PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Particle Physics & Quantum Fields · Beyond the Standard Model
Symmetry and Flavour, Ch Grand unification: running couplings, SU(5), proton decay and the hierarchy problem
Problem
The 5ˉ representation of SU(5) holds, under SU(3)×SU(2), one colour antitriplet that is a weak singlet and one colourless weak doublet, 5ˉ→(3ˉ,1)⊕(1,2). The antitriplet is the three colour states of dc (the charge-conjugate right-handed down quark); the doublet is the lepton doublet (ν,e−). Nothing is assumed about the charge of dc: call it x, the same for all three colours.
((a)) List the electric charges of the five states of the 5ˉ, using Q(ν)=0 and Q(e−)=−1 as the reference unit of charge.
((b)) A generator of a simple Lie group is traceless in every representation. Apply this to the electric charge generator Q on the 5ˉ to find x and hence the charge of the down quark d, in units of e.