ N. Valle${}^{\ref{943385},\ref{902885}}$ R.
Venditti${}^{\ref{902660},\ref{902877}}$ P. Verwilligen${}^{\ref{902877}}$ H.
Weber${}^{\ref{902858}}$ A. Zaza${}^{\ref{902660},\ref{902877}}$ D.
Zuliani${}^{\ref{903113},\ref{902884}}$
Signatories:
D. Acosta${}^{\ref{903156}}$ K. Agashe${}^{\ref{902990}}$ B.C.
Allanach${}^{\ref{907623}}$ P. Andreetto${}^{\ref{902884}}$ F.
Anulli${}^{\ref{902887}}$ A. Apresyan${}^{\ref{902796}}$ P.
Asadi${}^{\ref{1237813}}$ D. Athanasakos${}^{\ref{910429}}$ A.
Azatov${}^{\ref{904416},\ref{902888}}$ J.J. Back${}^{\ref{903734}}$ L.
Bandiera${}^{\ref{905268}}$ R. J. Barlow${}^{\ref{911708}}$ E.
Barzi${}^{\ref{902796},\ref{903092}}$ F. Batsch${}^{\ref{902725}}$ M.
Bauce${}^{\ref{902887},\ref{903168}}$ J. S. Berg${}^{\ref{902689}}$ J.
Berryhill${}^{\ref{902796}}$ A. Bersani${}^{\ref{902881}}$ K.M.
Black${}^{\ref{903349}}$ M. Bonesini${}^{\ref{902882},\ref{907960}}$ C.
Booth${}^{\ref{903196}}$ S. Bottaro${}^{\ref{903128},\ref{902886}}$ D.
Bowring${}^{\ref{902796}}$ A. Bross${}^{\ref{902796}}$ E.
Brost${}^{\ref{902689}}$ D. Buttazzo${}^{\ref{902886}}$ B.
Caiffi${}^{\ref{902881}}$ G. Calderini${}^{\ref{926589},\ref{903119}}$ P.
Cameron${}^{\ref{902689}}$ R. Capdevilla${}^{\ref{908474},\ref{903282}}$ F.
Casaburo G. Cavoto${}^{\ref{903168},\ref{902887}}$ L.
Celona${}^{\ref{902879}}$ G. Cesarini C. Cesarotti${}^{\ref{902835}}$ Z.
Chacko${}^{\ref{902990}}$ A. Chancé${}^{\ref{912490}}$ S.
Chen${}^{\ref{1471035}}$ Y.-T. Chien${}^{\ref{1275736}}$ M.
Chiesa${}^{\ref{943385},\ref{902885}}$ G.
Collazuol${}^{\ref{902884},\ref{903113}}$ D. J. Colling${}^{\ref{902868}}$ M.
Costa${}^{\ref{903128},\ref{902886}}$ N. Craig${}^{\ref{903307}}$ L. M.
Cremaldi A. Crivellin${}^{\ref{903370},\ref{905405}}$ D.
Curtin${}^{\ref{903282}}$ R. T. D’Agnolo${}^{\ref{1087875}}$ M.
Dam${}^{\ref{902882}}$ H. Damerau${}^{\ref{902725}}$ J. de
Blas${}^{\ref{903836}}$ E. De Matteis${}^{\ref{907142}}$ A.
Deandrea${}^{\ref{1743848}}$ J. Delahaye${}^{\ref{902725}}$ A.
Delgado${}^{\ref{903085}}$ C. Densham${}^{\ref{903174}}$ R.
Dermisek${}^{\ref{902874}}$ K. F. Di Petrillo${}^{\ref{902796}}$ J.
Dickinson${}^{\ref{902796}}$ J. Duarte${}^{\ref{903305}}$ P.
Everaerts${}^{\ref{903349}}$ L. Everett${}^{\ref{903349}}$ S.
Farinon${}^{\ref{902881}}$ J. F. Somoza${}^{\ref{902725}}$ G.
Ferretti${}^{\ref{902825}}$ F. Filthaut${}^{\ref{903075}}$ M.
Forslund${}^{\ref{910429}}$ R. Franceschini${}^{\ref{906528},\ref{907692}}$ P.
Franchini${}^{\ref{902948},\ref{903170}}$ M. Frigerio${}^{\ref{1508424}}$ E.
Gabrielli${}^{\ref{903287},\ref{902888}}$ I. Garcia Garcia${}^{\ref{903889}}$
F. Garosi${}^{\ref{904416}}$ A.S. Giannakopoulou${}^{\ref{903237}}$ D.
Giove${}^{\ref{907142}}$ L. Gladilin A. Glioti${}^{\ref{1471035}}$ S.
Goldfarb${}^{\ref{902999}}$ H.M. Gray${}^{\ref{903299},\ref{902953}}$ L.
Gray${}^{\ref{902796}}$ A. Greljo${}^{\ref{902668}}$ C.
Grojean${}^{\ref{902770},\ref{902858}}$ J. Gu${}^{\ref{903628}}$ J.
Haley${}^{\ref{903094}}$ C. Han${}^{\ref{903702}}$ T. Han${}^{\ref{903130}}$
J. Hauptman${}^{\ref{902893}}$ B. Henning${}^{\ref{1471035}}$ K.
Hermanek${}^{\ref{902874}}$ S. Homiller${}^{\ref{902835}}$ S.
Jana${}^{\ref{902841}}$ H. Jia${}^{\ref{903349}}$ C. Jolly${}^{\ref{903174}}$
Y. Kahn${}^{\ref{902867}}$ D. M. Kaplan${}^{\ref{902865}}$ I.
Karpov${}^{\ref{902725}}$ D. Kelliher${}^{\ref{903174}}$ W.
Kilian${}^{\ref{903203}}$ K. Kong${}^{\ref{902912}}$ P.
Koppenburg${}^{\ref{903832}}$ G. Krnjaic${}^{\ref{902796}}$ P.
Kyberd${}^{\ref{903940}}$ R. LOSITO${}^{\ref{902725}}$ S. Levorato W.
Li${}^{\ref{903156}}$ R. L. Voti${}^{\ref{902887}}$ Z. Liu${}^{\ref{903010}}$
M. Liu${}^{\ref{903142}}$ Q. Lu${}^{\ref{902835}}$ L. Ma${}^{\ref{904187}}$ Y.
Ma${}^{\ref{903130}}$ F. Maltoni${}^{\ref{910783},\ref{902674}}$ B.
Mansoulié${}^{\ref{912490}}$ L. Mantani${}^{\ref{907623}}$ S.
Mariotto${}^{\ref{903009},\ref{907142}}$ D. Marzocca${}^{\ref{902888}}$ K.
Matchev${}^{\ref{902804}}$ A. Mazzacane${}^{\ref{902796}}$ A.
Mazzolari${}^{\ref{905268}}$ N. McGinnis${}^{\ref{903290}}$ P.
Meade${}^{\ref{910429}}$ B. Mele${}^{\ref{902887}}$ P.
Merkel${}^{\ref{902796}}$ C. Merlassino${}^{\ref{903112}}$ R. K.
Mishra${}^{\ref{902835}}$ A. Mohammadi${}^{\ref{903349}}$ R. Mohapatra N.
Mokhov${}^{\ref{902796}}$ P. Montagna${}^{\ref{943385},\ref{902885}}$ R.
Musenich${}^{\ref{902881}}$ M.S. Neubauer${}^{\ref{902867}}$ D.
Neuffer${}^{\ref{902796}}$ H. Newman${}^{\ref{902711}}$ Y.
Nomura${}^{\ref{903299}}$ I. Ojalvo${}^{\ref{16750}}$ G.
Ortona${}^{\ref{902889}}$ P. Panci${}^{\ref{903129},\ref{902886}}$ M.
Palmer${}^{\ref{902689}}$ R. Paparella${}^{\ref{907142}}$ P.
Paradisi${}^{\ref{1513358},\ref{902884}}$ A. Perloff${}^{\ref{902748}}$ F.
Piccinini${}^{\ref{902885}}$ M. Pierini${}^{\ref{902725}}$ M.
Prioli${}^{\ref{907142}}$ M. Procura${}^{\ref{903326}}$ R.
Rattazzi${}^{\ref{1471035}}$ D. Redigolo${}^{\ref{902880}}$ L.
Reina${}^{\ref{902803}}$ J. Reuter${}^{\ref{902770}}$ R.A.
Rimmer${}^{\ref{904961}}$ F. Riva${}^{\ref{902813}}$ T.
Robens${}^{\ref{902678}}$ C. Rogers${}^{\ref{903174}}$ M.
Romagnoni${}^{\ref{905268}}$ L. Rossi${}^{\ref{903009},\ref{907142}}$ R.
Ruiz${}^{\ref{902756}}$ R. Ryne${}^{\ref{1189711}}$ F. Sala${}^{\ref{908583}}$
J. Salko${}^{\ref{902668}}$ E. Salvioni${}^{\ref{1513358},\ref{902884}}$ J.
Santiago${}^{\ref{909079},\ref{903836}}$ J.
Schieck${}^{\ref{903324},\ref{904536}}$ R. Schwiehorst${}^{\ref{903006}}$ D.
Sertore${}^{\ref{907142}}$ V. Sharma${}^{\ref{903349}}$ V.
Shiltsev${}^{\ref{902796}}$ J. Shu${}^{\ref{903895}}$ L.
Silvestris${}^{\ref{902877}}$ K. Skoufaris${}^{\ref{902725}}$ P.
Snopok${}^{\ref{902865}}$ F.J.P. Soler${}^{\ref{902823}}$ M.
Sorbi${}^{\ref{903009},\ref{907142}}$ G. Stark${}^{\ref{1218068}}$ M.
Statera${}^{\ref{907142}}$ D. Stratakis${}^{\ref{902796}}$ N.
Strobbe${}^{\ref{903010}}$ J. Stupak${}^{\ref{1273509}}$ S.
Su${}^{\ref{902647}}$ R. Sundrum${}^{\ref{912511}}$ A.
Sytov${}^{\ref{905268}}$ A. Taffard${}^{\ref{903302}}$ A.
Tesi${}^{\ref{902880}}$ J. Thaler${}^{\ref{1237813}}$ R.
Torre${}^{\ref{902881}}$ L. Tortora${}^{\ref{907692}}$ Y.
Torun${}^{\ref{902865}}$ S. Trifinopoulos${}^{\ref{902888}}$ R. U.
Valente${}^{\ref{907142}}$ N. Vignaroli${}^{\ref{902883}}$ L.
Vittorio${}^{\ref{903128},\ref{902886}}$ P.
Vitulo${}^{\ref{943385},\ref{902885}}$ E. Vryonidou${}^{\ref{902984}}$ C.
Vuosalo${}^{\ref{903349}}$ L.-T. Wang${}^{\ref{902729}}$ C.G.
Whyte${}^{\ref{904214}}$ Y. Wu${}^{\ref{903094}}$ A. Wulzer${}^{\ref{903113}}$
K. Xie${}^{\ref{903130}}$ A. Yamamoto${}^{\ref{902916}}$ K.
Yonehara${}^{\ref{902796}}$ H.-B. Yu${}^{\ref{903304}}$ M.
Zanetti${}^{\ref{903113}}$ J. Zhang Y. J. Zheng${}^{\ref{902912}}$ A.
Zlobin${}^{\ref{902796}}$ J. Zurita${}^{\ref{907907}}$

###### Abstract

In this paper we report on the current status of studies on the expected
performance for a detector designed to operate in a muon collider environment.
Beam-induced backgrounds (BIB) represent the main challenge in the design of
the detector and the event reconstruction algorithms. The current detector
design aims to show that satisfactory performance can be achieved, while
further optimizations are expected to significantly improve the overall
performance. We present the characterization of the expected beam-induced
background, describe the detector design and software used for detailed event
simulations taking into account BIB effects. The expected performance of
charged-particle reconstruction, jets, electrons, photons and muons is
discussed, including an initial study on heavy-flavor jet tagging. A simple
method to measure the delivered luminosity is also described. Overall, the
proposed design and reconstruction algorithms can successfully reconstruct the
high transverse-momentum objects needed to carry out a broad physics program.

## Executive Summary

Despite the success of the Standard Model (SM) of particle physics, several
observations point to the existence of physics beyond what is currently known.
Theoretical explanations of such phenomena most probably require new states or
forces that manifest at energies higher than what is currently accessible at
world leading facilities. The lack of a firm indication of the exact energy
scale of those phenomena further motivates an energy frontier research program
capable of reaching to the highest possible energy. In the landscape of
possible future energy-frontier accelerators, a circular $\mu^{+}\mu^{-}$
collider, also referred to simply as muon collider, is a particularly
attractive option for the future of energy frontier exploration. Such a
machine has the potential to deliver a vast physics program [1] in a
relatively compact accelerator complex.

The aim of this paper is to describe the expected performance of a multi-
purpose Muon Collider Detector (MCD) designed to reconstruct the products of a
muon-muon collisions with high accuracy. Several operating center-of-mass
energies are possible for the considered machine; in this paper a center-of-
mass energy of $1.5$ TeV is explored.

One of the biggest challenges for a MCD is to successfully disentangle the
products of the $\mu^{+}\mu^{-}$ collisions from an intense beam-induced
background (BIB) coming primarily from secondary and tertiary interactions of
the muon decay products. For this reason an essential part of the Machine
Detector Interface (MDI) at a Muon Collider is a pair of tungsten nozzles
cladded with borated polyethylene, which reduce the rate and energy of BIB
particles that reach the detector by several orders of magnitude. The nozzles
limit the maximum angular acceptance of the MCD to about $10^{\circ}$ from the
beamline axis.

The MCD design presented in this paper follows the classical cylindrical
layout typical for multipurpose detectors of symmetric collisions. It consists
of a silicon-based Vertex Detector, with innermost barrel layer at $22$ mm
from the beamline, followed by silicon-based Inner and Outer Trackers. High
granularity electromagnetic and hadronic calorimeters and an outer muon
spectrometer complement the main elements of the detector.

A detailed GEANT4 based simulation package is used to simulate the response of
the detector. A fast digitization and detailed reconstruction algorithms were
developed to study the expected performance of the MCD in the event
reconstruction. The BIB is overlaid to the $\mu^{+}\mu^{-}$ process of
interest.

Three approaches are studied for track reconstruction at the MCD: a Conformal
Tracking (CT) algorithm developed for the clean environment of the electron-
positron colliders and supplemented with either a Region of Interest approach
or a filter based on double pixel layers in the innermost layers of the
detector, and a Combinatorial Kalman Filter (CKF) algorithm developed for the
busy environment of hadron colliders. The results show a robust track
reconstruction for charged particles with the momentum above $1~{}$GeV
throughout the detector acceptance, with the potential for extending towards
softer particles.

Jets are reconstructed using a particle-flow approach and a
$k_{\textrm{T}}$-based clustering. An energy calibration is derived and
applied throughout the studies. The performance is evaluated on samples of
light, $b-$ and $c-$jets, resulting in a reconstruction efficiency ranging
from 82% at $p_{\textrm{T}}\approx 20$ GeV to 95% at higher $p_{\textrm{T}}$
with sub-percent fake rate in the region $|\eta|<1.5$. The jet energy
resolution ranges from about 50% to about $15\%$, depending on
$p_{\textrm{T}}$. The resulting di-jet invariant mass resolution allows good
separation between hadronically-decaying Higgs and $Z$ bosons.

A simple heavy-flavor tagging based on finding a displaced-vertex yields a
$b-$jet identification efficiency ranging from 50% to 80%, depending on
$p_{\textrm{T}}$, with a mis-tag rate of about $1-5\%$ (20%) for light
($c$-)jets. It is expected that more advanced algorithms would be able to
perform significantly better.

Prototypes of electron and photon reconstruction algorithms able to cope with
BIBs have been developed, yielding a successful reconstruction of
high-$p_{\textrm{T}}$ electrons and photons with relatively small loss of
efficiency and energy resolution. Muon reconstruction is performed by
combining a stand-alone track reconstructed in the muon system and a track
reconstructed in the tracker. BIBs mostly affect the forward endcap region.
Filtering hits based on their properties as well as based on standalone muon
tracks allows for negligible impact of BIB in the performance of muon
reconstruction. Reconstruction of other high- objects, as $\tau$ leptons, is
still in progress, but it is expected to pose challenges similar to the ones
that have been already solved for track and jet reconstruction as well as
heavy-flavor jets identification.

The precise determination of the delivered luminosity is of crucial importance
for the planned physics program. A method based on $\mu-$Bhabha scattering
measurement shows that a statistical precision of about $0.2\%$ can be
achieved for the integrated luminosity delivered per year of data-taking.

In conclusion, this manuscript uses detailed simulation of a proposed Muon
Collider detector that satisfies the basic requirements of reconstructing
high-$p_{\textrm{T}}$ objects needed to deliver an ambitious physics program.
The effects of BIB is taken into account, showing that its impact can be
successfully mitigated.

###### Contents

  1. 1 Introduction
  2. 2 Beam-Induced Background characterization
    1. 2.1 BIB properties at 1.5 TeV
    2. 2.2 Simulation in FLUKA
    3. 2.3 Implications of higher beam energies
  3. 3 Overview of the Detector design
    1. 3.1 Tracking Detector
  4. 4 Detector Simulation Software
    1. 4.1 Detector digitization
  5. 5 Charged particle reconstruction
    1. 5.1 Conformal Tracking in Regions of Interest
    2. 5.2 Conformal Tracking with Double Layers
    3. 5.3 Combinatorial Kalman Filter
  6. 6 Jets
    1. 6.1 Track filter
    2. 6.2 Calorimeter hit selection
    3. 6.3 Particle flow, jet clustering and fake jet removal
    4. 6.4 Jet momentum correction
    5. 6.5 Jet reconstruction performance
    6. 6.6 Invariant mass reconstruction of Higgs and Z bosons to b-quarks
    7. 6.7 Future prospects on jet reconstruction
    8. 6.8 Heavy-flavor jet identification
      1. 6.8.1 Secondary vertex tagging
      2. 6.8.2 SV-tagging performance
  7. 7 Photons and Electrons
    1. 7.1 Photon reconstruction and identification
    2. 7.2 Electron reconstruction and identification
  8. 8 Muons
  9. 9 Luminosity measurement
  10. 10 Conclusions

##  1 Introduction

The Standard Model (SM) of particle physics has been an extremely successful
theory to describe measurements at the energy frontier. In the last decades, a
vast physics programs at the energy frontier has produced discoveries that led
to the current incarnation of the Standard Model (SM) theory and stress-tested
it with precision measurements. Nevertheless, observations, as well as
theoretical motivations, unambiguously point to physics phenomena Beyond the
Standard Model (BSM). Theoretical explanations of such phenomena almost always
require new states or forces that manifest at energies higher than what
currently accessible at world leading facilities. Despite the lack of a clear
energy target that guarantees with certainty a discovery, multiple indirect
indications point to energy scales of several TeV as the most likely energy
regime where BSM phenomena would manifest. It is therefore of paramount
importance to develop a physics program for the energy frontier that allows us
to explore such an energy regime in the future, possibly with the ability of
adapting the target energy depending on future insights from accelerator and
non-accelerator based experiments.

In the landscape of possible future energy-frontier accelerators [2], a
circular $\mu^{+}\mu^{-}$ collider, also referred to simply as muon collider,
is a particularly attractive option for the future of energy frontier
exploration [3]. Such a machine has the potential to deliver a vast physics
program [1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21,
22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40] in
a relatively compact accelerator complex [41]. Such an option has been studied
in the past [42, 43, 44, 45, 46, 47, 48, 49], and it sparked renewed interest
thanks to advances in accelerator technology as well as given the current
needs of high-energy exploration of the field.

The aim of this paper is to describe the expected performance of a multi-
purpose muon collider detector (MCD). The study focuses on the reconstruction
algorithms and performance of high-$p_{\textrm{T}}$ objects that will be used
needed for a successful physics program. The results are built upon and
significantly extend the ones reported in Ref. [50]. The current design aims
at showing that satisfactory performance can be achieved in a muon collider
environment. Further optimization of the detector design and reconstruction
algorithms are expected to significantly improve the performance of a MCD and
will be subject of future work.

Several operating center-of-mass energies are possible for a future muon
collider [41]. In this paper, unless otherwise noted, a center-of-mass energy
of $1.5$ TeV is assumed. Table 1 reports additional accelerator parameters
that have been assumed for the simulations used for the results presented in
this manuscript. Most of the results presented in this manuscript do not
simulate the spread of collisions in the transverse and longitudinal
directions, unless otherwise stated; it is expected that this has a minor
impact on the performance, and it is discussed explicitly when relevant.

Parameter | Symbol | Value
---|---|---
Center-of-mass energy | $\sqrt{s}$ |  $1.5$ TeV
Muons per bunch | $N_{\mu}$ | $2\cdot 10^{12}$
Normalised transverse emittance | $\epsilon_{TN}$ | 25 $\pi\,\mu$m rad
Normalised longitudinal emittance | $\epsilon_{LN}$ | 7.5 MeV m
IP relative energy spread | $\delta_{E}$ | 0.1 %
IP beta function | $\beta^{*}_{x,y}$ | 1 cm
IP transverse beam size | $\sigma_{x,y}$ | 6 $\mu$m
IP longitudinal beam size | $\sigma_{z}$ | 10 mm
Table 1: Representative set of muon collider parameters used in the detailed
simulation presented in this paper.

One of the biggest challenges for a MCD is to successfully disentangle the
products of the $\mu^{+}\mu^{-}$ collisions from an intense beam-induced
background (BIB) coming primarily from secondary and tertiary interactions of
the muon decay products [51, 52, 53]. Section 2 summarizes the main
characteristics expected for such a background and also includes a discussion
on the expected background composition for higher center-of-mass energy muon
colliders.

BIB has a big influence on the detector design, which is summarised in Section
3. The technical challenges for the MCD and the most promising technologies
for fulfilling the physics requirements of the detector are described in a
separate manuscript [54], which also covers the readout and trigger schema.

Results in this manuscript are obtained using detailed simulations of the MCD
based on a Geant4 [55] framework. The software setup is described in Section
4. All simulations, unless otherwise stated, include overlaying on top of the
desired $\mu^{+}\mu^{-}$ collision the model of BIB described in Section 2.

Sections 5, 6, 7 and 8 describe the reconstruction algorithms and expected
performance for the basic objects needed to carry a comprehensive physics
program. Reconstruction of other objects, as $\tau$ leptons or missing
momentum, is still in progress, but it is expected to pose challenges similar
to the ones that have been already solved. A method for measuring the
delivered luminosity with high accuracy is described in Section 9.

##  2 Beam-Induced Background characterization

The unstable nature of muons (lifetime $\tau_{\mu}=$ 2.2 µ at rest) makes the
beam-induced background (BIB) a much more challenging issue at a Muon Collider
than it is at facilities that use stable-particle beams. In fact, $2\cdot
10^{12}$ muons per bunch in a 750GeV muon beam lead to about $4\cdot 10^{5}$
muon decays per meter. Interactions of the decay products with the accelerator
lattice produce even larger amounts of particles that eventually reach the
detector, making the reconstruction of clean $\mu^{+}\mu^{-}$ collision events
nearly impossible without dedicated BIB mitigation. A discussion of the
expected radiation levels due to BIB can be found in Ref. [54]. For this
reason an essential part of the Machine Detector Interface (MDI) at a Muon
Collider is a pair of tungsten (W) nozzles cladded with borated polyethylene
(BCH), which reduce the rate and energy of BIB particles reaching the detector
by several orders of magnitude.

A dedicated design of the MDI has been developed by MAP collaboration [52]
using the MARS15 software [56] to simulate muon decays within ±200 from the
interaction point (IP) in a collider ring with the parameters summarised in
Table 1. The exact shape and positioning of the nozzles was optimised
specifically for the $\sqrt{s}=$ 1.5TeV case, including the 10° opening angle
and 12 distance between the tips of the nozzles, as shown in Fig. 1. The same
approach has been adopted by the Muon Collider collaboration based on the
FLUKA simulation software to characterize BIB at higher center-of-mass
energies and to study other MDI designs.

Figure 1: Cross-sectional view of the accelerator lattice including the MDI as
designed by MAP collaboration for the $\sqrt{s}=$1.5 TeV Muon Collider and
visualised with FLUKA simulation software. Distinct colours represent
different materials of the MDI: tungsten (green), borated polyethylene
(violet), iron (mustard), concrete (gray). The black box in the center
encloses the detector volume, which is excluded from the standalone BIB
simulation process. Dimensions are reported in centimeters.

In order to include the effects of the BIB on detector performance the
particles induced by the muon decays are collected at the outer surface of the
MDI and before entering the detector volume, which is represented by a black
box on Fig. 1. This allows to later simulate their interaction with the
detector together with particles from the $\mu^{+}\mu^{-}$ collision. Thus,
BIB sample in this article refers to the collection of particles originating
from the muon decays before any interaction with the detector material.

###  2.1 BIB properties at 1.5 TeV

The exact BIB sample used for detector-performance studies at $\sqrt{s}=$
1.5TeV presented in this paper has been produced in MARS15 software, because
the FLUKA-based workflow was not fully validated at that time. Yet FLUKA
interface is used for visualisation of the accelerator geometry and BIB
trajectories in the MDI region thanks to its advanced graphical capabilities.
An example of simulated particle decays and their interaction with the
accelerator lattice is shown in Fig. 2, demonstrating multiple points along
the beamline where BIB particles originate from.

Figure 2: The top picture shows the tracks of secondary particles for a few
$\mu^{-}$ decays arriving from the right, while in the middle picture neutrons
are excluded. The bottom plot illustrates the tracks in the case of a single
$\mu^{-}$ decay in the proximity of the IP. Different particle types are
separated by colour: photons (red), neutrons (blue), $e^{-}$ (black), $e^{+}$
(yellow).

BIB particles at a Muon Collider have a number of characteristic features that
set them apart from typical backgrounds at $e^{+}e^{-}$ or $pp$ colliders,
namely their low momentum, displaced origin and asynchronous time of arrival.
Distributions of these three properties are presented in Fig. 3 for the three
dominant particle types exiting from the MDI surface: neutrons, photons and
electrons/positrons.

Figure 3: Kinematic properties of BIB particles entering the detector region:
momentum (left), position along the beam line (middle) and arrival time with
respect to the bunch crossing (right).

The most distinctive aspect of BIB particles at the Muon Collider is their
extremely large number and low momentum: about $4\cdot 10^{8}$ particles
exiting the MDI in a single bunch crossing depositing energy to the detector
in a diffused manner. The separation between the tips of the tungsten nozzles
leads to most of the BIB particles exiting at a significant distance from the
interaction point. Finally, there is a substantial spread in the arrival time
of the BIB particles with respect to the bunch crossing, ranging from a few
nanoseconds for electrons and photons to microseconds for neutrons due to
their smaller velocity.

Each of these aspects has different implications for the BIB signatures in
different parts of the detector, which depend on the position, spatial
granularity and timing capabilities of the corresponding sensitive elements.
Thus, a careful choice of detector technologies and reconstruction techniques
allows to mitigate the negative effects of the BIB, as demonstrated in the
later sections of this paper.

###  2.2 Simulation in FLUKA

We report in this section the most relevant BIB features computed at
$\sqrt{s}=1.5$ TeV by the Monte Carlo multi-particle transport code FLUKA [57,
58]. The complex FLUKA geometry is assembled by means of the LineBuilder [59]
using the optics file provided by the MAP collaboration. The accelerator
elements have been defined in Fluka Elements Database following the
information contained in this file and in MAP publications [60, 61]. The
results obtained by FLUKA are benchmarked against those provided by the MAP
collaboration and the detailed comparison is described in Ref. [53].

The results presented below are computed for one beam, given the symmetric
nature of the $\mu^{+}$$\mu^{-}$ collider. In particular, the primary
$\mu^{-}$ beam is simulated according to parameters reported in Table 1
travelling counterclockwise starting 200 away from the IP.

The major contributors to BIB are photons, neutrons and electrons/positrons.
The time at which BIB exits the machine in the IR is spread over a wide range
but the major part is concentrated around the beam crossing time ($t=0$), as
shown by the top panel of Fig. 4.

The bottom panel of Fig. 4 reports the longitudinal distribution of primary
$\mu^{-}$ decays generating the most relevant BIB families: the cumulative
function shows it is enough to consider decays within $\sim\qty{25}{}$ from
the IP. On the contrary, simulations show that to correctly account for the
secondary $\mu^{\pm}$, it is necessary to consider primary decays up
$\sim\qty{100}{}$ from the IP.

Figure 4:  Time distribution of BIB particles exiting the machine (top).
Longitudinal distribution of primary $\mu^{-}$ decay generating BIB particles
exiting the machine and cumulative function normalized to 1 of the total
distribution in black dotted line (bottom). Results by FLUKA considering
primary $\mu^{-}$ decays within 100 from the IP.

The kinetic energy distribution of most relevant BIB particle types is
reported in Fig. 5. Energy cutoffs have been applied in the simulation at 100
keV for $\gamma$, $e^{\pm}$, $\mu^{\pm}$, charged hadrons and at 10-14 GeV for
neutrons. The nozzles act in a very significant way in cutting out the high
energy BIB component: as we can notice the BIB particles entering the detector
hall have kinetic energy below few GeVs. Only charged hadrons and secondary
muons can reach much higher energies but their number is quite low, in the
order of 104 and 103, with respect to 107 photons, neutrons and 105 electrons,
positrons.

Most of the BIB exits the machine in the region around the IP and by
considering a time cut within -1 and 15 ns, which is the most relevant for the
detector measures, a big portion of photons and neutrons is removed, as
displayed by the left panel of Fig. 5.

Figure 5:  Lethargy plot (left) and longitudinal exit coordinate distribution
(right) of BIB particles, divided by particle type. Results by FLUKA
considering primary $\mu^{-}$ decays within 100 m from the IP. No time cut is
applied to distributions represented in dotted lines while in solid line only
particles exiting the machine between -1 and 15 are considered.

###  2.3 Implications of higher beam energies

FLUKA simulations at $\sqrt{s}=3$ TeV and $\sqrt{s}=10$ TeV are currently
under development. Since the MDI has not yet been optimized for those
energies, the one designed for $\sqrt{s}=1.5$ TeV has been adopted. In both
cases the preliminary results show a BIB with intensity of the same level as
in the $\sqrt{s}=1.5$ TeV configuration characterized by spatial and temporal
structures very similar to those presented in the previous section. A careful
optimization of machine lattice and MDI is expected to further suppress BIB in
the detector region.

##  3 Overview of the Detector design

The Muon Collider Detector (MCD) follows the classical cylindrical layout
typical for multipurpose detectors of symmetric collisions and the specific
geometry used for simulation studies in this work has the reference code
MuColl_v1. The rendering of the detector geometry is presented in Fig. 6, with
the dimensions of each subsystem summarised in Table 2. A cylindrical
coordinate system is used with its centre placed at the nominal interaction
point. The $Z$ axis is defined as the moving direction of the $\mu^{+}$ beam.
The $X$ axis is defined to point towards the inner part of the ring and the
$Y$ axis therefore pointing upwards. Cylindrical coordinates are often used
with $R$, $\theta$ and $\phi$ denoting the radial distance from the
interaction point, the polar and azimuthal angles respectively. Pseudo-
rapidity $\eta=-\log\big{[}\tan(\theta/2)\big{]}$ is also used in some cases
for convenience.

Figure 6: Rendering of the MCD geometry used for the presented simulation
studies, including the cone-shaped shielding nozzles (cyan) and the beryllium
beampipe (violet). Shown are the R-Z cross sections of the full detector
geometry (left) and two zoomed-in portions: up to ECAL (top right) and up to
Vertex Detector (bottom right). Muon Detector (violet and green) surrounds the
solenoid (cyan), which encloses the HCAL (magenta), ECAL (yellow) and the
Tracking Detector (green and black).

Starting from the Be beampipe with a radius of 22, the Vertex Detector is the
closest to the IP with its innermost layer having a radius of only 30. It is
followed by the Inner and Outer Trackers. The three sub-systems complete the
all-silicon Tracking Detector, which operates in the strong magnetic field of
3.57 provided by the superconducting solenoid, to reconstruct trajectories and
transverse momenta ($p_{\text{T}}$) of charged particles. High-granularity
sampling ECAL and HCAL calorimeters are arranged outside of the Tracking
Detector and provide measurements of electromagnetic and hadronic showers
respectively, with fine longitudinal segmentation: 40 layers in ECAL and 60
layers in HCAL. Outside of the superconducting solenoid is the Muon Detector
based on RPC technology with its iron yoke closing the magnetic field. It is
worth noting that accurate timing information from all sub-systems is an
essential component of the detector, as it allows rejecting a large fraction
of BIB.

Subsystem | Region | R dimensions [cm] |  $|$Z$|$ dimensions [cm] | Material
---|---|---|---|---
Vertex Detector | Barrel | $3.0-10.4$ | $65.0$ | Si
| Endcap | $2.5-11.2$ | $8.0-28.2$ | Si
Inner Tracker | Barrel | $12.7-55.4$ | $48.2-69.2$ | Si
| Endcap | $40.5-55.5$ | $52.4-219.0$ | Si
Outer Tracker | Barrel | $81.9-148.6$ | $124.9$ | Si
| Endcap | $61.8-143.0$ | $131.0-219.0$ | Si
ECAL | Barrel | $150.0-170.2$ | $221.0$ | W + Si
| Endcap | $31.0-170.0$ | $230.7-250.9$ | W + Si
HCAL | Barrel | $174.0-333.0$ | $221.0$ | Fe + PS
| Endcap | $307.0-324.6$ | $235.4-412.9$ | Fe + PS
Solenoid | Barrel | $348.3-429.0$ | $412.9$ | Al
Muon Detector | Barrel | $446.1-645.0$ | $417.9$ | Fe + RPC
| Endcap | $57.5-645.0$ | $417.9-563.8$ | Fe + RPC
Table 2: Boundary dimensions of individual subsystems of the Muon Collider
Detector concept as defined in the geometry MuColl_v1.

The design of the MuColl_v1 detector is largely based on the CLICdet geometry
developed by CLIC collaboration, with the details about technologies used for
each component documented in Ref. [62]. The only subsystem with significant
changes to its design is the Tracking Detector, which is described in the
following.

###  3.1 Tracking Detector

The tracking detector is assumed to follow the typical design of an all-
silicon tracker. The geometry is shown in Fig. 7. It consists of three sub-
detector; the Vertex Detector, Inner and Outer Trackers. All three are split
into a central barrel section and a forward end-cap section. The Vertex
Detector is made from 4 double-layers with a 2 gap to improve secondary vertex
resolution. The forward coverage is limited by the presence of the tungsten
nozzle shield.

Figure 7: View of the tracking detector projected on $Z-R$ (left) and
transverse plane (right). The transverse plane view is zoomed into the Vertex
Detector to demonstrate the double-layer arrangement.

The Tracking Detector returns 4-dimensional hit coordinates; the three
positional dimensions and precision timing information. The timing information
plays a critical role in reducing the effective hit density from BIB. Fig. 8
shows the timing distribution (corrected for time-of-flight) of hits from a
$\mu^{+}\mu^{-}$ collision compared to those induced by BIB. The spatial and
timing resolutions assumed in the detector simulations are summarized in Table
3.

Figure 8: Comparison of hit-time distributions in the Tracking Detector between BIB particles (solid lines) and signal muons (filled areas) corrected by the time of flight of a photon from the IP, taking into account time resolution of each sub-detector. | Vertex Detector | Inner Tracker | Outer Tracker
---|---|---|---
Cell type | pixels | macropixels | microstrips
Cell Size |  25 $\times$ 25 |  50 $\times$ 1 |  50 $\times$ 10
Sensor Thickness |  50 |  100 |  100
Time Resolution |  30 |  60 |  60
Spatial Resolution |  5 $\times$ 5 |  7 $\times$ 90 |  7 $\times$ 90
Table 3: Assumed spatial and time resolution in different sub-systems of the
Tracking Detector. There is no difference between the barrel and end-cap
regions.

##  4 Detector Simulation Software

Full simulation of a single $\mu^{+}\mu^{-}$ collision event involves several
stages:

  1. 1.

generation of all stable particles entering the detector;

  2. 2.

simulation of their interaction with the passive and sensitive material of the
detector;

  3. 3.

simulation of the detector’s response to these interactions;

  4. 4.

application of data-processing and object-reconstruction algorithms that would
happen in a real experiment.

The first stage of generating stable input particles is handled by standalone
software, such as Monte Carlo event generators for the $\mu^{+}\mu^{-}$
interaction and FLUKA or MARS15 for the BIB particles. The rest of the
simulation process is performed inside the iLCSoft framework [63] previously
used by the CLIC experiment [64] and now forked for developments of Muon
Collider studies [65]. Particle interactions with the detector material are
simulated in GEANT4 [55], while detector response and event reconstruction are
handled inside the modular Marlin framework [66]. The detector geometry is
defined using the DD4hep detector description toolkit [67], which provides a
consistent interface with both the GEANT4 and Marlin environments. More detail
about the software structure and computational optimization methods used for
simulating such a large number of BIB particles is documented in Ref. [68].

###  4.1 Detector digitization

Response of each sensitive detector to the corresponding energy deposits
returned by GEANT4 is simulated by dedicated digitization modules implemented
as individual Marlin processors.

The Tracking Detector uses Gaussian smearing to account for the spatial and
time resolutions of local hit coordinates on the sensor surface and the time
coordinate. The assumed resolution values are listed in Table 3. Acceptance
time intervals, individually configured for each sub-detector, are used for
replicating the finite readout time windows in the electronics of a real
detector and to reject hits from asynchronous BIB particles. The result of
this simplified approach is one-to-one correspondence between the GEANT4 hits
and digitized hits, which ignores the effect of charge distribution across
larger area due to the Lorentz drift and shallow crossing angles with respect
to the sensor surface. These effects are taken into account in the more
realistic tracker digitization code that is currently under development and
will allow stronger BIB suppression based on cluster-shape analysis.

ECAL and HCAL detectors are digitized using realistic segmentation of
sensitive layers into cells by summing all energy deposits in a single cell
over the configured integration time of ±250. Time of the earliest energy
deposit is consequently assigned to the whole digitized hit. The same
digitization approach is used also for the Muon Detector.

##  5 Charged particle reconstruction

In the magnetic field of MCD a charged particle will follow a helix trajectory
aligned with the $Z$ axis. The radius of curvature in the transverse plane is
proportional to the magnetic field (3.57) and the transverse momentum of the
particle. Deviations from a perfect helix can occur due to multiple-
scattering, ionizing energy losses and bremsstrahlung. The first two are a
direct function of the detector material. The amount of radiation lengths that
a particle traverses through the tracking detector when starting from the
nominal collision point is shown in Fig. 9.

Figure 9: Radiation length of the tracking detectors, as seen along a line
defined by the nominal interaction point and the polar angle $\theta$.

The charged particle trajectory is reconstructed from hits positions in the
silicon tracking detector. The reconstructed object is called a track. A track
consists of a set of hits (one per layer) and five fitted parameters
describing the helix. A track reconstruction algorithm can roughly be broken
up into two steps: pattern recognition to identify the hits belonging to a
single track and fitting the hit coordinates by a track model to deduce the
relevant track parameters.

Track reconstruction at the Muon Collider is complicated by the presence of a
huge number of hits in the silicon sensor originating from the Beam Induced
Background (BIB-hits). The density of BIB hits is ten times larger than the
expected contribution from pile-up events at a High Luminosity LHC detector.
Table 4 compares the hit density between the MCD and ATLAS Inner TracKer (ITk)
upgrade for HL-LHC operation [69, 70]. The increase in possible hit
combinations creates a challenge for the hit pattern recognition step. It is
crucial to reduce the amount of hits at the input to a track reconstruction
algorithm through alternate means, such as precision timing information. The
BIB hits are out-of-time with hard collision hits after the time-of-flight
correction has been applied. By applying a $-3\sigma/+5\sigma$ time window,
the hit density can be reduced by a factor of two as seen in Fig. 10.

ATLAS ITk Layer | ITk Hit Density [^2] | MCD Equiv. Hit Density [^2]
---|---|---
Pixel Layer 0 | 0.643 | 3.68
Pixel Layer 1 | 0.022 | 0.51
Strips Layer 1 | 0.003 | 0.03
Table 4: Comparison of the hit density in the tracking detector between the
ATLAS ITk upgrade for HL-LHC and the MCD with full BIB overlay. The hit
densities for layers at equivalent radii are shown. The MCD numbers are after
timing cuts. Figure 10: Hit density in the different layers of the tracking
detectors in a single event with full BIB overlay. The density before (blue)
and after (yellow) applying the timing cut is shown.

The spatial distribution of BIB-hits is also unique. They are different from
hits created by pile-up. Pile-up hits come from real charged particle tracks
originating from multiple vertices in the collision region. On the other hand,
BIB-hits come from a diffuse shower of soft particles originating from the
nozzles. The compatibility of a track with a trajectory originating from the
luminous region provides an important handle for differentiating "real" tracks
of charged particles produced in the primary collision and "fake" tracks
generated from random combinations of BIB-hits.

The remainder of this section describes three approaches that were studied for
track reconstruction at the MCD. The first two use the Conformal Tracking (CT)
algorithm developed for the clean environment of the electron-positron
colliders [71]. However in the presence of BIB, the CT algorithm takes weeks
to reconstruct a single event and is impractical for large-scale production of
simulated data. To ease the computational effort, the input hits are first
reduced by either defining a Region of Interest (Section 5.1) or by exploiting
the double-layered Vertex Detector to select only hit pairs pointing to the
collision region (section 5.2). The third approach (Section 5.3) uses the
Combinatorial Kalman Filter (CKF) [72, 73, 74] algorithm developed for the
busy environment of hadron colliders. It can perform track reconstruction in a
reasonable time without requiring any additional filtering of input hits.

It should be noted that the CT and CKF algorithms have very different software
implementations that are responsible for much of the difference in their
performance. The CKF algorithm is implemented using the A Common Tracking
Software (ACTS)[75] library that is heavily optimized for efficient computing.
The same is not true for the CT algorithm implemented directly in iLCSoft with
less emphasis on computational efficiency. It is possible that part of the
computational improvements come from code quality alone. For example, the ACTS
Kalman Filter implementation is a factor 200 faster than the default iLCSoft
implementation given the same inputs. This demonstrates the advantage of an
experiment-independent track reconstruction library developed by a dedicated
team with strong computing expertise.

The expected performance is assessed using a sample of single muons
originating at the interaction point. BIB-hits are overlaid, unless explicitly
noted. Two set of samples are used; one set is generated with the muon having
a fixed momentum $(p)$ of $1$, $10$ and $100$ GeV and a uniform distribution
in $\theta$. The second set is generated at discrete values of
$\theta=13^{\circ},30^{\circ},89^{\circ}$ and uniform transverse momentum
distribution in the $1-100$ GeV range. The chosen $\theta$ values correspond
to particles expected to leave hits entirely in the endcap system, some in the
barrel and some in the endcap, and entirely in the barrel region,
respectively.

###  5.1 Conformal Tracking in Regions of Interest

The Conformal Tracking algorithm is based on Ref [71] and implementation in
iLCSoft. The algorithm is designed and optimized to find charged particles in
very clean environments, as the ones in $e^{+}e^{-}$ colliders. The conformal
mapping technique [76] is combined with cellular automaton approach [77] to
increase the acceptance of non-prompt particles.

Due to the very large hit multiplicity from BIB-hits, running such an
algorithm for the full event is prohibitive in terms of CPU and memory
resources. Instead a region-of-interest approach is used, where hits to be
considered are pre-selected based on existing objects reconstructed in either
the calorimeters or the muon system. To assess tracking performance only hits
within a cone of $\Delta R<0.5$ around the signal muon were selected as input
to the CT algorithm, where $\Delta R=\sqrt{\Delta\phi^{2}+\Delta\eta^{2}}$.

Fig. 11 (left) shows the reconstruction efficiency as a function of
$p_{\textrm{T}}$; a muon is considered reconstructed if at least half of the
hits associated to the track have been originated by the muon. Optimal
reconstruction efficiency is achieved throughout the $p_{\textrm{T}}$
spectrum, with a somewhat smaller efficiency for very forward particles due to
their proximity with the nozzle and much larger expected occupancy in the
region. The latter is expected to be recovered by a more dedicated tuning of
the algorithm or by using one of the algorithms described in the next
sections.

Fig. 11 (right) shows transverse momentum resolution as a function of polar
angle $\theta$. The resolution is computed by comparing reconstructed and
generated $p_{\textrm{T}}$ and shown divided by $p_{\textrm{T}}^{2}$. A
localized degradation of the resolution can be seen around $\theta\approx
35^{\circ}$, corresponding to the barrel-endcap transition; in addition, the
feature is enhanced by the non-physical lack of spread of the muon originating
point. It is expected that a more realistic simulation of the luminous region
as well as future optimizations of the tracker layout will mitigate such
localized degradation to negligible levels.

Fig. 12 shows the resolution on the impact parameter $D_{0}$ (left) and the
resolution on the longitudinal impact parameter $Z_{0}$ (right) as a function
of the polar angle $\theta$. Similarly to the case of the resolution on
$p_{\textrm{T}}$, the resolution on $D_{0}$ and $Z_{0}$ slightly degrades in
the barrel-endcap transition region, around $\theta\approx 35^{\circ}$.

Figure 11: Tracking performance for single-muon events overlaid with BIB.
Left: track reconstruction efficiency is shown as a function of
$p_{\textrm{T}}$. Right: momentum resolution $\Delta p_{\textrm{T}}$ is shown
divided by the $p_{\textrm{T}}^{2}$ as a function of $\theta$.

Figure 12: Tracking performance for single-muon events overlaid with BIB.
Left: impact parameter resolution as a function of $\theta$. Right:
longitudinal impact parameter resolution as a function of $\theta$

###  5.2 Conformal Tracking with Double Layers

The Vertex Detector is constructed using double-layers (DL). A double-layer
consists of two silicon detector layers separated by a small distance (2mm for
MCD). This concept will also be used by the CMS Phase-II tracking detector[78]
to reduce the hit combinatorics for a fast track reconstruction in their
trigger system. It works by selecting only those hits that can form a pair
with a hit from the neighboring layer that in aligned with the interaction
point (IP). If there is no 2nd hit in the double-layer to form a consistent
doublet the hit is discarded. This approach is particularly effective for
rejecting BIB hits, because BIB electrons are very likely to either stop in
the first layer due to their very low momentum, or to cross the double-layer
at shallow angles, creating doublets that are not aligned with the IP. The DL
filtering implemented in our simulation software is based on the angular
distance between the two hits of a doublet when measured from the interaction
point, as demonstrated in Fig. 13. For simplicity the two variables used for
filtering are the polar ($\Delta\theta$) and azimuthal ($\Delta\phi$) angle
differences.

In practice there are several limitations to the precision of alignment that
can be imposed by the DL filtering while maintaining high efficiency for
signal tracks. The first is driven by the finite spatial resolution of the
pixel sensors, which limits the minimum resolvable displacement between the
two hits of a doublet.

Its position needs to be known beforehand and any uncertainty will result in
an inefficiency. The latter point is also important for displaced particles
(ie: secondary vertices of $b$-mesons) that do not originate from the IP.

For simplicity the two variables used for filtering are the polar
($\Delta\theta$) and azimuthal ($\Delta\phi$) angular distances. Fig. 15 shows
the distributions of $\Delta\theta$ and $\Delta\phi$ in single-muon events
with a realistic beamspot in the first layer of the vertex detector. The
distribution in other double-layers for $p=\qty{1}{{\rm GeV}}$ muons are shown
in Fig. 14. The bi-modal nature of the $\Delta\phi$ distribution for low
energy muons is the result of the circular path that charged particle take in
the transverse plane. This biases DL filtering toward charge particles with
higher $\pT$ unless very loose cuts are used.

Table 5 lists the two operating points (loose and tight) used to filter hits
for the CT algorithm in two stages. Fig. 16 shows the hit multiplicity (mostly
BIB-hits) as a function of VXD layer.

The loose working point targets high efficiency reconstruction of
$p>\qty{1}{{\rm GeV}}$ muons with the realistic beamspot size (Fig. 15). It
reduces the number of hits in the innermost double-layer by a factor of two.
This reduces the CT reconstruction time from $\approx\qty{1}{week/event}$ to
$\approx\qty{2}{days/event}$. While significant, this is still not practical
for sample production. Instead the loose working point is used with a special
CT configuration as a first stage of a two-stage reconstruction process. The
first stage reconstructs high- tracks to precisely determine the IP position,
which can be used for

The tight working point is optimized for the scenario when the interaction
point is precisely known (Fig. 14). It has a hit survival rate of
$\approx{}2\%$ in the inner-most layer. This reduces the hit multiplicity
enough for the CT algorithm to complete in $\approx\qty{2}{min/event}$. It is
used as the second stage of the track reconstruction algorithm, with the IP
determined from the first-stage.

The two-stage doublet-layer filtering provides a computationally viable method
to track reconstruction in the presence of the BIB. However its efficiency is
very limited for reconstructing non-prompt particles, including $b$-meson
decays.

Figure 13: Illustration of the doublet-layer filtering used for the rejection
of BIB-induced hits in the Vertex Detector. Horizontal black lines represent
double layers of pixel sensors that are crossed by signal (green) and BIB
(grey) particle tracks. Hit doublets created by a signal particle are
perfectly aligned with the beamspot position, but have a sizeable angular
difference when measured from the center of the detector. Hit doublets created
by BIB particles are characterized by larger angular difference due to their
shallow crossing angle and more displaced origin.

Figure 14: Expected $\Delta\phi$ (left) and $\Delta\theta$ (right)
distributions for hit doublets in the barrel region of the Vertex Detector.
Simulated are single-muon tracks with $p_{\textrm{T}}=\qty{1}{{\rm GeV}}$
radiated from the nominal interaction point in the centre of the detector.
Angular spread is increasing in layers at smaller radii.

Figure 15: Left: dependence of $\Delta\phi$ in hit doublets on the transverse momentum of muon tracks. Two separate peaks become visible for low-$p_{\textrm{T}}$ tracks corresponding to $\mu^{+}$ and $\mu^{-}$. Right: dependence of $\Delta\theta$ distribution from $p_{\textrm{T}}=\qty{1}{{\rm GeV}}$ muon tracks on the longitudinal displacement of the interaction point by 10. Both plots are obtained from hits in the innermost double-layer of the Vertex Detector in the barrel region. Figure 16: Expected reduction of hit multiplicity in the Vertex Detector achieved by applying the loose or tight double-layer filtering. | Barrel | Endcap
---|---|---
| Layer IDs | 0,1 | 2,3 | 4,5 | 6,7 | 0,1 | 2,3 | 4,5 | 6,7
Loose DL selections | Max. $\Delta\phi$ (mrad) | 2.8 | 2.0 | 1.7 | 1.5 | 2.1 | 1.7 | 1.6 | 1.5
Max. $\Delta\theta$ (mrad) | 35 | 18 | 10 | 6.5 | 3.5 | 1.5 | 0.7 | 0.5
Hit surival fraction | 55% | 18%
Tight DL selections | Max. $\Delta\phi$ (mrad) | 3.0 | 2.0 | 1.6 | 1.5 | 2.2 | 1.8 | 1.7 | 1.6
Max. $\Delta\theta$ (mrad) | 0.5 | 0.4 | 0.3 | 0.25 | 0.2 | 0.18 | 0.12 | 0.1
Hit survival fraction | 2% | 2%
Table 5: Angular selection on double-layers for hit suppression. A loose
(left) and tight (right) selection is shown. The hit survival rate represents
the fraction of hits (mostly from BIB) surviving the selections.

###  5.3 Combinatorial Kalman Filter

The Combinatorial Kalman Filter (CKF) algorithm seeded using hit triplets was
implemented using the ACTS library v13.0.0. The implementation is a Marlin
Processor that serves as a drop-in replacement for the existing tracking
Processors. In addition to providing an alternate algorithm designed for large
hit multiplicity, ACTS also provides a heavily optimized code for fast
computation. The triplet seeding + CKF used in this section reconstructs
tracks at the rate of 4min/event. It provides the first practical and
comprehensive tracking solution for the MCD.

The seeds for the CKF algorithm are formed from hit triplets in the four
layers of the Vertex Detector. Only hits in the outer half of doublet layer
are considered. Several heuristics are use to determine if each triplet is
compatible with a track and can be used as a seed. The seeding algorithm is
configured using the ACTS default values, with the exception of

  * •

radial distance between hits is between 5 and 80,

  * •

minimum estimated is 500MeV,

  * •

maximum forward angle of 80°,

  * •

extrapolated collision region is within 1mm of detector center,

  * •

average radiation length per seed is 0.1,

  * •

allowed amount of scattering is 50σ,

  * •

the middle hits in each seed are unique.

This configuration has not been fully optimized. For example, the size of the
collision region is smaller than the expected beam-spot size. Also one needs
to explore the use of the Outer Tracker for seeding. Its lower BIB-hit
multiplicity can allow the loosening of some requirements.

Around 150,000 seeds are found per event. The efficiency of the seeding
algorithm is shown in Fig. 17. Seeds are found for more than 90% of the muons
with >2GeV. Loosening the collision region definition increases the amount of
fake seeds and reduces the seed finding efficiency due to the seed overlap
removal. The latter can be addressed, at a cost in run time, by allowing
multiple seeds to share the same middle hit.

Figure 17: Seed finding efficiency for events containing a single muon with
BIB overlay for several definition collision region ("rough $z_{0}$") as a
function of truth (left) and $\lambda$ (right).

The CKF is run inside-out, meaning that the track extrapolation starts from
the inner-most seed hit and continues outward in the radial direction. The
initial track parameters are estimated from the seed. The CKF algorithm has
only two tunable parameters that are set as follows;

  * •

maximum one (closest) hit added at each layer,

  * •

hit search window at each layer of 10χ^2.

As with the seeding algorithm, these values are not optimized. The
consideration of only a single hit at each layer means that the CKF algorithm
will not branch to consider multiple track candidates for a single seed. This
seems very tight, however a good tracking reconstruction efficiency is still
seen. This underlines the difference between BIB and pile-up; the BIB-hits are
a "random" background that is not compatible with the trajectory of a track.

Fig. 18 shows the track reconstruction efficiency as a function of particle
and $\theta$ for single muon events with $p=\qty{10}{{\rm GeV}}$. Muons with
$p_{\textrm{T}}>\qty{2}{{\rm GeV}}$ are reconstructed with 90% efficiency or
greater even in the presence of BIB. There is a considerable amount of fake
tracks ($\approx 100,000$ per event). Fig. 19 compares the distribution and
the associated number of hits between real and fake tracks. Fake tracks are
mostly at low and are associated with a significantly smaller number of hits.
The latter further underlines the randomness of the BIB-hits and provides a
handle for reducing the fake rate.

Figure 18: Track reconstruction efficiency for events containing a single muon
with (blue) and without (orange) BIB overlay as a function of truth (left) and
$\theta$ (right).

Figure 19: Track (left) and $N_{\text{hit}}$ (right) distributions between
real (blue) and fake (orange) tracks in single muon events.

Fig. 20 shows the track reconstruction efficiency in a sample containing Higgs
bosons decaying to two $b$-quarks. This is a practical demonstration CKF
performance on a realistic event. Maintaining a good track reconstruction
efficiency as a function of production radius is important for identifying
secondary vertices of $b$-mesons. The tight configuration maintains above 70%
efficiency up to 5. While not perfect, it is a significant improvement over
the inefficiency of doublet-layer filtering required for CT.

Figure 20: Track reconstruction efficiency for events containing a Higgs boson
decaying to two $b$-quarks with (blue) and without (orange) BIB overlay as a
function of truth $p_{\textrm{T}}$ (left), $\theta$ (right) and production
radius (bottom).

##  6 Jets

Hadronic jets are fundamental objects in a Muon Collider experiment. As an
example, the measurements of the tri-linear and quadri-linear Higgs couplings
rely on the reconstruction and identification of $b$ jets from the Higgs boson
decay, as the dominant decay signature of the Higgs boson is into $b$-quarks
pair, with a branching ratio of $58\%$. Jet reconstruction is one of the most
difficult reconstruction tasks at a Muon Collider, since almost all sub-
systems are involved, and the impact of the BIB is significant in all of them,
with different features in different sub-detectors. The jet reconstruction
algorithm employed is described in this section, and its performance is
discussed. The algorithm has been designed to reconstruct jets in the presence
of the BIB, but it is far from being fully optimized, and further studies are
needed in the future. However we are going to demonstrate that even at this
stage the jet reconstruction can achieve a decent performance. Given the high
level of details of the full simulation (including the BIB impact) this give
us the confidence that measurements with jets are possible at a Muon Collider,
and further dedicated advancements on MDI, detectors and reconstruction
algorithms would go in the direction of improving the physics reach. The
algorithm works as follows:

  1. 1.

tracks are reconstructed using the Combinatorial Kalman Filter algorithm and
are filtered depending on the number of hits in the sub-systems;

  2. 2.

calorimeter hits are selected by requiring a hit time window and an energy
threshold;

  3. 3.

tracks and calorimeter hits are used by the PandoraPFA algorithm to obtain
reconstructed particles;

  4. 4.

the reconstructed particles are clustered into jets with the $k_{t}$
algorithm;

  5. 5.

requirements are applied to remove fake jets;

  6. 6.

a jet energy correction is applied.

The jet performance has been evaluated on simulated samples of $b\bar{b}$,
$c\bar{c}$ and $q\bar{q}$ dijets, where $q$ stands for a light quark ($u$,$d$
or $s$). These samples have been generated with an almost uniform dijet
$p_{T}$ distribution from 120 to 200GeV. Samples of $\mu^{+}\mu^{-}\to H(\to
b\bar{b})+X$ and $\mu^{+}\mu^{-}\to Z(\to b\bar{b})+X$ at $\sqrt{s}=3$ TeV are
also used to study the dijet invariant mass resolution. The BIB at
$\sqrt{s}=1.5$ TeV has been overlayed with all the samples.

###  6.1 Track filter

Tracks are reconstructed using the Combinatorial Kalman Filter algorithm
described in Section 5.3. Since more than 100k fake tracks per event are found
in this way, they are filtered depending on the number of hits in the Vertex
Detector and Inner Tracker. In Fig. 21 hit distributions for tracks from
$b$-jets and BIB are compared.

Figure 21: Distributions of the number of track hits in Vertex Detector and
Inner Tracker, for $b$-jet tracks and BIB combinatorial tracks. Distributions
are normalized to the same area. The distribution in the Vertex Detector peaks
at even numbers due to the double-layer structure of the detector.

A number of Vertex Detector hits greater than 3 and a number of Inner Tracker
hits greater than 2 are required for each track, reducing the average number
of BIB combinatorial tracks per event from more than 100k to less than 100.

###  6.2 Calorimeter hit selection

Calorimeter hits are filtered depending on the normalized hit time, defined as
$t_{N}=t-t_{0}-cD$, where $t$ is the absolute hit time, $t_{0}$ is the
collision time, $c$ is the speed of light, and $D$ is the hit distance from
the origin of the reference system. A time window of $\pm 250$ ps is applied
to remove most of the BIB hits but preserving the signal, as can be seen in
Fig. 22.

Figure 22: Normalized hit time in ECAL barrel, for $b$-jets and BIB. Both
distributions are normalized to the same area. Notice that this time is not
smeared for the detector time resolution. The time window of ±250 applied in
the jet reconstruction is shown.

As rule of thumb we can assume that a time window of width $\Delta$ is
applicable if the Full Width at Half Maximum (FWHM) of the cell time
resolution is below $\Delta/3$. Therefore, in this particular case, we assume
a FWHM of at least 167 ps, that should be achievable by state-of-the-art
calorimeter technologies.

Several calorimeter hit energy thresholds have been tested, in order to reduce
hits produced by BIB. It has been found that a threshold of 2 MeV applied to
both ECAL and HCAL is a compromise between jet performance and computing time.
In fact this requirement reduces the average number of ECAL Barrel hits from
1.5M to less than 10k. The computing time of the jet algorithm exponentially
grows with the number of calorimeter hits, therefore with the current
resources we are not able to reduce the thresholds far below 2 MeV. However
the impact of reducing the thresholds at the close value of 1 MeV is described
in Section 6.7.

###  6.3 Particle flow, jet clustering and fake jet removal

Calorimeter hits and tracks are given as input to the PandoraPFA algorithm,
that produces as output reconstructed particles known as particle flow
objects. The PandoraPFA algorithm is described in detail in [79]. Then
particle flow objects are clustered into jets by the $k_{T}$ algorithm. A cone
radius of $R=0.5$ is used.

At this point an average number of 13 fake jets per event is reconstructed,
which need to be removed by applying additional quality criteria. The number
of tracks in the jet has been found to be the most discriminating feature
between $b$-jets and fake jets, as shown in Fig.23 (left).

Figure 23: Number of tracks associated to real $b$-jets and to fake jets from
BIB (left). Jet fake rate as a function of the jet $p_{\textrm{T}}$ (right),
obtained after requiring at least one track associated to the jet.

It can be seen that most of the fake jets from BIB have no tracks associated
to them. Therefore, requiring at least one track allows to greatly reduce the
rate of fake jets. In Fig. 23 (right) the fake jet rate as a function of jet
$p_{\textrm{T}}$ is presented. It has been obtained using a sample of
$b\bar{b}$ dijets and defined as the average number of reconstructed fake jets
per event, which is well below 1%.

###  6.4 Jet momentum correction

The jet 4-momentum is defined by the sum of 4-momenta of particles that belong
to the jet. The jet axis is identified by the jet momentum direction. We
define the fiducial region of the jet reconstruction by selecting jets with
pseudorapidity $|\eta|<2.5$, where
$\eta=-\mathrm{log}[\mathrm{tan}(\theta/2)]$, with $\theta$ angle between the
jet axis and beam axis.

In order to recover the energy lost by reconstruction inefficiencies, as well
as to take into account BIB contamination, a correction to the jet 4-momentum
is applied. This correction has been determined by comparing the reconstructed
jet $p_{T}$ with the corresponding truth-level jet $p_{T}$. Truth-level jets
are defined as jet clustered by applying the $k_{t}$ algorithm to Monte Carlo
particles. Reconstructed and truth-level jets are matched if their $\Delta R$
distance is below 0.5, _i.e._ :

$\Delta R=\sqrt{(\Delta\eta)^{2}+(\Delta\phi)^{2}}<0.5,$

where $\Delta\eta$ and $\Delta\phi$ are respectively the pseudo-rapidity the
azimuthal angle differences between the reconstructed-level jet axis and
truth-level jet axis. If more than one reconstructed jet is matched to the
same truth-level jet than the one with lower $\Delta R$ is chosen.

The correction is evaluated in five equal-width intervals of reconstructed
$|\eta|$ between 0 and 2.5. Each pseudo-rapidity interval is further divided
in 19 equal-width intervals of reconstructed $p_{T}$ between 10 and 200 GeV.
For each interval the average and standard deviation of the truth-level jet
$p_{T}$ distribution is calculated. Transfer functions as then obtained in
each $\eta$ interval by fitting the average truth-level jet $p_{T}$ as a
function of reconstructed jet $p_{T}$. Examples of transfer functions are
shown in Fig. 24. These functions are then used obtain the scale factor that
is applied to each component of the reconstructed 4-momentum.

Figure 24: Transfer functions used for jet momentum corrections. The average
truth-level jet $p_{T}$ as a function of the reconstructed jet $p_{T}$ is
shown, for $0<|\eta|<0.5$ (left) and $1.5<|\eta|<2.0$ (right). Error bars
represent the standard deviation of the truth-level jet in each interval.

###  6.5 Jet reconstruction performance

The $b$-jet reconstruction performance is first evaluated on the simulated
sample of $b\bar{b}$ dijets. In Fig. 25 (left) the $b$-jet efficiency as a
function of the true $b$-jet $\eta$ is shown. It is worth noting that this is
only the jet reconstruction efficiency, while the flavor tagging performance
is studied in Section 6.8. The efficiency is around 90$\%$ in the central
region, but there is a drop for $|\eta|>2$, where the efficiency is 20$\%$. It
has been verified that this drop is mainly due to the fake jet removal
requirement, since many jets without reconstructed tracks are found in the
forward region. In Fig. 25 (right) the $b$-jet efficiency as a function of the
true $b$-jet $p_{T}$ in the region $|\eta|<1.5$ is presented, and it goes from
82$\%$ at low $p_{T}$ to 95$\%$ at higher $p_{T}$.

Figure 25: Efficiency of $b$-jet reconstruction as a function of truth-level
jet $\eta$ (left) and as a function of the truth-level jet $p_{T}$ (right, for
$|\eta|<1.5$).

The relative difference between reconstructed and true $b$-jet $\eta$ is shown
in Fig. 26 (right): the standard deviation of this distribution, directly
related to the jet-axis angular resolution, is 14$\%$. The $b$-jet $p_{T}$
resolution as a function of the true $b$-jet $p_{T}$ is shown in Fig. 26
(right).

Figure 26: Left: relative difference between reconstructed and true jet
pseudo-rapidity. Right: $b$-jet $p_{T}$ resolution as a function of the jet
$p_{T}$.

The resolution goes from 35$\%$ for jet $p_{T}$ around 20 GeV to 20$\%$ for
high jet $p_{T}$.

The jet reconstruction performances obtained separately in the $b\bar{b}$,
$c\bar{c}$ and $q\bar{q}$ samples and in the central region ($|\eta|<1.5$) are
compared in Fig. 27. It can be seen that both the efficiencies and $p_{T}$
resolutions are of the same order, however some differences exist between the
jet flavours: it has been checked that these are mainly due to different jet
$\eta$ distributions in the three samples.

Figure 27: Left: Jet reconstruction efficiency (left) and jet $p_{T}$
resolution (right) as a function jet $p_{T}$ for $b$-jets, $c$-jets and light
jets in the central region $|\eta|<1.5$. It has been checked that differences
between the jet flavours are mainly due to different jet $\eta$ distributions
in the three samples.

###  6.6 Invariant mass reconstruction of Higgs and Z bosons to b-quarks

The jet reconstruction is applied to the simulated samples of $H\rightarrow
b\bar{b}$ and $Z\rightarrow b\bar{b}$ to probe the dijet invariant mass
reconstruction. The invariant mass separation between these two processes is
of paramount importance for physics measurements at the muon collider. In this
study both jets are required to have $p_{T}>40$ GeV and $|\eta|<2.5$. In Fig.
28 the dijet invariant mass distributions for $H\rightarrow b\bar{b}$ and
$Z\rightarrow b\bar{b}$ are shown. The distributions are fitted with double
gaussian functions, and the shapes are compared. A relative width, defined as
the standard deviation divided by the average value of the mass distribution,
of 27$\%$(29$\%$) for $H\rightarrow b\bar{b}$($Z\rightarrow b\bar{b}$) is
found. It can be seen that a significant separation is achieved.

Figure 28: Dijet invariant mass distributions for $H\rightarrow b\bar{b}$ (top
left) and $Z\rightarrow b\bar{b}$ (top right) are shown. Distributions are
normalized to the same area, and are fitted with double gaussian functions.
The shapes are compared is the bottom plot.

###  6.7 Future prospects on jet reconstruction

Before discussing the heavy-flavour jet identification, we notice that, at
this stage, the jet reconstruction algorithm can be improved in several ways.
In this Section some guidelines are given:

  * •

track filter: it has been verified that the track filter has a different
impact in the central and the forward region, in particular the efficiency in
the forward region is lower. An optimized selection should be defined,

  * •

calorimeters threshold: the hit energy threshold has been set to the
relatively high value of 2 MeV, as a compromise between computing time and jet
reconstruction performance. This is a major limitation in the jet performance
as can be seen in Fig. 29 (left), where the $H\rightarrow b\bar{b}$ dijet
invariant mass, reconstructed without the presence of the BIB, is compared
between 2 MeV and 200 KeV thresholds. However reducing this threshold is not
an easy task, given the large number of calorimeter hits selected from the BIB
that contaminate the jet reconstruction. This is shown in Fig. 29 (right),
where can be clearly seen that the performance at 1 MeV threshold is degraded
with respect to 2 MeV. To tackle this problem an optimized algorithm should be
developed: as an example thresholds that depend on the sensor depth could by
applied, since the longitudinal energy distribution released by the BIB is
different from the signal jets as shown in Fig. 30 (right). A generalization
of this idea could be the application of a multivariate-algorithm trained to
select signal hits and reject BIB hits,

  * •

fake jet removal: the fake jet removal applied in this study has an impact in
reducing the jet efficiency in the forward region. Moreover this issue is
highly dependent from the calorimeter thresholds. A fake removal tool based on
machine learning and with jet sub-structure observables as input should be
developed to solve this task.

Figure 29: Left: $H\rightarrow b\bar{b}$ dijet invariant mass, reconstructed
without the presence of the BIB and with 2 MeV and 200 KeV calorimeter hit
energy thresholds. Right: $H\rightarrow b\bar{b}$ dijet invariant mass
reconstructed with 2 MeV and 1 MeV thresholds. Distributions are normalized to
the same area. Figure 30: Distribution of the ECAL barrel hits distance from
the interaction point (weighted for the hit energy), for both $b$-jets and
BIB. Both distributions are normalized to the same area.

###  6.8 Heavy-flavor jet identification

The $b$-jet identification algorithm described in this section relies on the
reconstruction of the secondary vertices inside the jet cones, that are
compatible with the decay of the heavy-flavour hadron.

For the secondary vertices reconstruction, tracks are reconstructed with the
Conformal Tracking algorithm described in Section 5.1, where the region of
interests are defined by cones with $\Delta R=0.7$ around the jet axes. In
order to reduce the computational time, the Double Layer filter (see Section
5.2) is applied to reduce the number of hits due to BIB. The effect of this
filter on the secondary vertex reconstruction is then corrected to obtain the
efficiency prior to its application, as will be explained in Section 6.8.2.

####  6.8.1 Secondary vertex tagging

The vertexing algorithm employed for the primary and secondary vertex
reconstruction is described in detail in [80]. In order to reduce the amount
of combinatorial tracks due to BIB, cuts are applied to the reconstructed
tracks given as input to the algorithm. The main steps of the algorithm and
cuts to the input tracks are here summarized:

  1. 1.

primary vertex finding: tracks with $|D_{0}|\leq\qty{0.1}{}$ and
$|Z_{0}|\leq\qty{0.1}{}$ are selected and given in input to the primary vertex
fitter. Furthermore, to reduce the number of combinatorial short BIB tracks,
each track is required to have at least four hits in the vertex detector;

  2. 2.

tracks selection for secondary vertex finder: tracks not used to reconstruct
the primary vertex are given in input to the secondary vertex fitter. Fig. 31
shows the distributions of the total number of hits in the vertex detector for
combinatorial BIB tracks and for tracks matched at Monte Carlo level with
particles generated by $b$ or $c$ hadrons decays. A minimum number of 4 hits
in the vertex detector is required in order to keep most of the the tracks
from $b$ and $c$ hadrons decay, while rejecting most of BIB tracks.

Figure 31:  Number of hits in the vertex detector of BIB combinatorial tracks
(red) and of tracks matched with Monte Carlo truth particles coming from $b$
or $c$ hadrons (blue and black respectively) decay. Distributions are
normalized to the unity area.

Other cuts on the track , the maximum track $Z_{0}$ (longitudinal impact
parameter) and $D_{0}$ (transverse impact parameter), and the $D_{0}$ and
$Z_{0}$ errors are applied in order to further reduce the amount of BIB
tracks. As an example, Fig. 32 (left) shows the $p_{T}$ distributions of BIB
combinatorial tracks and tracks from $b$\- or $c$-hadrons decays. By requiring
$p_{T}>0.8~{}{\rm GeV}$ about 80 $\%$ of the BIB tracks are rejected, while
about 85-90 $\%$ of the tracks from $b$\- or $c$-hadrons are kept. The $Z_{0}$
distributions of tracks coming from $b$\- and $c$-hadrons and combinatorial
BIB tracks are shown in Fig. 32 (right). The selection of tracks with
$|Z_{0}|\leq\qty{5}{mm}$ is applied to reject the tails at large $Z_{0}$ that
characterize BIB tracks;

Figure 32:  Left: $p_{T}$ distribution of tracks coming from $b$ (blue) or $c$
(black) hadrons decay and combinatorial BIB (red) tracks. Right: $Z_{0}$
distribution of tracks coming from $b$ (blue) or $c$ (black) hadrons decay and
combinatorial BIB (red) tracks. Distributions are normalized to the number of
tracks.

  3. 3.

secondary vertex finding: tracks passing the requirements are used to build
two-tracks vertex candidates, that must satisfy the following requirements:
the invariant mass must be below 10 GeV and must be smaller than the energy of
each track, the position with respect to the primary vertex must lie in the
same side of the sum of the tracks momenta, and the $\chi^{2}$ of the tracks
with respect the secondary vertex position must be below a threshold value,
set to 5. Track pairs are also required not to be compatible with coming from
the decay of neutral long lived particles. Additional tracks are iteratively
added to the two-tracks vertices if they satisfy the above requirements.
Tracks associated to more than one secondary vertex are assigned to the vertex
with the lowest $\chi^{2}$ and removed from other vertices.

####  6.8.2 SV-tagging performance

In order to evaluate the $b$-tagging efficiency and mis-identification, a
truth-level flavour is associated to the reconstructed jets with the following
steps:

  * •

first, the truth-level jets are matched with the quarks from Monte Carlo to
determine its flavour, requiring a distance $\Delta$R $<$ 0.5 between the
truth-level jet axis and quark momentum. If more than one truth-level jet is
found to match with the same quark, the one with the lowest $\Delta$R is
chosen;

  * •

then, the flavour of the reconstructed jets is determined by matching them
with the truth-level jets, by requiring a $\Delta$R distance between the jets
axes below 0.5;

  * •

if the reconstructed jet do not match with any truth-level jet, it is labeled
as fake.

The characteristics of secondary vertices inside reconstructed jets have been
studied in order to reduce the mis-identification of $c$, light and fake jets.
Fig. 33 shows the distribution of the secondary vertices proper lifetime
($\tau$) for $b$ jets, $c$ jets and light+fake jets. A cut on $\tau>$ 0.2 ns
rejects $\sim 30\%$ of both $c$ and light+fake jets, while keeping $90\%$ of
$b$ jets. A reconstructed jet is identified (tagged) as $b$ jet, if at least
one secondary vertex with $\tau>$ 0.2 ns is found inside its cone ($\Delta R<$
0.5).

The $b$-tagging efficiency is defined as:

$\epsilon_{b}=\frac{N_{b,SV}}{N_{b}}$ (1)

where $N_{b,SV}$ is the number of tagged and truth-matched $b$ jets, while
$N_{b}$ is the total number of truth-matched $b$ jets. Then, the mistag on $c$
and light+fake jets is calculated as:

$\mathrm{mistag}_{c,\mathrm{light}}=\frac{N_{c(\mathrm{light}),SV}}{N_{c(\mathrm{light})}}$
(2)

where $N_{c(\mathrm{light}),SV}$ is the number of tagged and truth-matched
$c$(light+fake) jets, while $N_{c(\mathrm{light})}$ is the total number of
truth-matched $c$(light+fake) jets.

The effect of the Double Layer Filter on the secondary vertex finding
efficiency have been evaluated reconstructing $b\bar{b}$, $c\bar{c}$ and
$q\bar{q}$ dijet samples without BIB, with and without the Double Layer
filter. The following ratio has been calculated as a function of of the jet
$p_{T}$ and of the angle of the jet axis with respect to the beam ($\theta$):

$r(p_{T},\theta)=\frac{N_{noBIB,NoDL}}{N_{noBIB,DL}},$ (3)

where $N_{noBIB,NoDL}$ is the number of tagged jets without double layer
filter and $N_{noBIB,DL}$ is the number of tagged jets with the double layer
filter. The final tagging efficiencies are then corrected for this ratio,
assuming that its value does not change in the presence of the BIB. The
$b$-tagging efficiency as a function of the jet $p_{T}$ and $\theta$ is shown
in Fig. 34. The efficiency is around $50\%$ at low $p_{T}$ and increases up to
$70-80\%$ at high $p_{T}$. At low $\theta$ the efficiency is lower ($\sim
30\%$) probably due to the presence of the nozzles. The mistag for $c$ jets is
shown in Fig. 35 and is found to be around $20\%$. As for $b$-jet efficiency,
the $c$ mistag increases in the central region of the detector. Fig. 36 (left)
shows the mistag for the light and fake jets versus jet $p_{T}$, up to 90 GeV,
and is found to be lower than $1\%$ below 50 GeV, while increases to $5\%$ at
higher jet $p_{T}$. Fig. 36 right shows the light+fake jet mistag as a
function of $\theta$.

Figure 33: Distribution of the secondary vertex proper lifetime for $b$, $c$
and light jets tagged. Distributions are normalized to the unit area.

Figure 34:  Left: $b$-tagging efficiency as a function of Right: $b$-tagging
efficiency as a function of the angle between the jet and the beam axes.

Figure 35: Mistag in $c\bar{c}$ dijet samples as a function of $p_{T}$ (left)
and $\theta$ (right).

Figure 36:  Mistag in light dijet samples as a function of $p_{T}$ (light) and
$\theta$ (right).

As for jet reconstruction, the $b$-jet identification algorithm needs further
improvements, but a solid starting point has been set. In particular it will
take advantage of the advancements in the vertex detector and track
reconstruction. Given the impact of the BIB, the features shown in this
section are not sufficient to setup an effective $c$-tagging algorithm,
therefore this case should be studied by defining a dedicated strategy for the
muon collider environment.

##  7 Photons and Electrons

###  7.1 Photon reconstruction and identification

The photon reconstruction and identification performance of the muon collider
detector is assessed in a sample of 100000 events with a single photon per
event. Photons were generated in the nominal collision vertex at the center of
the detector, uniformly distributed in energy between 1 and 1500 GeV, in polar
angle between $10^{\circ}$ and $170^{\circ}$, and in the full azimuthal angle
range. The sample was then processed with the detector simulation and fully
reconstructed. 40000 events were also reconstructed with the beam-induced
background overlaid.

The reconstruction procedure includes the CKF tracking and the calorimeter
reconstruction. Prior to track reconstruction, the tracker hits were processed
with the Doublet-Layer filter (see Section 5.2). Moreover, to get rid of most
of the fake tracks due to the spurious hits from the background, a track
quality selection is applied before the track refitting step, which requires
at least three hits in the vertex detector and at least two hits in the inner
tracker. To reject part of the background hits in the calorimeters, an energy
threshold of 2 MeV is applied to both the ECAL and HCAL hits. Photons are
reconstructed and identified with the Pandora Particle Flow algorithm. A
detailed description of the package can be found in Ref. [79].

The energy threshold of the calorimeter hits and the presence of the beam-
induced background affect the energy scale of the reconstructed photons. A
correcting factor is applied to the reconstructed photon energy to make the
detector response uniform as a function of the photon energy and the photon
polar angle. The correction was calculated from a comparison of the
reconstructed photon energy with the photon energy at generation level in an
independent subsample of events.

Figure 37: Photon reconstruction efficiency as a function of the photon energy
(left) and the photon polar angle (right).

Fig. 37 shows a comparison of the photon reconstruction efficiency as a
function of the generated photon energy and polar angle $\theta$ with and
without the BIB. The efficiencies are defined as the fraction of generated
photons in the range $10^{\circ}$ and $170^{\circ}$ that are matched to a
reconstructed photon within a cone
$R=\sqrt{(\Delta\phi)^{2}+(\Delta\theta)^{2}}<0.05$.

The dip in the region between the barrel and endcap calorimeters, which is
reflected in the efficiency below 400GeV .

effect of BIB on energy resolution standard deviation $\sigma_{\Delta E}$ of a
Gaussian fit to the distribution of residuals $E_{\gamma}^{RECO}-E_{\gamma}$.
It may be seen that the BIB degrades significantly the photon reconstructed
energy in the region between the barrel and the endcap.

Figure 38: Energy resolution of the reconstructed photond as a function of the
photon energy (left) and the photon polar angle (right).

Figure 39: Fraction of photons misidentified as electrons as a function of the
photon energy (left) and the photon polar angle (right).

Fig. 39 reports the fraction of photons that are reconstructed and identified
as electons.

.

###  7.2 Electron reconstruction and identification

Performance for electron reconstruction and identification is obtained in a
sample of 50000 events with a single electron per event produced at the
nominal collision point. The generated electrons are uniformly distributed in
energy between 1 and 1500 GeV, in polar angle between $10circ$ and $170circ$,
and in azimuthal angle over the whole range. The sample was processed with the
detector GEANT4 simulation and the Marlin reconstruction algorithms. A
subsample of 20000 events was also reconstructed with the beam-induced
background overlaid.

Electrons are identified by means of an angular matching of the
electromagnetic clusters, reconstructed in the same way as Section 39, with
tracks reconstructed with the CKF algorithm, as described in Section 5.3 in a
$R=0.1$ cone. The Double-Layer filter was used to clean the tracker hits
upstream the track reconstruction and tracks are required to have
$\chi^{2}/\textrm{ndof}<10$. In the presence of the beam-induced background,
the energy thresholds of the calorimeter hits play a dominant role for an
efficient and precise cluster reconstruction. In this study, a threshold of 5
MeV was set. Futher developments and more sophisticated algorithms are needed
to improve the cluster reconstruction, particularly in the lower energy
regime.

Figure 40: Comparison of the electron reconstruction efficiency as a function
of the electron energy (left) and the electron polar angle (right) in the case
of no beam-induced background and beam-induced background added to the event.

The electron reconstruction and identification efficiencies as a function of
the electron generated energy and polar angle are shown in Fig. 40.

The drop at $40\circ$ in the left panel is caused by a tracking inefficiency
introduced by the Double-Layer filter in the transition region between the VXD
barrel and endcap.

##  8 Muons

Preliminarily, the muon reconstruction has been performed, following CLIC
procedure, within the Pandora PFA framework [81], that allows to investigate
cluster topologies. A cluster is defined as a combination of hits (one hit per
layer) inside a cone and on neighbouring layers. A detailed description of the
muon reconstruction algorithm is reported in [64].

The efficiency, defined as the ratio between generated particles associated
with a cluster and total generated particles, has been evaluated for single
muons with transverse momentum uniformly distributed in the range 100MeV-1TeV
and polar angle $\qty{8}{\degree}<\theta<\qty{172}{\degree}$. No BIB was
overlaid. The cluster efficiency is higher than 99% for $\pT>\qty{10}{{\rm
GeV}}$ and higher than 98% for $\qty{8}{\degree}<\theta<\qty{172}{\degree}$.

Muon track $p_{T}$ resolution is shown in Fig. 41 where $\Delta p_{T}$ is the
difference between the generated muon $p_{T}$ and the $p_{T}$ of the
corresponding Pandora PFA reconstructed track. It results to be less than
$10^{-4}$ GeV-1 for $p_{T}>30$ GeV and around a factor of 7 better in the
barrel region compare to the endcap [82].

Figure 41: Muon track $p_{T}$ resolution as a function of $p_{T}$.

Figure 42: Left: BIB muon hit spatial distribution in the first layer of the
muon endcap. In red the hits not associated to a cluster. The blue circle
corresponds to region $\theta<8^{\circ}$, while the purple to
$\theta<10^{\circ}$. Right: Number of hits per bunch crossing in each layer of
the muon system. Different cuts are applied.

Running the Conformal Tracking algorithm on a full event and Pandora requires
an enormous computational effort to reconstruct a single event when BIB is
overlaid. Moreover, Pandora reconstruction is fully driven by the tracker
where the BIB hit density is very high as already discussed before. At the
same time, the BIB occupancy in the muon system is very low compared to other
detectors. BIB hits are concentrated in the endcaps around the beam axis
(Fig.42 left) and a simple geometrical cut, combined for example with a cut on
the track transverse momentum, allows to get rid of almost all the BIB hits
(Fig.42 right). This suggests using standalone muon objects to seed the global
muon track reconstruction.

A processor has been developed to clusterize muon hits inside a cone with a
certain angular aperture $\Delta
R_{\mu}=\sqrt{(\Delta\phi)^{2}+(\Delta\eta)^{2}}$. At least hits on five
layers are required to make a stand-alone muon track. Reconstructed hits in
the Vertex Detector, Inner Tracker and Outer Tracker are then filtered using
the region-of-interest approach described in Section 5.1: only hits in a cone
of aperture $\Delta R_{t}$ (see below) around the muon stand-alone direction
are selected. These hits are then passed to the CT algorithm. At the moment no
information coming from the ECAL and HCAL calorimeters is used.

Figure 43: Muon reconstruction efficiency as a function of the polar angle in
a sample of single muons with no BIB overlaid (left) and in a sample with
multi-muons in the final state both with and without BIB (right).

Figure 44: Muon reconstruction efficiency as a function of transverse momentum
in a sample of single muons with no BIB overlaid (left) and in a sample with
multi-muons in the final state both with and without BIB (right).

Two different samples have been used to evaluate the performance of the
processor:

  1. 1.

single muons with transverse momentum uniformly distributed in the range 100
MeV-700 GeV and polar angle $8^{\circ}<\theta<172^{\circ}$ with no BIB
overlaid

  2. 2.

HZ channel resulting in six muons in the final state at a center-of-mass
energy of $1.5\text{\,}\mathrm{T}\mathrm{e}\mathrm{V}$ both with and without
BIB.

The angular apertures chosen to maximize track purity and mitigate the
background are $\Delta R_{\mu}=0.02$ and $\Delta R_{t}=0.05$.

Figure 45: Muon track transverse momentum resolution as a function of in a
sample of single muons with no BIB overlaid (left) and in a sample with multi-
muons in the final state both with and without BIB (right).

Efficiencies are defined as the fraction of generated muons that are matched
to a track within a cone of aperture 0.01. Figure 43 shows the efficiency as a
function of the polar angle $\theta$ both for the first (left) and second
sample (right). An efficiency loss is evident in both cases at
$\theta<15^{\circ}$ and $\theta>165^{\circ}$ due to a limit of the CT
algorithm, while the two dips around $\theta\sim 52^{\circ}$ and $\theta\sim
132^{\circ}$ are due to tracks with hits both in endcap and barrel that
require a specific approach, currently under study.

To study the efficiency as a function of transverse momentum the acceptance
region was restricted to $15^{\circ}<\theta<165^{\circ}$ to take into account
only the region with a high efficiency for CT. Figure 44 shows an efficiency
greater than 85% for transverse momentum higher than
$80\text{\,}\mathrm{G}\mathrm{e}\mathrm{V}$ for both the samples. The drop of
the curve at low $p_{T}$ is due to known effects currently under study, that
mostly concern the inefficiency of reconstruction of muons in the region
between barrel and endcap and tracks with a high curvature for which the
choice of $\Delta R_{t}$ has to be optimized. The muon track transverse
momentum resolution (Fig. 45) is comparable with that obtained with CT and
Pandora without BIB.

In general the presence of the BIB does not strongly affect the performance:
the efficiency is lower only in the endcaps where all the BIB hits are
concentrated, and the $p_{T}$ resolution is just a few percent worse.

##  9 Luminosity measurement

A precise determination of the luminosity is of crucial importance for any
physics absolute cross section measurement since it directly translates to its
error. LHC experiments have dedicated detectors, so-called luminometers,
installed often the backward/forward region of the detector that are used in
combination to the van Der Meer scan method to precisely measure the
instantaneous luminosity. The $e^{+}e^{-}$ experiments like Belle2[83] and
BESIII[84], measure the integrated luminosity by counting the number of events
of a process whose cross section is theoretically known with high precision.
The most used one is the Bhabha scattering ($e^{+}e^{-}\rightarrow
e^{+}e^{-}$) where, for example, the theoretical uncertainty on $\sigma$ at
$\sqrt{s}=1-10$ GeV is 0.1% at large angle [85]. Due to the reduced acceptance
in the backward/forward region because of the nozzles shielding structure at
muon collider, the large angle muon Bhabha ($\mu^{+}$ $\mu^{-}$ $\rightarrow$
$\mu^{+}$ $\mu^{-}$) is proposed as a method for the luminosity measurement
instead. A first evaluation of the expected precision has been performed by
using a sample of $\mu$-Bhabha generated with Pythia at the center-of-mass
energy of 1.5TeV. The events are then simulated through the detector and
reconstructed by using the Pandora Particle Flow algorithm [79].

Muons with emission angle $\theta$ respect to the beam-pipe
$30^{\circ}<\theta<150^{\circ}$ are selected. The beam-induced background
effects in this angular region are negligible, as shown in Section 8. Two
opposite sign charge muons, both with transverse momentum
$p_{\textrm{T}}>\qty{130}{{\rm GeV}}$ and angle between muons direction pair
$\Delta\theta>3.08$ rad are required. Finally, the di-muon invariant mass has
to be $1440<M_{\mu\mu}<1560$ GeV. These selection requirements reject almost
all the physics background contributions and allow to keep the entire signal
sample at $\sqrt{s}=1.5$ TeV. A different center of mass energy, muon
reconstruction and sample selection need to be re-optimized.

By assuming an instantaneous luminosity $\mathcal{L}=1.25\times 10^{34}$
cm-2s-1 and considering a year of data taking ($10^{7}$ s) the statistical
uncertainty obtained at $\sqrt{s}=1.5$ TeV is 0.2% [86]. This study
demonstrates that the method can be applied at the muon collider from the
experimental point of view. The expected total uncertainty on the luminosity
measurement using this method strongly depends on the accuracy on the
theoretical cross section of $\mu$-Bhabha scattering at large angles, and at
high center of mass energy( several TeV). Further theoretical developments are
needed in this area.

##  10 Conclusions

A $\mu^{+}\mu^{-}$ collider offers a viable path towards a rich and scalable
physics program at the energy frontier. One of the biggest challenges in
designing a detector for the muon collider environment is the presence of
beam-induced background. In this manuscript, we presented the performance of a
detector design, including appropriate shielding near the interaction point,
that can successfully cope with the expected beam-induced background and
reconstruct with high accuracy the main physics observables needed for
carrying out the expected physics program. Expected performance for
reconstructing charged particles, jets, electrons, muons and photons has been
presented, in addition to preliminary results on tagging heavy-flavor jets and
measuring the delivered integrated luminosity.

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