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SU(3) Symmetry in Hadron Physics

Explore the origin of SU(3) symmetry, the limitations of a simple extension from SU(2), and the graphical method for finding states in baryon and meson spectra. Delve into fundamental representation differences and the challenges with baryon wave functions. Uncover the complexities of quark-antiquark pairs in mesons and the unique properties of hadronic states. Gain insights into the symmetric combinations that survive in hadron physics.

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SU(3) Symmetry in Hadron Physics

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  1. Flavour SU(3) • Origin of SU(3) • Why a simple extension of SU(2) is not enough • Extending the Graphical method of finding states • Application to Baryon and Meson spectrum

  2. Fundamental SU(3) Representation Fundamental SU(2) Representation Y I3 +2/3 1 1/2 -1/2 I3 1 0 1/2 -1/3 0 -2/3 -1/2 -1 Fundamentally Different….see?

  3. Unlike SU(2), antiquarks can’t fit on the graph the same way as quarks Y Y Quarks Antiquarks +2/3 +2/3 1/2 -1/2 I3 1 0 -1/3 -1/2 I3 1 0 1/2 -1/3 -2/3 -2/3

  4. Use Same Graphical method to combine quark- antiquark pairs into mesons Y Y Quarks Antiquarks +2/3 +2/3 1/2 -1/2 I3 1 0 -1/3 -1/2 I3 1 0 1/2 -1/3 -2/3 -2/3

  5. Use Same Graphical method to combine quark- antiquark pairs into mesons Y +1 Caveat: only one of the states at S=I3=0 is fully antisymetric. -1/2 I3 1 0 1/2 -1

  6. Baryon Woes • We have one decplet, two Octets, and one singlet 10s  8ms  8ma  1a. • The decuplet is fully symmetric • The first octet is symmetric in the interchange of the first two quarks. • The second octet is symmetric in the interchange of the second two quarks (therefore antisymmetric in the interchange of the first two quarks). • The singlet is fully antisymmetric in the interchange of any two quarks. • BUT: Nature only has 10s  8s!WHY????

  7. Baryon Woes: The Wave Function • Not a problem in Mesons….a quark and an antiquark are fundamentally distinguishable particles. We have to pay attention to the form of the baryon wave function (antisymmetric, fermion): • Y = y(space)(flavour)(spin)(colour) • (colour) is always antisymmetric • This is related to the fact that no hadron is observed to have any net ‘colour’ charge. • So y(space)(flavour)(spin)  symmetric • Assume, for the moment, that there is no orbital angular momentum between the quarks in the baryon. (l = 0) • Then y(space) will be symmetric.

  8. Baryon Woes: The Wave Function • Y = y(space)(flavour)(spin)(colour) • y(space)l=0symmetric. • So we need (flavour)(spin) to be symmetric also. • Look at (spin) • Combine 3 spin 1/2 objects using SU(2)…the group that correctly handles spins. • 222 = 4s2ms2ma • 4s is symmetric, 2ms is symmetric in the exchange quarks 1 2, 2ma is antisymmetric in the exchange of quarks 1  2. • Combine with 10s  8ms  8ma  1a • (10s  8ms  8ma  1a) (4s2ms2ma)  Keep symmetric ones • (10s 4s) and two (8  2) combinations survive.

  9. Hadrons Are Looking Up! • We can understand/predict a whole bunch of stuff! • Why the Baryons cluster like they do. • IF the strong force is a ‘colour SU(3)c’ group • Why the l=0 Decuplet MUST have j=3/2 spin. • Why the l=0 Octets MUST have j=1/2 spin. • Predicts structure of l0 states as well. (many not found yet). • Can get mass splittings and magnetic moments! • Why the Mesons only form Octets + singlets. • The symmetry we must invoke for more quarks: • SU(4) for Charm…up to SU(6) to get to top quarks. • But this makes NO sense because even SU(4) is broken beyond all recognition.

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