SU(3) symmetry and Baryon wave functions
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Transcript of 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)
• Idea: extend isospin symmetry to three flavors (Gell-Mann, Ne’eman 1961)
• SU(3) flavour and color symmetry groups
q q̄
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)
• 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
Symmetry and conservation
• 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)
Isospin
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: 3λ λ8
• Only need 2 axes (quantum numbers) : (I3,Y)
All other combinations give zero
Quarks: Anti-Quarks:
• First combine two quarks:
• a symmetric sextet and anti-symmetric triplet
• Add the third quark
Combining uds quarks for baryons
1. Building with sextet:
2. Building with the triplet:
• In summary, the combination of three uds quarks decomposes into:
Symmetric decupletMixed symmetry
octet
Mixed symmetry octet
Totally anti-symmetric singlet
combination of three uds quarks in strangeness, charge and isospin axes
Octet Decuplet
Charge: Q=I3+1/2 YHypercharge: 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)
Quarks: Anti-Quarks:
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 I3
c=0, Yc=0
– Invariant under SU(3) colour transformation– Ladder operators are yield zero
• 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 colour wave-function
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:
qu=+2/3
qd,s=-1/3
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 2006WWW.hep.phy.cam.ac.uk/~thomson/lectures/lectures.html