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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

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
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