Beta Phase: Square45 is currently in beta testing. Expect some features or content to be incomplete or missing.
45

Three Generations of Fermions

The Standard Model posits three generations of fundamental fermions (quarks and leptons), each with distinct masses and mixing angles.
📜

The statement of the theorem

The Standard Model Lagrangian LSM\mathcal{L}_{SM} is constructed using three generations of left-handed fermion doublets Li=(νi)L,(e)LL_i = \begin{pmatrix} \nu_i \end{pmatrix}_L, \begin{pmatrix} e \end{pmatrix}_L and right-handed singlets eiRe_{iR}, and three generations of quark doublets Qi=(u)L,(d)LQ_i = \begin{pmatrix} u \end{pmatrix}_L, \begin{pmatrix} d \end{pmatrix}_L and right-handed singlets uiR,diRu_{iR}, d_{iR}. The kinetic and interaction terms are given by the sum over generations i=1,2,3i=1, 2, 3: \nLfermion=∑i=13[LˉiiγμDμLi+QˉiiγμDμQi+eˉiRiγμDμeiR+uˉiRiγμDμuiR+dˉiRiγμDμdiR]+LYukawa\mathcal{L}_{fermion} = \sum_{i=1}^{3} \left[ \bar{L}_i i \gamma^{\mu} D_{\mu} L_i + \bar{Q}_i i \gamma^{\mu} D_{\mu} Q_i + \bar{e}_{iR} i \gamma^{\mu} D_{\mu} e_{iR} + \bar{u}_{iR} i \gamma^{\mu} D_{\mu} u_{iR} + \bar{d}_{iR} i \gamma^{\mu} D_{\mu} d_{iR} \right] + \mathcal{L}_{Yukawa}