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SU(3) symmetry and Baryon wave functions

SU(3) symmetry and Baryon wave functions. Sedigheh Jowzaee PhD seminar- FZ Juelich, Feb 2013. Introduction. Fundamental symmetries of our universe Symmetry to the quark model: Hadron wave functions Existence (mesons) and qqq (baryons)

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SU(3) symmetry and Baryon wave functions

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  1. SU(3) symmetry and Baryon wave functions Sedigheh Jowzaee PhD seminar- FZ Juelich, Feb 2013

  2. Introduction • Fundamental symmetries of our universe • Symmetry to the quark model: • Hadron wave functions • Existence (mesons) and qqq (baryons) • Idea: extend isospin symmetry to three flavors (Gell-Mann, Ne’eman 1961) • SU(3) flavour and color symmetry groups

  3. Unitary Transformation • Invariant under the transformation • Normalization: U is unitary • Prediction to be unchanged: Commutation U & Hamiltonian • Define infinitesimal transformation (G is called the generator of the transformation)

  4. Symmetry and conservation • Because U is unitary G is Hermitian, corresponds to an observable • In addition: G is conserve Symmetry conservation law For each symmetry of nature there is an observable conserved quantity • Infinitesimal spatial translation: , Generator px is conserved • Finite transformation

  5. Isospin • Heisenberg (1932) proposed : (if “switch off” electric charge of proton ) There would be no way to distinguish between a proton and neutron (symmetry) • p and n have very similar masses • The nuclear force is charge-independent • Proposed n and p should be considered as two states of a single entity (nucleon): Analogous to the spin-up/down states of a spin-1/2 particle Isospin: n and p form an isospin doublet (total isospin I=1/2 , 3rd component I3=±1/2)

  6. Flavour symmetry of strong interaction • Extend this idea to quarks: strong interaction treats all quark flavours equally • Because mu≈md (approximate flavour symmetry) • In strong interaction nothing changes if all u quarks are replaced by d quarks and vs. • Invariance of strong int. under u d in isospin space (isospin in conserved) • In the language of group theory the four matrices form the U(2) group • one corresponds to multiplying by a phase factor (no flavour transformation) • Remaining three form an SU(2) group (special unitary) with det U=1 Tr(G)=0 • A linearly independent choice for G are the Pauli spin matrices

  7. The flavour symmetry of the strong interaction has the same transformation properties as spin. • Define isospin: , • Isospin has the exactly the same properties as spin (same mathematics) • Three correspond observables can not know them simultaneously • Label states in terms of total isospin I and the third component of isospin I3 : generally d u u d System of two quarks: I3=I3(1)+I3(2) , |I(1)-I(2)| ≤ I ≤ |I(1)+I(2)|

  8. Combining three ud quarks • First combine two quarks, then combine the third • Fermion wave functions are anti-symmetric • Two quarks, we have 4 possible combinations: (a triplet of isospin 1 states and a singlet isospin 0 state ) • Add an additional u or d quark

  9. Grouped into an isospin quadruplet and two isospin doublets • Mixed symmetry states have no definite symmetry under interchange of quarks 1 3 or 2 3

  10. Combining three quark spin for baryons • Same mathematics

  11. SU(3) flavour • Include the strange quark • ms>mu/md do not have exact symmetry u d s • 8 matrices have detU=1 and form an SU(3) group • The 8 matrices are: • In SU(3) flavor, 3 quark states are :

  12. SU(3) uds flavour symmetry contain SU(2) ud flavour symmetry • Isospin • Ladder operators • Same matrices for u s and d s • and 2 other diagonal matrices are not independent, so de fine as the linear combination:

  13. Only need 2 axes (quantum numbers) : (I3,Y) All other combinations give zero Quarks: Anti-Quarks:

  14. Combining uds quarks for baryons • First combine two quarks: • a symmetric sextet and anti-symmetric triplet • Add the third quark

  15. 1. Building with sextet: 2. Building with the triplet: • In summary, the combination of three uds quarks decomposes into: Mixed symmetry octet Symmetric decuplet Totally anti-symmetric singlet Mixed symmetry octet

  16. combination of three uds quarks in strangeness, charge and isospin axes OctetDecuplet Charge: Q=I3+1/2 Y Hypercharge: Y=B+S (B: baryon no.=1/3 for all quarks S: strange no.)

  17. SU(3) colour • In QCD quarks carry colour charge r, g, b • In QCD, the strong interaction is invariant under rotations in colour space SU(3) colour symmetry • This is an exact symmetry, unlike the approximate uds flavor symmetry • r, g, b SU(3) colour states: (exactly analogous to u,d,s flavour states) • Colour states labelled by two quantum numbers: I3c(colour isospin), Yc(colour hypercharge) Quarks: Anti-Quarks:

  18. Colour confinement • All observed free particles are colourless • Colour confinement hypothesis: only colour singlet states can exist as free particles • All hadrons must be colourless (singlet) • Colour wave functions in SU(3) colour same as SU(3) flavour • Colour singlet or colouerless conditions: • They have zero colour quantum numbers I3c=0, Yc=0 • Invariant under SU(3) colour transformation • Ladder operators are yield zero

  19. Baryon colour wave-function • Combination of two quarks • No qq colour singlet state Colour confinement bound state of qq does not exist • Combination of three quarks • The anti-symmetric singlet colour wave-function qqq bound states exist

  20. Baryon wave functions • Quarks are fermions and have anti-symmetric total wave-functions • The colour wave-function for all bound qqq states is anti-symmetric • For the ground state baryons (L=0) the spatial wave-function is symmetric (-1)L • Two ways to form a totally symmetric wave-function from spin and isospin states: 1. combine totally symmetric spin and isospin wave-function 2. combine mixed symmetry spin and mixed symmetry isospin states - both and are sym. under inter-change of quarks 1 2 but not 1 3 , … - normalized linear combination is totally symmetric under 1 2, 1 3, 2 3

  21. Baryon decuplet • The spin 3/2 decuplet of symmetric flavour and symmetric spin wave-functions Baryon decuplet (L=0, S=3/2, J=3/2, P=+1) • If SU(3) flavour were an exact symmetry all masses would be the same (broken symmetry)

  22. Baryon octet • The spin 1/2 octet is formed from mixed symmetry flavor and mixed symmetry spin wave-functions Baryon octet (L=0, S=1/2, J=1/2, P=+1) • We can not form a totally symmetric wave-function based on the anti-symmetric flavour singlet as there no totally anti-symmetric spin wave –function for 3 quarks

  23. Baryons magnetic moments • Magnetic moment of ground state baryons (L = 0) within the constituent quark model: μl =0 , μs ≠0 • Magnetic moment of spin 1/2 point particle: • for constituent quarks: • magnetic moment of baryon B: qu=+2/3 qd,s=-1/3

  24. Baryons magnetic moments • magnetic moment of the proton: • further terms are permutations of the first three terms 

  25. Baryons: magnetic moments • result with quark masses: • Nuclear magneton

  26. Thank you Reference: University of Cambridge, Prof. Mark Thomson’s lectures 7 & 8, part III major option, Particle Physics 2006 WWW.hep.phy.cam.ac.uk/~thomson/lectures/lectures.html

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