SU(3) symmetry and Baryon wave functions
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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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SU(3) symmetry and Baryon wave functions

Sedigheh Jowzaee

PhD seminar- FZ Juelich, Feb 2013


  • 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

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)

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


  • 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)

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

  • 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)|

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

    • 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

    Combining three quark spin for baryons

    • Same mathematics

    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 :

    • 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:

    • Only need 2 axes (quantum numbers) : (I3,Y)

      All other combinations give zero



    Combining uds quarks for baryons

    • First combine two quarks:

    • a symmetric sextet and anti-symmetric triplet

    • Add the third quark

    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

    combination of three uds quarks in strangeness, charge and isospin axes


    Charge: Q=I3+1/2 Y

    Hypercharge: Y=B+S (B: baryon no.=1/3 for all quarks

    S: strange no.)

    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)



    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

    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

    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

    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)

    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

    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:



    Baryons magnetic moments

    • magnetic moment of the proton:

    • further terms are permutations of the first three terms 

    Baryons: magnetic moments

    • result with quark masses:

    • Nuclear magneton

    Thank you

    Reference: University of Cambridge, Prof. Mark Thomson’s lectures 7 & 8, part III major option, Particle Physics 2006

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