Table of Links Abstract and 1. Introduction Abstract and 1. Introduction 2 Muons vs. Protons 2.1 Muon annihilation 2.1 Muon annihilation 2.2 Vector boson fusion 2.2 Vector boson fusion 2.3 Annihilation vs. VBF 2.3 Annihilation vs. VBF 2.4 Signal vs. background 2.4 Signal vs. background 3 Muon Colliders Are Gauge Boson Colliders 3 Muon Colliders Are Gauge Boson Colliders 3.1 From the effective vector approximation to PDFs 3.1 From the effective vector approximation to PDFs 3.2 PDFs with broken electroweak symmetry 3.2 PDFs with broken electroweak symmetry 3.3 Impact of subleading logs 3.3 Impact of subleading logs 3.4 Finite mass effects 3.4 Finite mass effects 4 Physics 4.1 Electroweak symmetry breaking 4.1 Electroweak symmetry breaking 4.2 Dark matter 4.2 Dark matter 4.3 Naturalness 4.3 Naturalness 5 Complementarity 5.1 EDMs 5.1 EDMs 5.2 Flavor 5.2 Flavor 5.3 Gravitational waves 5.3 Gravitational waves 6 Summary and Future Directions 6 Summary and Future Directions Acknowledgments Acknowledgments A. Simplified Models A. Simplified Models A.1 Standard Model A.1 Standard Model A.2 Supersymmetry A.2 Supersymmetry A.3 Vector-like quarks A.3 Vector-like quarks A.4 Higgs portal A.4 Higgs portal A.5 Hidden valleys A.5 Hidden valleys A.6 Axion-like particles A.6 Axion-like particles References References 4.2 Dark matter The predominance of apparently non-baryonic matter in the universe remains one of the few unambiguous indicators of physics beyond the Standard Model, and identifying the microscopic properties of dark matter is a central goal of multiple fields. Among the many candidates for particle dark matter, the Weakly Interacting Massive Particle (WIMP) paradigm has long been one of the most compelling. Within this paradigm, dark matter candidates arising as the lightest member of an electroweak (EW) multiplet form a particularly simple class of models [37, 111, 112]. The thermal relic abundance of such “minimal” dark matter is fixed strictly in terms of the quantum numbers of the electroweak multiplet in question, picking out a high mass scale between 1 - 23 TeV for SU(2)L representations ranging from doublets to septuplets. This makes minimal dark matter a motivated but difficult scenario for colliders in light of the high mass scale. Additionally, it is challenging from the detector point of view, because the typically small splittings of the EW multiplets suppress the amount of visible energy (and hence, missing momentum) in a typical event. Nevertheless, the abundant electroweak cross sections and relatively low irreducible backgrounds at a muon collider make it well positioned to search for minimal dark matter, to the point where a muon collider of sufficient energy could potentially render a decisive verdict on the scenario. In this section, we summarize the studies performed in Ref. [33], adapting their projections to the optimistic and conservative luminosity targets presented here Of course, minimal dark matter candidates may also arise from real or complex scalar multiplets carrying electroweak quantum numbers. Scalars admit more renormalizable couplings to the Standard Model, most notably through Higgs portal operators of the schematic form χχ†HH† . These couplings can induce significant tree-level mass splittings after electroweak symmetry is broken, introducing a high degree of model dependence. We leave this case for future study. As noted above, the leading interactions between a minimal dark matter multiplet and the Standard Model are strictly controlled by the multiplet’s EW quantum numbers. These interactions control the thermal relic abundance of cold dark matter resulting from freeze-out. Assuming that this is the sole source of the dark matter’s abundance, matching current observations [118] determines the dark matter’s “thermal target” mass. The various fermionic dark matter candidates and their corresponding thermal target masses are enumerated in Table 4. It bears emphasizing that the perturbative calculation of the thermal target mass is subject to large corrections from both Sommerfeld enhancement [119–121] and bound state effects [122, 123]. For the purposes of this discussion, we mainly use the thermal targets presented in Ref. [37], which themselves are primarily obtained by including Sommerfeld corrections to results in Ref. [117]. A notable exception is the quintuplet Majorana fermion, for which bound state effects are significant; these lift the thermal target from 9 TeV to 14 TeV [123]. This is in contrast to the triplet Majorana fermion, for which bound state effects do not shift the thermal target relative to the Sommerfeld calculation [123]. For the septuplet Majorana fermion, we obtain an approximate target by using the fact that the degrees of freedom decrease by a factor of two relative to the Dirac fermion, pushing the thermal target higher by a factor of √2 relative to the Dirac case [117]. Needless to say, all thermal targets quoted here are subject to residual theoretical uncertainties. Experimental coverage of these targets is a compelling goal for a future collider program. High-energy muon colliders are exceptionally well-positioned in this regard. There are a number of promising channels in which to search for minimal dark matter, including monophoton, mono-muon, and VBF di-muon final states with an inclusive missing mass signature [33]. Alternately, the production of charged particles in the multiplet followed by decay into dark matter (and soft tracks) gives rise to a promising disappearing track signature, where the small mass splitting due to EW corrections translates into a macroscopic distance traversed by the charged particles before they decay. This channel’s performance is subject to considerable uncertainties owing to the currently-unknown beam-induced backgrounds at a muon collider, but may significantly enhance the reach [34]. Here we summarize the performance of each channel, following [33]. • Disappearing tracks: In this channel, the charge ±1 particle in the EW multiplet is pair produced and decays into the dark matter plus soft particles with a long lifetime due to the small radiative mass splitting. For the cases considered here, cτ ranges from 0.37 cm to 5.6 cm. If the charged particle hits several layers of the tracker before decaying, this results in the unique signature of a disappearing track, a potentially low-background process. However, making accurate projections for the reach of this channel at a muon collider is hampered by our current ignorance of tracker design and beam-induced backgrounds. Here we present an estimate based on the combination of a singlet displaced track plus another tagging object such as a photon (with the expectation that this will be required to suppress backgrounds), requiring tens of signal events for discovery. Requiring two displaced tracks would necessarily provide further background suppression, albeit with a significant loss of rate; this channel would be useful to study further in the event that backgrounds for the single-track channel prove to be prohibitive. Focusing on the single-track final state with an additional tagging object, the mono-photon channel with one disappearing track will have the largest signal rate, significantly extending the reach for all odd-dimensional cases. However, this channel fails to reach the kinematic threshold owing to the boost required for the charged particle to leave enough hits in the tracker before decaying.11 The triplet enjoys the greatest increase in sensitivity from this channel, coming close to the kinematic threshold, while for the doublet, this channel is stronger than the mono-muon channel. • Disappearing tracks: The 5σ discovery reach of muon colliders operating at various center of mass energies is summarized in Fig. 15 for the optimistic (conservative) integrated luminosity scaling scenarios defined in Table 1. The sensitivity obtained by the combination of missing mass searches in the mono-photon, mono-muon, and VBF di-muon channels is shown separately from the sensitivity of the displaced track search. When combining the missing mass channels in the optimistic luminosity scaling scenario (left panel), the overall reach does not extend to the kinematic limit mχ ∼ √ s/2 (most notably for multiplets with n ≤ 3) due to the low signal-to-background ratio. It is possible to cover (with 2σ confidence) the thermal targets of the doublet and Dirac triplet with a 10 TeV muon collider, while a 30 TeV option would suffice for the Majorana triplet. The thermal targets of Dirac (Majorana) quintuplet would be covered by muon colliders operating at 30 (100) TeV, while a 100 TeV collider would also cover the thermal target for the septuplet. Rather than considering the reach of the benchmark collider energies, it is also interesting to note the minimum collider energy that would cover a given multiplet, assuming integrated luminosity scales with s. From this perspective, a Majorana triplet can be reached by a 20 TeV muon collider (still assuming integrated luminosity scales with s). A Majorana quintuplet can be covered by a 50 TeV muon collider, while a septuplet can be covered by a 70 TeV muon collider. The thermal targets of all the minimal multiplets considered here could be discovered at the 5σ level by a muon collider operating at 75 TeV. Finally, we emphasize that the disappearing track signal has excellent potential, bringing the reach close to the kinematic threshold mχ ∼ √ s/2 on the basis of the current study [33]. For instance, a 10 TeV muon collider alone could reach the thermal target of both doublet and triplet cases with a disappearing track search, motivating further studies and careful consideration for detector design. Authors:
(1) Hind Al Ali, Department of Physics, University of California, Santa Barbara, CA 93106, USA;
(2) Nima Arkani-Hamed, School of Natural Sciences, Institute for Advanced Study, Princeton, NJ, 08540, USA;
(3) Ian Banta, Department of Physics, University of California, Santa Barbara, CA 93106, USA;
(4) Sean Benevedes, Department of Physics, University of California, Santa Barbara, CA 93106, USA;
(5) Dario Buttazzo, INFN, Sezione di Pisa, Largo Bruno Pontecorvo 3, I-56127 Pisa, Italy;
(6) Tianji Cai, Department of Physics, University of California, Santa Barbara, CA 93106, USA;
(7) Junyi Cheng, Department of Physics, University of California, Santa Barbara, CA 93106, USA;
(8) Timothy Cohen, Institute for Fundamental Science, University of Oregon, Eugene, OR 97403, USA;
(9) Nathaniel Craig, Department of Physics, University of California, Santa Barbara, CA 93106, USA;
(10) Majid Ekhterachian, Maryland Center for Fundamental Physics, University of Maryland, College Park, MD 20742, USA;
(11) JiJi Fan, Department of Physics, Brown University, Providence, RI 02912, USA;
(12) Matthew Forslund, C. N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY 11794, USA;
(13) Isabel Garcia Garcia, Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA;
(14) Samuel Homiller, Department of Physics, Harvard University, Cambridge, MA 02138, USA;
(15) Seth Koren, Department of Physics and Enrico Fermi Institute, University of Chicago, Chicago, IL 60637, USA;
(16) Giacomo Koszegi, Department of Physics, University of California, Santa Barbara, CA 93106, USA;
(17) Zhen Liu, Maryland Center for Fundamental Physics, University of Maryland, College Park, MD 20742, USA and School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA;
(18) Qianshu Lu, Department of Physics, Harvard University, Cambridge, MA 02138, USA;
(19) Kun-Feng Lyu, Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong S.A.R., P.R.C;
(20) Alberto Mariotti, Theoretische Natuurkunde and IIHE/ELEM, Vrije Universiteit Brussel, and International Solvay Institutes, Pleinlaan 2, B-1050 Brussels, Belgium;
(21) Amara McCune, Department of Physics, University of California, Santa Barbara, CA 93106, USA;
(22) Patrick Meade, C. N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY 11794, USA;
(23) Isobel Ojalvo, Princeton University, Princeton, NJ 08540, USA;
(24) Umut Oktem, Department of Physics, University of California, Santa Barbara, CA 93106, USA;
(25) Diego Redigolo, CERN, Theoretical Physics Department, Geneva, Switzerland and INFN Sezione di Firenze, Via G. Sansone 1, I-50019 Sesto Fiorentino, Italy;
(26) Matthew Reece, Department of Physics, Harvard University, Cambridge, MA 02138, USA;
(27) Filippo Sala, LPTHE, CNRS & Sorbonne Universite, 4 Place Jussieu, F-75252 Paris, France
(28) Raman Sundrum, Maryland Center for Fundamental Physics, University of Maryland, College Park, MD 20742, USA;
(29) Dave Sutherland, INFN Sezione di Trieste, via Bonomea 265, 34136 Trieste, Italy;
(30) Andrea Tesi, INFN Sezione di Firenze, Via G. Sansone 1, I-50019 Sesto Fiorentino, Italy and Department of Physics and Astronomy, University of Florence, Italy;
(31) Timothy Trott, Department of Physics, University of California, Santa Barbara, CA 93106, USA;
(32) Chris Tully, Princeton University, Princeton, NJ 08540, USA;
(33) Lian-Tao Wang, Department of Physics and Enrico Fermi Institute, University of Chicago, Chicago, IL 60637, USA;
(34) Menghang Wang, Department of Physics, University of California, Santa Barbara, CA 93106, USA. Authors: Authors: (1) Hind Al Ali, Department of Physics, University of California, Santa Barbara, CA 93106, USA; (2) Nima Arkani-Hamed, School of Natural Sciences, Institute for Advanced Study, Princeton, NJ, 08540, USA; (3) Ian Banta, Department of Physics, University of California, Santa Barbara, CA 93106, USA; (4) Sean Benevedes, Department of Physics, University of California, Santa Barbara, CA 93106, USA; (5) Dario Buttazzo, INFN, Sezione di Pisa, Largo Bruno Pontecorvo 3, I-56127 Pisa, Italy; (6) Tianji Cai, Department of Physics, University of California, Santa Barbara, CA 93106, USA; (7) Junyi Cheng, Department of Physics, University of California, Santa Barbara, CA 93106, USA; (8) Timothy Cohen, Institute for Fundamental Science, University of Oregon, Eugene, OR 97403, USA; (9) Nathaniel Craig, Department of Physics, University of California, Santa Barbara, CA 93106, USA; (10) Majid Ekhterachian, Maryland Center for Fundamental Physics, University of Maryland, College Park, MD 20742, USA; (11) JiJi Fan, Department of Physics, Brown University, Providence, RI 02912, USA; (12) Matthew Forslund, C. N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY 11794, USA; (13) Isabel Garcia Garcia, Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA; (14) Samuel Homiller, Department of Physics, Harvard University, Cambridge, MA 02138, USA; (15) Seth Koren, Department of Physics and Enrico Fermi Institute, University of Chicago, Chicago, IL 60637, USA; (16) Giacomo Koszegi, Department of Physics, University of California, Santa Barbara, CA 93106, USA; (17) Zhen Liu, Maryland Center for Fundamental Physics, University of Maryland, College Park, MD 20742, USA and School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA; (18) Qianshu Lu, Department of Physics, Harvard University, Cambridge, MA 02138, USA; (19) Kun-Feng Lyu, Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong S.A.R., P.R.C; (20) Alberto Mariotti, Theoretische Natuurkunde and IIHE/ELEM, Vrije Universiteit Brussel, and International Solvay Institutes, Pleinlaan 2, B-1050 Brussels, Belgium; (21) Amara McCune, Department of Physics, University of California, Santa Barbara, CA 93106, USA; (22) Patrick Meade, C. N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY 11794, USA; (23) Isobel Ojalvo, Princeton University, Princeton, NJ 08540, USA; (24) Umut Oktem, Department of Physics, University of California, Santa Barbara, CA 93106, USA; (25) Diego Redigolo, CERN, Theoretical Physics Department, Geneva, Switzerland and INFN Sezione di Firenze, Via G. Sansone 1, I-50019 Sesto Fiorentino, Italy; (26) Matthew Reece, Department of Physics, Harvard University, Cambridge, MA 02138, USA; (27) Filippo Sala, LPTHE, CNRS & Sorbonne Universite, 4 Place Jussieu, F-75252 Paris, France (28) Raman Sundrum, Maryland Center for Fundamental Physics, University of Maryland, College Park, MD 20742, USA; (29) Dave Sutherland, INFN Sezione di Trieste, via Bonomea 265, 34136 Trieste, Italy; (30) Andrea Tesi, INFN Sezione di Firenze, Via G. Sansone 1, I-50019 Sesto Fiorentino, Italy and Department of Physics and Astronomy, University of Florence, Italy; (31) Timothy Trott, Department of Physics, University of California, Santa Barbara, CA 93106, USA; (32) Chris Tully, Princeton University, Princeton, NJ 08540, USA; (33) Lian-Tao Wang, Department of Physics and Enrico Fermi Institute, University of Chicago, Chicago, IL 60637, USA; (34) Menghang Wang, Department of Physics, University of California, Santa Barbara, CA 93106, USA. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv available on arxiv [11] In this case, further sensitivity may be obtained from using timing information [124]. However, the large out-of-time contribution from beam-induced backgrounds requires more detailed studies.

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