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
4.3 Naturalness
The hierarchy problem is a prime motivator that new physics should be accessible at colliders, as it strongly correlates the mass scale of additional degrees of freedom with those of the Standard Model. Precisely what degrees of freedom appear at scales indicated by the hierarchy problem is much less definite; in recent years it has become increasingly clear that there exists a plethora of solutions with wildly varying signatures. Nonetheless, the spectrum of solutions can be usefully divided into two categories: solutions of the “big” hierarchy problem, namely those reaching from the weak scale all the way to the putative scale of quantum gravity, and solutions of the “little” hierarchy problem, extending from the weak scale to the highest scales directly probed (thus far) by experiments.
There are two known solutions to the “big” hierarchy problem: compositeness and supersymmetry. The lack of weak scale evidence for either solution suggests the existence of a mass gap between the Higgs and whatever physics resolves the hierarchy problem. Such a mass gap implies a significant degree of fine tuning, somewhere between the percent and per mille level depending on the details of the UV completion. While it is entirely possible that the weak scale is finely tuned – after all, we do not actually know how Nature computes fine tuning – a robust commitment to naturalness could suggest the existence of additional physics that bridges the gap between the weak scale and the appearance of supersymmetry or compositeness. Resolutions to this “little” hierarchy problem need only span an order of magnitude in energy in order to reconcile the paucity of new physics at the weak scale with the expectations of naturalness. In contrast to the sparsity of qualitative solutions to the big hierarchy problem, there are innumerable solutions to the little hierarchy problem consistent with current data, ranging from dynamical mechanisms that relax the Higgs mass [125] to symmetry-based mechanisms that reside in “dark” hidden sectors [98,126]. Their signatures are equally diverse, often falling outside the scope of conventional collider signals.
Solutions to the hierarchy problem reduce the UV sensitivity of the Higgs mass parameter, making it possible to understand what sets the value of the weak scale and, ultimately, why electroweak symmetry is broken in the first place. To systematically test the naturalness of the weak scale, one would ideally like to pursue two lines of experimental inquiry: leveraging precision to directly test solutions to the “little” hierarchy problem at or around the weak scale, and leveraging energy to reach the scale at which states associated with resolution of the “big” hierarchy problem begin to appear. The great advantage of a high-energy muon collider is that, provided sufficient energy and luminosity, it may achieve both goals. On the one hand, the relatively low background rate and clean environment make it a promising tool for discovering new light states with weak (or no) Standard Model quantum numbers. On the other hand, the high c.m. energy gives it the reach to discover states well above the weak scale. In what follows, we illustrate this potential by considering aspects of muon collider sensitivity to representative solutions of the “big” and “little” hierarchy problems.
4.3.1 The “big” hierarchy problem: supersymmetry
The many superpartners predicted by supersymmetry give rise to a host of experimental signatures; see e.g. [127] for an overview. Here we will focus on the three states most closely tied to the naturalness of the Higgs potential: the higgsino, stop, and gluino, which respectively contribute to the Higgs potential at tree level, one loop, and two loops; these are the calling cards of “natural supersymmetry” [128–132]. In addition, we will explore a unique opportunity available to high-energy lepton colliders: probing low-scale supersymmetry breaking sectors through a direct search for the gravitino.
4.3.1.1 Higgsinos, stops, and gluinos
Not all supersymmetric final states are so distinctive, however. For the higgsino, if Standard Model radiative corrections are the only source of mass splitting, then there is little phase space for missing energy without additional initial-state radiation. The mass reach then becomes more sensitive to backgrounds. Of course, if there is additional splitting in the higgsino multiplet, e.g., due to mixing with a partially decoupled bino or wino, then the final state rapidly becomes more distinctive and the estimate of the reach follows the same logic as above.
With this in mind, we estimate the reach for various superpartners, beginning with the higgsino. The mass of the higgsino is the most immediate measure of fine tuning in the Higgs potential, since supersymmetry relates the masses of the higgsino and Higgs doublets at tree level. In general the contribution of the higgsino to the tuning of the weak scale[12] is parametrically of the form
Now we turn to the stop, whose contribution to the tuning of the weak scale at leading-logarithmic order is
Apart from considerations of fine tuning, the stop mass is also central to the supersymmetric prediction of the observed Higgs mass. Famously, accommodating mh ∼ 125 GeV
in the minimal supersymmetric extension of the Standard Model without significant mixing between the stop gauge eigenstates suggests that the stops lie around or above ∼ 5 - 10 TeV; see, e.g., [136] for a review. A high-energy muon collider operating at √s = 30 TeV could cover the typical scale of the stop mass suggested by the observed Higgs mass at large values of tan β, subject to further dependence on mixing angles and the remaining sparticle spectrum. At moderate values of tan β, the stops can be heavier, ∼ 100 - 1000 TeV in wellmotivated models. However, in such scenarios the electroweakinos may be accessible (e.g., via the searches discussed in Sec. 4.2), and the lightest sfermions, such as the right-handed stau, may be an order of magnitude lighter than the stops and could be directly accessible. Thus, the full range of searches for superparticles could cover a substantial portion of the parameter space motivated by the measured Higgs mass.
Taken together, these projections suggest that a high-energy muon collider operating at √s ∼ 20 - 30 TeV is capable of probing a natural supersymmetric explanation of the weak scale beyond the per mille level, with meaningful sensitivity to both of the states most important for understanding the scale and origin of electroweak symmetry breaking: the higgsinos and stops. Setting aside fine tuning, which may or may not be a sharp guide to the scale of new physics, a collider operating at √s ∼ 30 TeV could reach the scale suggested by the Higgs mass prediction of the MSSM, regardless of the the stop sector mixing.
4.3.1.2 The gravitino
Although impressive in reach, the potential for a high-energy muon collider to probe Standard Model superpartners represents a continuation of the already-vast search program currently under way at the LHC. But a muon collider offers more than incremental progress in the search for supersymmetry; it would be perhaps the first collider with the potential to directly discover supersymmetry through its universal feature, the goldstone fermion of spontaneous supersymmetry breaking.
A universal prediction of spontaneous breaking of supersymmetry in the rigid limit is the presence of a massless Majorana fermion, the goldstino. When gravity effects are taken into account, the goldstino is eaten through super-Higgs mechanism by the spin 3/2 gravitino [137], which obtains a mass
In low-energy supersymmetry breaking scenarios, the stable gravitino is dominantly produced in the early universe through gluon-gluino scattering processes as computed in Refs. [138–141]. The gravitino yield can be written as
Since the gravitino freezes out when it is still relativistic, the matter power spectrum is going to be damped at small scales [146]. In Ref. [147] a combination of CMB data from WMAP and Lyman-α forest data was used to set an upper bound on the gravitino mass (and the SUSY-breaking scale)
This bound will presumably be improved with current Planck data and even further with future cosmological surveys.
Given the upper bound on the SUSY-breaking scale, the gluino can be made to lie above the 2 TeV reach of the LHC provided
where the lower bound is given by the current LHC bound on the gluinos plus perturbativity of the SM couplings, while the upper bound is from cosmological constraints on warm dark matter.
Within the light gravitino window defined in Eq. (54), we can safely assume that the superpartner spectrum is decoupled and the gravitino interactions with the SM are described by a universal EFT with couplings controlled by the supersymmetry breaking scale only [150–152]. Collider searches for direct gravitino production can hence provide a robust limit on the supersymmetry breaking scale, independent of the specific details of the superpartner spectrum. Traditionally, this has not been an emphasis of the supersymmetry search program at hadron colliders because direct searches for colored superpartners always exceed the sensitivity of direct gravitino searches. In contrast, at high-energy lepton colliders the reduced background and rapid growth of signal cross sections with √s makes this a competitive channel for the discovery of supersymmetry.
We conclude that a future high energy muon collider can almost certainly push up the lower bound of the ultralight gravitino window by one order of magnitude. An improvement of the cosmological bounds with respect to the ones derived from WMAP data could allow the light gravitino window to be completely closed in the future. More broadly, this exemplifies the ability of a high-energy muon collider to directly probe the mechanism of supersymmetry breaking.
4.3.2 The “big” hierarchy problem: compositeness
4.3.3 The “little” hierarchy problem
As we have seen, a high-energy muon collider operating at tens of TeV could provide satisfying coverage of known solutions to the “big” hierarchy problem. It is no less suited to constraining or discovering solutions to the “little” hierarchy problem, whose subtle signatures will remain largely untouched by the LHC. There are a plethora of such solutions which operate by relying on novel field theoretical [98, 99, 155–164] or cosmological [125, 165–168] ingredients. Here we will focus on solutions involving “neutral naturalness” such as the Twin Higgs [98] or Hyperbolic Higgs [162], in which the partner particles are entirely neutral under the Standard Model. Such models feature (at least) four possible avenues to discovery at colliders: Higgs coupling deviations from mixing between scalars; direct production of SM singlet partner particles; displaced vertices from exotic Higgs decays; and direct production of the “radial mode” associated with spontaneous breaking of the discrete symmetry
A more exotic collider signature of models of neutral naturalness is the prediction of displaced decays of dark particles into Standard Model states [126]. The scalar mixing produces a non-zero branching ratio of Higgses into neutral partner states, which gives a portal for energy to be transferred to the partner sector at a collider. Some of the produced partner sector states are unstable to decaying through an off-shell Higgs into Standard Model states, bearing out “Hidden Valley” type phenomenology [169, 170]. This produces spectacular signatures – for example vertices displaced from the beamline from which Standard Model jets appear – and this allows search strategies with very low background [124, 171]. The prospects for probing these signatures in neutral naturalness models at future lepton colliders have been studied in the context of Higgs factories [172,173], and a branching ratio reach is projected which is competitive with or better than the LHC forecasts. Dedicated study for a higher-energy muon collider has yet to be performed.
Apart from fine tuning considerations, Fig. 20 illustrates the correlation between direct searches and Higgs coupling deviations in the Twin Higgs parameter space. Interestingly, a high energy muon collider of √ s = 14 TeV will be able to fully probe the parameter space corresponding to a % deviation in Higgs couplings (sin γ ' 0.1) by means of direct searches for the extra scalar. This may be taken as further motivation to reach such a high c.m. energy at a future collider. Conversely, moving into the region with small λ∗, the deviations in the Higgs couplings will be difficult to observe, but direct production at a muon collider could still cover a large portion of this parameter space.
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
