Wing Sheung Chan

The Standard Model and lepton flavour violation 5 In this section, the Standard Model will be introduced. For the benefits of some readers, we will begin with a very brief introduction to the basic concepts of quantum field theory, upon which the foundation of the SM is built. After that, we will go through the particle contents of the SM and describe their interactions, with a focus on the electroweak sector as it is most relevant to this thesis. 1.2.1. Quantum field theory In quantum field theory, analogous to classical field theory, the evolution of a field can be determined by the Lagrangian density L (or simply Lagrangian for short). Consider a classical free scalar field φ ( x , t ) as an example, the Lagrangian density can be written as L = 1 2 ( ∂ µ φ ) ( ∂ µ φ ) − 1 2 m 2 φ 2 , (1.5) where m is a real constant. The equation of motion for the field can then be derived using the principle of least action by requiring vanishing variation of the action S : δS = δ Z L d 3 x d t = 0 . (1.6) In our example, this gives rise to the famous Klein-Gordon equation: ∂ t φ − ∇ 2 φ + m 2 φ = 0 . (1.7) The formulation of QFT is similar to its classical counterpart but differs in two main ways, each of which has important implication that makes quantum fields fundamentally different from classical fields. The first difference is that fields in QFT are promoted into field operators in a process known as canonical quantisation or second quantisation. For our free scalar field example, the field operator would be φ ( x ) = Z 1 √ 2 E a p e − ip µ x µ + a † p e ip µ x µ d 3 p (2 π ) 3 , (1.8) where x and p are position and momentum four-vectors, E = p | p | 2 + m 2 is the time- component of p , and a † p and a p are the creation and annihilation operators for momentum p , respectively. This implies that there exists a vacuum state | 0 i where a p | 0 i = 0 for all p , and that there can be an arbitrary number of particles, which can be interpreted as excitations of the field, at any given time. This contrasts classical theories, where “vacuum” is interpreted as the state in which all components of a field are zero, and the number of particles is always conserved. The second difference is that evolutions, or paths, that do not correspond to the least action can also contribute to the transition amplitude between two states at two given times. This can be formulated using Feynman path integrals, where the overall transition