Wing Sheung Chan

4 The Standard Model and lepton flavour violation and actions are expressed in units of the reduced Planck constant ~ = 6 . 626 070 15 × 10 − 34 J s 2 π ≈ 1 . 055 × 10 − 34 Js ≈ 6 . 582 × 10 − 16 eV s . (1.2) Consequently, length, time, mass, energy and momentum all share a same common uni t † . Furthermore, the unit of electric charge is chosen such that the vacuum permittivity ε 0 = 1 . This simplifies the expression of the electromagnetic coupling to be the same as the positron charge e . In particle physics, the common unit of mass, energy and momentum is often chosen to be electronvolt (eV) or decadic multiples of it, while for practical reasons, lengths and times in collider experiments are often still expressed in metres and seconds. Also, it might be worth noting that it is customary to express scattering cross sections (luminosities) in decadic multiples of barn (inverse barn), where barn is a unit of area defined as 1 b = 10 − 28 m 2 . (1.3) Einstein notation will be used throughout this chapter, implying summation over index variables that appear twice in a single term. Upper indices represent components of contravariant vectors, and lower indices represent components of covariant vectors. In the case of spacetime four-vectors, indices are raised or lowered by the Minkowski metric tensor with the ( + − − − ) signature g µν = g µν =   1 0 0 0 0 − 1 0 0 0 0 − 1 0 0 0 0 − 1   . (1.4) Spacetime four-vectors are always represented by notations in italic face, while their spatial components are represented by the same notations in bold face. 1.2. The Standard Model The Standard Model [2– 5] was developed over decades in the last century by many physicists based on previous discoveries and understandings. Ever since it was forged into its current form in the 1970s, the model has successfully explained and predicted many phenomena and has withstood myriad experiments to high precision and accuracy. Today, even after half a century since its birth, it remains the best description of Nature at small length scales and the foundation of particle physics. † Technically, the unit of length and time is the inverse of the unit of mass, energy and momentum.