Wing Sheung Chan

The Standard Model and lepton flavour violation 17 where ν 1 , ν 2 and ν 3 are the neutrino mass eigenstates. The PMNS matrix is also unitary and can be parameterised by three mixing angles, θ 12 , θ 23 and θ 13 , and one phase angle, δ : U PMNS =   c 12 c 13 s 12 c 13 s 13 e − iδ − s 12 c 23 − c 12 s 13 s 23 e iδ c 12 c 23 − s 12 s 13 s 23 e iδ c 13 s 23 s 12 s 23 − c 12 s 13 c 23 e iδ − c 12 s 23 − s 12 s 13 c 23 e iδ c 13 c 23   , (1.26) where c jk = cos θ jk and s jk = sin θ jk . These parameters have been measured by various neutrino oscillation experiments [22] . They take on different values depending on the mass ordering assumption: the Normal Ordering where m ν 1 < m ν 2 < m ν 3 or the Inverted Ordering where m ν 3 < m ν 1 < m ν 2 . The current best-fit values are summarised in Table 1.8. Unlike the CKM matrix which is approximately diagonal, the PMNS matrix has much larger off-diagonal elements. Table 1.8.: Current best-fit values of the mixing angles and the phase angle of the PMNS matrix [22] . Parameter Best-fit value ± 1 σ [ ◦ ] Normal Ordering Inverted Ordering θ 12 34 . 5 +1 . 2 − 1 . 0 34 . 5 +1 . 2 − 1 . 0 θ 23 47 . 7 +1 . 2 − 1 . 7 47 . 9 +1 . 0 − 1 . 7 θ 13 8 . 45 +0 . 16 − 0 . 14 8 . 53 +0 . 14 − 0 . 15 δ 218 +38 − 27 281 +23 − 27 One noteworthy clarification is that while quark flavours usually refer to the quark mass eigenstates, neutrino flavours always refer to the interaction eigenstates. This confusing terminology is related to the facts that the neutrino masses are not exactly known due to experimental limitations and that historically neutrinos were discovered in weak interaction processes and were therefore defined by the flavours of their charged lepton interaction partners. In this thesis, from this point onward, neutrino mixing and the PMNS matrix will be considered as part of the SM, with neutrino masses treated as Dirac masses. In the literature, this is also known as the Neutrino Minimal Standard Model ( ν MSM) [23] with zero Majorana masses. The scenario where neutrinos have Majorana masses will be discussed later in Section 1.3.1. 1.2.6. Incompleteness of the Standard Model Despite its success in explaining many physical phenomena, the SM is still an incomplete theory. This is known as a fact due to a number of unsolved problems in physics to which the SM fails to provide satisfactory answers.