2 Muons vs. Protons
Before getting into the detailed physics case, this section will describe the physics of the initial state at a high energy muon collider. Naively, the advantage of a lepton collider is that the colliding beams are composed of elementary particles (so that the collision is relatively clean), which are in momentum eigenstates (so that the c.m. energy for each collision is known). This can be contrasted against proton colliders, where the beams are composed of composite states, so that the partonic c.m. energy varies from collision to collision. To make predictions in this case, one convolves the hard process of interest with universal PDFs. Additionally, the smashed protons leave a trail of debris in their wake, the so-called underlying event. As we will argue in this section, making predictions for a muon collider whose beams carry TeVs of energy has aspects in common with both better known types of machines: one must use PDFs, but the collision yields a small number of particles in the initial state that can be modeled reasonably well using perturbation theory.[1]
At the theoretical level, the situation for a muon collider is simplified with respect to a proton collider since perturbative control can be maintained at every step of the calculation.[2] For example, the boundary conditions for the proton PDFs are set at a scale where QCD is non-perturbative, implying that one must rely on inputs from experiment to numerically determine the proton PDFs. All of the complications that stem from this fact are avoided when studying muon PDFs. The muon colliders we discuss here have energy in the TeV to 100 TeV range, and so the masses of the weak gauge bosons can be treated as a small perturbation, i.e., it is typically reasonable to treat them as massless so that the PDF formalism applies; see Sec. 3.4 for a brief discussion of finite mass effects. And since the electroweak gauge couplings are relatively small, working with leading order unresummed PDFs provides a reasonable approximation to the resummed result; we will demonstrate the minimal impact of next-to-leading-log corrections in Sec. 3.3 below. Interesting complications arise due to electroweak symmetry breaking, but other than treating the mass versus gauge eigenbasis for the electroweak bosons consistently as we do below, these tend to have a small numerical effect on the cross-section predictions. There are additionally subtleties associated with capturing the physics of the longitudinal gauge boson modes, and the interplay with the Goldstone equivalence theorem and unitarity; we will not comment on this further and will simply use the splitting functions in the “Goldstone Equivalence Gauge” computed in [53]. Finally, while it is beyond the scope of this work, we note that one can also include the effects of QCD into the muon PDFs, as was recently described in [54].
In the rest of this section, we will first write down the formalism used to solve for the PDFs to leading logarithmic order using leading order splitting functions. This will provide us with a framework to explore the accuracy that can be achieved when taking different approximations. Our goal will be to demonstrate that the leading log (unresummed) PDFs provide a reasonable approximation to the more complete all log order results that result from integrating the DGLAP evolution equations. Given that the leading log PDFs are easy to understand and can be expressed analytically, we advocate that these are all that are required to make predictions for a future muon collider in the energy range of interest here, unless high precision calculations are needed.
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.
[1] Throughout this paper, we treat the muon beams as stable. Everything we say here is independent of this assumption, as long as our amazing accelerator colleagues can figure out how to provide us with a robust muon beam to play with.
[2] Of course, some of these techniques are also relevant for past and proposed electron-position experiments, e.g., when predicting VBF initiated processes. However, the small mass of the electron effectively bounds the maximum energy for circular machines to be near the electroweak scale.
