Particle Physics Lecture Notes on Group Theory
Group Theory
For particle physicists Lecture 5
Rachel Dowdall
University of Glasgow, UK
University of Glasgow – p. 1/2
Last Lecture
SU (3) Flavour symmetry Classification of mesons and baryons This time: Finish the classification of baryons Colour SU (3) symmetry Young’s tableaux
University of Glasgow – p. 2/2
Wave functions
The wave function for a particle is the product Ψ = φFlavour χspin ξcolour ηspace Overall, this should be symmetric under exchange of quarks for mesons and antisymmetric for baryons We considered φFlavour last lecture If we consider ground states (zero ang. momentum L), then ηspace is symmetric. Let’s now consider ξcolour , this will turn out to be antisymmetric
University of Glasgow – p. 3/2
SU (3) Colour symmetry
Colour was proposed as an additional quantum number to explain how states that looked the same could co-exist Quarks can have three “colours” r, g, b, antiquarks have r, g, b Processes do not depend on the colour, i.e. invariant under SU (3)C We don’t observe colour in nature (confinement) so we think that bound states of quarks must be in a colour singlet, or colourless state - Not proven theoretically why this is true e.g. mesons qq have 3 ⊗ 3 = 8 ⊕ 1, so must be in the 1 rep 1 ξmeson = √ (rr + gg + bb) 3 Just like for flavour SU (3) except that SU (3)C is an exact symmetry Baryons qqq have 3 ⊗ 3 ⊗ 3 = 10 ⊕ 8 ⊕ 8 ⊕ 1 and the colourless state is 1 ξbaryon = √ (rgb − rbg + gbr − grb + brg − bgr) 6 Exercise: Check this is antisymmetric
University of Glasgow – p. 4/2
Colour
We are not allowed to have qq or qqq as these do not contain a singlet Can have exotics qqqq, qqqqq but these have not been found Since particles are in the singlet state the wave function ξcolour is antisymmetric So far, we have ξcolour ηspace is antisymmetric So χspin φflavour must be symmetric for baryons Aside: We say the standard model has SU (3) × SU (2) × U (1) symmetry - Colour is the SU (3) part The Lagrangian for QCD is an
For particle physicists Lecture 5
Rachel Dowdall
University of Glasgow, UK
University of Glasgow – p. 1/2
Last Lecture
SU (3) Flavour symmetry Classification of mesons and baryons This time: Finish the classification of baryons Colour SU (3) symmetry Young’s tableaux
University of Glasgow – p. 2/2
Wave functions
The wave function for a particle is the product Ψ = φFlavour χspin ξcolour ηspace Overall, this should be symmetric under exchange of quarks for mesons and antisymmetric for baryons We considered φFlavour last lecture If we consider ground states (zero ang. momentum L), then ηspace is symmetric. Let’s now consider ξcolour , this will turn out to be antisymmetric
University of Glasgow – p. 3/2
SU (3) Colour symmetry
Colour was proposed as an additional quantum number to explain how states that looked the same could co-exist Quarks can have three “colours” r, g, b, antiquarks have r, g, b Processes do not depend on the colour, i.e. invariant under SU (3)C We don’t observe colour in nature (confinement) so we think that bound states of quarks must be in a colour singlet, or colourless state - Not proven theoretically why this is true e.g. mesons qq have 3 ⊗ 3 = 8 ⊕ 1, so must be in the 1 rep 1 ξmeson = √ (rr + gg + bb) 3 Just like for flavour SU (3) except that SU (3)C is an exact symmetry Baryons qqq have 3 ⊗ 3 ⊗ 3 = 10 ⊕ 8 ⊕ 8 ⊕ 1 and the colourless state is 1 ξbaryon = √ (rgb − rbg + gbr − grb + brg − bgr) 6 Exercise: Check this is antisymmetric
University of Glasgow – p. 4/2
Colour
We are not allowed to have qq or qqq as these do not contain a singlet Can have exotics qqqq, qqqqq but these have not been found Since particles are in the singlet state the wave function ξcolour is antisymmetric So far, we have ξcolour ηspace is antisymmetric So χspin φflavour must be symmetric for baryons Aside: We say the standard model has SU (3) × SU (2) × U (1) symmetry - Colour is the SU (3) part The Lagrangian for QCD is an