Table of Links
2 Muons vs. Protons
3 Muon Colliders Are Gauge Boson Colliders
3.1 From the effective vector approximation to PDFs
3.2 PDFs with broken electroweak symmetry
4 Physics
4.1 Electroweak symmetry breaking
5 Complementarity
6 Summary and Future Directions
4 Physics
We turn next to the physics potential of a high-energy muon collider, focusing on some of the central themes – electroweak symmetry breaking (EWSB), naturalness, and dark matter (DM) – that have motivated new physics since the inception of the Standard Model. These considerations provide sharp goalposts for a future collider, indicating energies and luminosities that would enable such a collider to comprehensively explore the underpinnings of the Standard Model.
4.1 Electroweak symmetry breaking
The discovery of the Higgs completed the particle content of the SM. However, this discovery has also reinforced the puzzles associated with the Higgs field’s role in the SM, generating as much confusion as clarity. The Higgs is the linchpin of the SM, responsible for all of the masses of elementary particles as well as flavor mixings, via EWSB. The majority of the SM parameters associated with the Higgs are not determined by gauge invariance, and their values must be measured. Moreover, the very fact that EWSB occurs via the Higgs is put into the SM by hand, in that we must specify the potential. Before we declare that the SM is complete, we must measure all of its parameters.
Unfortunately, the path to completing the goal of measuring all the SM parameters is often regarded as requiring two different colliders after the HL-LHC, due to the reliance on two qualitatively different types of observables. The first is to probe the Higgs couplings to other SM particles; we note that the light flavor Yukawa couplings have yet to be measured at all. The second is to explore the Higgs potential itself. To study the couplings of light flavors to the Higgs requires an extremely clean collider environment, which favors lepton colliders, such as the low-energy Higgs factories that have been proposed. Their advantage is clearly illustrated in Fig. 4, which shows that Higgs production is a relatively large fraction of the total cross section at lepton colliders, once its production via gauge bosons is kinematically allowed. Even at these colliders, there should be sufficient luminosity to probe the Yukawa couplings of the charm quark, while other light flavors pose a significant challenge; the capability to tag and measure light flavors is a subject of ongoing research [58]. However, lepton colliders offer at least the promise of measurements that would be overwhelmingly difficult at hadron colliders, where precision measurement of SM Higgs branching fractions must overcome vast numbers of u, d, s, c and gluon background jets. Furthermore, a future lepton collider running at the Higgs mass pole could measure the s-channel resonance production to directly probe the lepton Yukawa coupling, precisely (at a muon collider) or with an upper limit of a few times the SM prediction (at an e +e − collider) [59–62].
Pursuing such a program of future Higgs measurements would not simply complete the SM, it could likely open the first window to physics beyond the SM. The Higgs boson is unique; it is the only apparently elementary scalar among all the particles observed in the universe. Its distinct properties provide many compelling reasons to investigate it further. The Higgs provides the only source of flavor physics in the SM; the most relevant, invariant portal to other BSM sectors or dark matter; the unitarization of scattering amplitudes in the SM; a window on early universe cosmology via the EW phase transition (EWPT), and potentially EW baryogenesis (EWBG); and, last but not least, the naturalness puzzle. We are strongly motivated to determine whether the Higgs is solely responsible for EWSB, and whether it is (partially) composite. In this section, we will discuss a muon collider’s role in addressing these topic.[7] All of these questions can be attacked by measuring the Higgs’s properties with sufficient precision. Many of them benefit from the large Higgs production rate and cleanliness of a high energy muon collider, as well as the dynamical range of c.m. energy that such a collider achieves by virtue of being a vector boson collider. An apt analogy for the path that started with finding the Higgs and continues by investigating it in sufficient detail is provided by cosmology. While the expansion of the universe was known since Hubble, it was not until many decades later that the right observable was found and measured precisely enough that the accelerated expansion of the universe was conclusively discovered. We are now just beginning to acquire experimental knowledge of the Higgs boson’s properties, at a relatively coarse level. We must move toward the new era of precision Higgs physics, which, like precision cosmology, offers the hope of revolutionizing our understanding of the universe.
4.1.1 Higgs coupling sensitivity estimates from on-shell Higgs processes
There are many new measurements of Higgs properties that are accessible via higher energies and cleaner environments, and we will explore a sampling of these in the subsequent sections. The desire to improve Higgs precision will drive one of the core programs for any future collider. Therefore, it is important to understand how precisely a high energy muon collider could measure Higgs properties on its own, as well as in combination with other colliders. The answer to this question depends both on the theoretical framework and experimental details, which leave an enormous range of possibilities that are beyond the scope of this paper to explore. In order to make a first quantitative estimate rather than simply stating that a large number of Higgs particles would be produced, we will make a number of simplifying assumptions. In this section, we focus on the processes in which on-shell Higgs bosons are produced. At a high energy muon collider, off-shell Higgs processes will in some cases offer an even more powerful probe of Higgs properties, a topic to which we will return in Sec. 4.1.2.
First, we will adopt the common κ fits for Higgs precision [65, 66]. This is not an endorsement of this methodology compared to any other, but a pragmatic choice for the sake of making comparisons, as all future collider proposals have an example of this type of fit (kappa-0 framework) [67]. The inputs to such a fit are the uncertainties on the cross section measurements in exclusive channels. These depend upon the signal cross section and physics backgrounds, as well as machine backgrounds, detector capabilities, and possible additional theoretical assumptions. The machine backgrounds and detector capabilities are particularly interesting in the context of high-energy muon colliders, as previously discussed. The BIB at muon-colliders serves both as a background to measurements and a driver of detector design. There is no optimized detector design available at all our benchmark c.m. energies, due to the fact that the BIB depends on the accelerator complex within roughly 25 m on each side of the interaction point. Therefore, we will simply choose our energy and luminosity benchmarks to be 10 TeV and 10/ab, and using the Muon Collider detector card for the Delphes fast simulation [68]. This choice of “detector” does not serve as a final word, but allows us to begin exploring how the physics requirements interact with detector design. We do not include BIB, as current full-simulation studies show that it appears to be under control, especially at higher energies [69,70], and there are potential ways to reduce its effects further. Furthermore, for this toy study we do not include physics backgrounds.
While our assumptions may seem like too drastic of a simplification, there still is useful sensitivity information despite having made these naively non-conservative estimates. Our signal rates using Delphes have a rather small acceptance, given that the detector card limits physics objects to |η| < 2.5 except for forward muons. There are a number of motivations for the detector card inspired from a hybrid of CLIC and FCC-hh for efficiencies and reconstruction [71], but these are not optimized for a particular physics target or energy. Additionally, the general acceptance roughly coincides with having BIB-suppressing
tungsten nozzles [72,73], with a 10° opening angle motivated by 1.5 TeV c.m. muon collider studies [73,74]. The nozzle opening should be able to be reduced at higher energies since the radiation will be more forward [35] (and timing should also mitigate BIB effects). Physics backgrounds, of course, potentially matter a great deal more than BIB. However, they are significantly reduced at a lepton collider as shown in Fig. 4. Detailed sensitivity studies including physics backgrounds have been performed for 3 TeV lepton colliders for CLIC Higgs studies [75], and thus serve as a proof of principle (or potential floor) for our signal-driven sensitivities. There are a variety of studies that can and should be done in the future, but we hope this serves as a useful starting point by showing the effects of acceptance and efficiency via fast simulation. From the perspective of signal and BIB, we take this to be a conservative starting point.
We stress again that the performance in the various channels shown in Table 2 is in no way optimized and needs further study. However, with these putative sensitivities, we can perform a simple 10-parameter κ fit to compare this benchmark to other proposed colliders. Of course, the muon collider can be enhanced with complementary measurements from other colliders as well. Therefore we also perform the fit with the HL-LHC or a 250 GeV e +e − collider included. Here we took the CEPC input with full correlation matrix for different channels [6, 79] to represent the 250 GeV e +e − collider. The results and discussion on complementarity with other lepton collider Higgs factories would be similar. We present the results of these fits in Table 3.
4.1.2 Flavor and exotic couplings
4.1.3 The Higgs potential and the electroweak phase transition
One of the most intriguing aspects of EWSB is its role in the early universe. Because we can not directly observe the early universe before the time of formation of the CMB other than through gravitational waves, we must make use of particle physics to draw inferences about what occured. Many interesting and yet unmeasured epochs in cosmology are directly intertwined with EWSB. For example, the evolution of neutrinos in the universe and the properties of the cosmic neutrino background depend crucially on the W and Z boson masses. The masses of SM particles arise from EW symmetry breaking, and so may have turned on during the EWPT in a thermal history in which EW symmetry was restored at even earlier times and hotter temperatures. If the EWPT was strongly first order and other sources of CP
violation exist – both of which require new physics beyond the SM – then EW baryogenesis could explain the matter/antimatter asymmetry in our universe.
4.1.4 Additional Higgs bosons
As a final case study demonstrating the potential for a high-energy muon collider to illuminate the physics of EWSB, we consider the search for additional “Higgs bosons” that acquire their Standard Model couplings by mixing with the Higgs. This is exemplified by one of the simplest extensions of the Higgs sector, a real scalar singlet with renormalizable couplings to the SM Higgs.[9] This encodes a large class of BSM theories which address the stability of the electroweak scale [98, 101], or relate the baryon asymmetry in the universe today with the EWPT [102–106]. More generally, given the diversity of vector bosons and fermions in the SM, it is natural to ask if the scalar sector possesses similar depth.
A scalar singlet SM extension is a very useful benchmark to assess the capabilities of future colliders [28,107], since it manifests itself in a two-fold way: indirectly, as modification of the Higgs decay rates, and directly, in single and double production channels. Both these effects are controlled by the same small set of parameters – notably the singlet mass and its mixing with the Higgs boson – allowing for an immediate comparison of the direct and indirect reach. As we shall see, the ability of a very high energy lepton collider to discover heavy resonances is crucial to overcoming the limitations of Higgs precision measurements, which are inevitably constrained by systematic uncertainties, and allows the exploration of entirely new territory involving weakly interacting new physics in the 10 TeV range.
The singlet phenomenology is dictated by the following Lagrangian
Production modes and decay channels
Single production proceeds via mixing with the Higgs. Its cross section is proportional to the mixing angle and is only logarithmically sensitive to the mass (at high energies), and
it can be simply written as
Sensitivity of a muon collider
where the factor proportional to α = 3% takes into account possible systematic uncertainties.
The results for the various collider benchmarks are shown in Fig. 13 (right).
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
[7] The Higgs potential also lies at the root of deep questions about the stability of the universe, which we will leave for future investigations.
[9] The sensitivity of muon colliders to extended Higgs sectors with electroweak doublets was recently studied in [30].