wave functions PhD seminar- FZ Juelich, Feb 2013

Transcription

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 q q 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: Prediction to be unchanged: U is unitary Define infinitesimal transformation Commutation U & Hamiltonian (G is called the generator of the transformation)

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

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, 3 rd component I 3 =±1/2)

6 Flavour symmetry of strong interaction Extend this idea to quarks: strong interaction treats all quark flavours equally Because m u m d (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 I 3 : generally d u u d System of two quarks: I 3 =I 3 (1) +I 3 (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 m s >m u /m d 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 λ and 2 other diagonal matrices are not independent, so de fine λ 3 8 as the linear combination:

13 Only need 2 axes (quantum numbers) : (I 3,Y) Quarks: Anti-Quarks: All other combinations give zero

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: Symmetric decuplet Mixed symmetry octet 2. Building with the triplet: Mixed symmetry octet Totally antisymmetric singlet In summary, the combination of three uds quarks decomposes into:

16 combination of three uds quarks in strangeness, charge and isospin axes Octet Decuplet Charge: Q=I 3 +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: I c (colour isospin), c 3, Y (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 I 3c =0, Y c =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 wavefunctions 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 Thank you

24 Baryons magnetic moments Magnetic moment of ground state baryons (L = 0) within the constituent quark model: μ l =0, μ s 0 Magnetic moment of spin 1/2 point particle: for constituent quarks: q u =+2/3 magnetic moment of baryon B: q d,s =-1/3

25 Baryons magnetic moments magnetic moment of the proton: further terms are permutations of the first three terms

26 Baryons: magnetic moments result with quark masses: Nuclear magneton