PhD thesis - CP3-Origins

Introduction to Elementary Particle Physics 13The first term contains kinetic terms for the gauge bosonsL Gauge = − 1 4 B µνB µν − 1 4 W i µνW iµν − 1 4 W a µνW aµν , (2.30)whereB µν = ∂ µ B ν − ∂ ν B µ , (2.31)W i µν = ∂ µ W i ν − ∂ ν W i µ + gɛ ijk W j µW k ν , (2.32)G a µν = ∂ µ G a ν − ∂ ν G a µ + g s f abc G b µG c ν. (2.33)The structure constants are defined as [τ i , τ j ] = iɛ ijk τ k and [λ a , λ b ] = if abc λ c , where τ iand λ a are the generators of SU(2) and SU(3) groups respectively. The second termcontains kinetic terms for the fermionsL = ¯Q L i /DQ L + ū R i /Du R + ¯d R i /Dd R + ¯L L i /DL L + ē R i /De R , (2.34)where the covariant derivative isD µ = ∂ µ − ig ′ Y B µ − ig τ i2 W i µ − ig sλ a2 Ga µ. (2.35)Here Y denotes the hypercharge of a respective particle. Of course the last two terms in thecovariant derivative can be absent if the particle is not charged under the correspondingforce.At this stage all the fermions are massless. A Majorana mass term is not possiblebecause all the fermions carry hypercharge. A Dirac mass term is not allowed becausethe left-handed and right-handed fermions are not in complex conjugated representations.Thus the fermion sector possesses five accidental global U(3) symmetries for the rightand left-handed quarks and leptons.The Yukawa Lagrangian violates these symmetriesL Y = −Y iju¯Q i LɛHu j R − Y ijdwhere Y u , Y d and Y e are 3 × 3 matrices andH =¯Q i LHd j R − Y e ij ¯L i LHe j R+ h.c., (2.36)⎛⎞⎠ . (2.37)h 0⎝ h+