Muon Physics: a Pillar of the Standard Model

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Muon Physics: a Pillar of the Standard Model Typeset with jpsj2.cls <ver.1.2> Full Paper Muon Physics: A Pillar of the Standard Model B. Lee Roberts ∗ Department of Physics Boston University Boston, MA 02215, USA (Received October 5, 2018) Since its discovery in the 1930s, the muon has played an important role in our quest to understand the sub-atomic theory of matter. The muon was the first second-generation standard-model particle to be discovered, and its decay has provided information on the (V ector − Axial V ector) structure of the weak interaction, the strength of the weak in- teraction, GF , and the conservation of lepton number (flavor) in muon decay. The muon’s anomalous magnetic moment has played an important role in restricting theories of physics beyond the standard standard model, where at present there is a 3.4 σ difference between the experiment and standard-model theory. Its capture on the atomic nucleus has provided valu- able information on the modification of the weak current by the strong interaction which is complementary to that obtained from nuclear β decay. KEYWORDS: muon, weak decay, muon capture, magnetic moment, lepton flavor violation 1. Introduction cause of the modification of the Coulomb potential in- The muon was first observed in a Wilson cloud cham- side the nucleus, and these x rays have provided infor- ber by Kunze1 in 1933, where it was reported to be “a mation on nuclear root-mean-square charge radii. The particle of uncertain nature.” In 1936 Anderson and Ned- Lamb shift in muonic hydrogen, ∆E2P −2S , which is being dermeyer2 reported the presence of “particles less mas- measured at at the Paul Scherrer Insitut (PSI), is given 7 2 3 sive than protons but more penetrating than electrons” by 209.974(6) 5.226Rp +0.036Rp meV, where Rp is the proton rms− charge radius. This experiment should in cosmic rays, which was confirmed in 1937 by Street and Stevenson,3 Nishina, Tekeuchi and Ichimiya,4 and provide a precise measurement of Rp. The weak nuclear by Crussard and Leprince-Ringuet.5 The Yukawa theory capture, called ordinary muon capture (OMC), of the muon on the atomic nucleus following the cascade to the of the nuclear force had predicted such a particle, but − 1S ground state, µ +Z Z− + νµ, is the analog this “mesotron” as it was called, interacted too weakly N → 1 N with matter to be the carrier of the strong force. Today to the weak capture of a K-shell electron by the nucleus, we understand that the muon is a second generation lep- and provides information on the modification of the weak ton, with a mass about 207 times the electron’s. Like the interaction by the hadronic matter. The muon mass of 106 MeV restricts the muon to electron, the muon obeys quantum electrodynamics, and ∼ can interact with other particles through the electromag- decay into the electron, neutrinos, and photons. Thus muon decay is a purely leptonic process, and the domi- netic and weak forces. Unlike the electron which appears − − to be stable, the muon decays through the weak force. nant decay mode is µ e + νµ +¯νe. This three-body arXiv:0704.2394v1 [hep-ex] 18 Apr 2007 → The muon lifetime of 2.2 µs permits one to make pre- decay tells us that the individual lepton number, electron cision measurements of its properties, and to use it as a and muon, is conserved separately, and that the two fla- 10 tool to study the semileptonic weak interaction, nuclear vors (kinds) of neutrinos are distinct particles. Here − − + + properties, as well as magnetic properties of condensed the µ and e are “particles” and the µ and e are the matter systems. The high precision to which the muo- antiparticles. In the 1950s, it became possible to make nium (µ+e− atom) hyperfine structure can be measured pions, and thus muons, in the laboratory. The energeti- + and calculated makes it a significant input parameter in cally favorable decay µ eγ was searched for and not 8 → −5 the determination of fundamental constants.6 In this re- found to a relative branching ratio of < 2 10 . Also searched for was the neutrinoless capture of× a µ− on an view, I will focus on the role of the muon in particle − − physics. atomic nucleus, µ + e + , which was not found at the level of 5 N10 →−4. SuchN processes are said to A beam of negative muons can be brought to rest in ∼ × matter, where hydrogen-like atoms are formed, with a “violate lepton flavor,” and continue to be the object of nuclear charge of Z. The Bohr radius for a hydrogen- present and planned studies reaching to sensitivities of −14 −16 like atom is inversely proportional to the orbiting par- 10 and 10 , respectively. 2 2 ticle’s mass (rn = [n ~c]/[mc Zα] ), so that for the The muon, like the electron, is a spin 1/2 lepton, with lowest quantum numbers of high-Z muonic atoms, the a magnetic moment given by muon is well inside of the atomic electron cloud, with q q~ (gs 2) ~µs = gs( )~s; µ = (1+ a) ; a − ; the Bohr radius of the 1S atomic state well inside the 2m 2m ≡ 2 nucleus. The 2P 1S x-ray energies are shifted be- (1) → where the muon charge q = e, and gs, the Land´e g- ± ∗E-mail address: [email protected] 1 2 J. Phys. Soc. Jpn. Full Paper B. Lee Roberts factor is slightly greater than the Dirac value of 2. The middle equation above is useful from a theoretical point of view, as it separates the magnetic moment into two pieces: the Dirac moment which is unity in units of the appropriate magneton, e~/2m, and is predicted by the Dirac equation; and the anomalous (Pauli) moment, (a) (b) (c) where the dimensionless quantity a is referred to as the anomaly. The muon anomaly, like the electron’s, arises Fig. 1. Muon decay in (a) the standard model; (b) the Fermi from radiative corrections that are discussed below. theory; (c) radiative corrections to the Fermi theory. When the muon was discovered, it was an unexpected surprise. Looking at this from our 21st century perspec- 14 tive, it is easy to forget how we reached what is now called electroweak gauge coupling g is related to GF by the “standard model” of subatomic physics, which incor- 2 GF g porates three generations of leptons, e, µ and τ and their = 2 (1 + ∆r) (2) neutrinos; three generations of quarks; the electro-weak √2 8MW gauge bosons, γ W and Z; and the gluons that carry the where ∆r represents the weak boson mediated tree-level strong force. When this author joined the field as a grad- process and its radiative corrections.15 In the standard uate student in the mid 1960s, none of this was clear. model, the Fermi constant is related to the vacuum ex- 2 Quarks were viewed by many as a mathematical device, pectation value of the Higgs field by GF =1/(√2 v ). not as constituent particles. Even after quarks were in- While the Fermi theory is nonrenormalizeable, the ferred from deep inelastic electron scattering off the pro- QED radiative corrections are finite to first order in GF , ton, we only knew of the existence of three of them. While and to all orders in the fine-structure constant α. This 14 the V A structure of the weak interaction was first in- gives the relationship between GF and the muon life- − ferred from nuclear β decay, the study of muon decay has time, τµ, provided a useful laboratory in which to study the purely 1 G2 mµ5 leptonic weak interaction, to search for physics beyond F = 3 (1 + ∆q) (3) the standard model, such as additional terms in the in- τµ 192π teraction besides the standard-model V A structure, as where ∆q is the sum of phase space, and QED and − well as looking for standard model forbidden decays like hadronic radiative corrections. More properly one should µ eγ. For many years, the experimental value of the write G since new physics contributions could make G → µ muon’s anomalous magnetic moment has served to con- different for the three leptons.16 strain physics beyond the standard model, and continues The MuLan experiment at PSI has recently re- that role today. ported a new measurement of the muon lifetime 2.197 013(21)(11) µs ( 11 parts per million (ppm)),17 G 2. Muon Decay and F to be compared with± the previous world average − − 18 The muon decay µ e νµν¯e is purely leptonic. 2.197 03(4) µs (19 ppm). The new world average → Since mµ << MW , muon decay can be described by muon lifetime of 2.197 019(21) µs gives, assuming a local four-fermion (contact) interaction. While non- only standard-model physics in muon decay, GF = renormalizeable, at low energies it provides an excel- 1.166 371(6) 10−5 GeV2 (5 ppm). This experiment × lent approximation to the full electroweak theory. The should eventually reach a precision of 1 ppm on τµ. weak Lagrangian is written as a current-current interac- tion, where the leptonic current is of the (V A) form, 3. Nuclear Muon Capture − uγ¯ λ(1 γ )u. The weak capture of a muon on a proton has much in − 5 Michel11 first wrote down a parameterization of muon common with nuclear β decay. As for other low-energy decay, defining five parameters, ρ, η, ξ, δ and h, which weak processes, the interaction can be described as a are combinations of the different possible couplings al- current-current interaction with the (V A) leptonic cur- λ − lowed by Lorentz invariance in muon decay.
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