PHYS 745G PresentationSymmetries & Quarks

Shakil MohammedDepartment of Physics & Astronomy

Overview A Brief Overview of Symmetries & GroupsThe SU(2) GroupThe SU(3) GroupQuark-Antiquark States: MesonsThree Quark States: BaryonsMagnetic Moments

Isospin: Quantum number related to the Strong Interactions

For a two-nucleon system, the spin singlet and triplet states are:

Symmetries in Physics

Similarly, each nucleon has an isospin, I = ½, with I3=±½ for protons and neutrons. Then the spin states are:

The base states

The Group SU(2)Generators

Pauli Matrices

Pauli Matrices are Hermitian

The 2×2 matrices known as U(2) and traceless 2×2 form a subgroup SU(2) in two dimension

Combining representations:• Composite system from 2 systems having angular momentum jA and jB

• Combined operator• With a basis,

Where C = Clebsh-Gordan coefficients and M=mA+mB. The C’s are calculated by using

Symbolically, For a third spin-1/2,

SU(2) of Isospin• The nucleon having an internal degree of freedom with two allowed states – Isospin• Isospin generators satisfy,• Generators are denoted as Ii = ½ τi, where

Isospin for Antiparticles

• The antinucleaon states with operator C

• Applying C to the state,

• If we want to transform the antiparticle doublet the same way as particle doublet, then

• A composite system of a nucleon-antinucleon pair

• The set of 3×3 matrices with detU = 1 for the group SU(3)• Fundamental representation of SU(3) is a triplet• The color charges of Quark R, G, B form a SU(3) symmetry group. They are denoted by λi, with i = 1,2,…,8.• The diagonal matrices are:

With eigenvalues:

The Group SU(3)

Hypercharge: Y = B + S

Charge Qe: Q = I3 + Y/2

Quark-Antiquark States: Mesons

• For 3 flavors of Quarks, q = u, d, s – 9 possible combinations of Quark-Antiquark

• Among 9 combinations – 8 states are in SU(3) Octet and 1 state in SU(3) singlet

• The 8 states transform among themselves, but do not mix with singlet state

The states uu*, dd*, ss* labeled A, B and C have I3 = Y = 0.

•The singlet combination C = √1/3(uu*+dd*+ss*)

• State A, a member of the isospin triplet (du*,A,-ud*)

A= √1/2(uu*-dd*)

• Isospin singlet state B (by requiring orthogonality to both A and C)

B= √1/6(uu*+dd*-2ss*)

• The excited states of mesons correspond to the observed meson states• Parity of Meson, P = -(-1)L • The particle-antiparticle conjugation operator C is given by,

C = -(-1)S+1(-1)L = (-1)L+S • In each nonet of the meson, there are two isospin doublets

Mesons of Spin 0

Mesons of Spin 1

Three Quark States: Baryons

• There are 27 possible qqq combinations involved in the SU(3) decomposition• First, the two qq combinations arrange themselves into two SU(3) multiplets having 6 symmetric and 3 anti-symmetric states

• Next, we add the 3rd Quark triplet such that,

• For the pA part,pA = √1/2(ud-du)u

• For the S partΔ = √1/3[uud+(ud+du)u]

• The remaining part requires orthogonality and thus,

pS = √1/6[(ud+du)u-2uud]

For the case of Spins,• Baryon spin multiplets with S = 3/2, 1/2, 1/2

• Replacing u →↑ and d →↓we can have the spin multiplets,

• Next, we combine the SU(3) flavor decomposition with SU(2) spin decomposition

The spin ½ baryon octet

The spin 3/2 baryon decuplet

For the case of Color

• The 3 possible values of color are R, G, B

• The quarks form fundamental triplet of an SU(3) color symmetry

• The color wavefunction of a baryon is,(qqq)col.singlet = √1/6(RGB-RBG+BRG-BGR+GBR-

GRB)

Example: Wavefunction of spin-up proton

• In ground state, l=l’=0 for the qqqThe parity in ground state = (-1)l+l’

• In 1st excited state, l=1, l’=0 or l=0, l’=1•The first excited state contains (1+8+10) flavor multiplets of S = ½ baryons and octet of S = 3/2 baryons• The spins combine with L = 1 to give

Multiplets 1, 8, 10 with JP=1/2- and JP=3/2-

Three octets with JP=1/2-,3/2-,5/2-

• The magnetic moment operator is given as

• Where, the magnetic moment for Quark is

• For proton (in the non-relativistic approx.)

Magnetic Moments