PHYS 745G Presentation Symmetries & Quarks 4/15/2017 7:15 PM PHYS 745G Presentation Symmetries & Quarks Shakil Mohammed Department of Physics & Astronomy Symmetries & Quarks

Overview A Brief Overview of Symmetries & Groups The SU(2) Group Quark-Antiquark States: Mesons Three Quark States: Baryons Magnetic Moments

Symmetries in Physics Isospin: Quantum number related to the Strong Interactions For a two-nucleon system, the spin singlet and triplet states are:

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

The Group SU(2) Generators Pauli Matrices The base states 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 Group SU(3) 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:

Hypercharge: Y = B + S Charge Qe: Q = I3 + Y/2 Quark-Antiquark States: Mesons Hypercharge: Y = B + S Charge Qe: Q = I3 + Y/2

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 part Δ = √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

The 3 possible values of color are R, G, B 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 qqq The 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-

Magnetic Moments The magnetic moment operator is given as Where, the magnetic moment for Quark is For proton (in the non-relativistic approx.)

Thank You! Questions?