Phys 450Spring 2003

Quarks

Experience the strong, weak, and EM interactions

There are anti-quarks as well

Quark masses are not well-defined

Quarks carry color (RGB) Color is the charge of the

strong interaction (SI) Free quarks do not exist? Quarks form bound states

through the SI to produce the hadron spectrum of several hundred observed particles

These bound states are colorless

Structureless and pointlike

Phys 450Spring 2003

Quarks

Experience the strong, weak, and EM interactions

There are anti-quarks as well

Quark masses are not well-defined

Quarks carry color (RGB) Color is the charge of the

strong interaction (SI) Free quarks do not exist? Quarks form bound states

through the SI to produce the hadron spectrum of several hundred observed particles

These bound states are colorless

Structureless and pointlike

Phys 450Spring 2003

Quark Content

Here are some particles for which you should know the quark content

p = uud, n = udd Δ’s = uuu, uud, udd, ddd π = ud, (uu + dd)/√2, du K0 = ds, K0 = sd, there are also K+, K-

Λ = uds, Ω- = sss J/ψ = cc, Υ = bb (the “oops Leon”) D0 = cu, D0 = uc, there are also D+, D-

B0 = db, B0 = bd, there are also B+, B-

Note there are no bound states of the top quark This is because the top quark decays before it

hadronizes

Phys 450Spring 2003

Hadrons

Hadrons == particles that have strong interactions Baryons (fermions) Mesons (bosons)

Baryons == 3 quarks (or antiquarks) p = uud, n = ddu, Λ = uds, Ω- = sss

Mesons == quark plus antiquark π+ = u(d-bar), π- = d(u-bar), π0 = (u(u-bar)+d(d-bar))/√2)

Hadrons can decay via the strong, weak, or electromagnetic interaction

Phys 450Spring 2003

Quark Model

By the 1960’s scores of “elementary particles” had been discovered suggesting a periodic table “The discoverer of a new particles used to be

awarded the Nobel Prize; now, he should be fined $10000” – Lamb

Underlying structure to this spectrum was suggested by Gell-Mann in the 1960’s First through the “Eightfold Way” and later

through the quark model It took approximately a decade for

physicists to accept quarks as being “real” Discovery of J/ψ and deep inelastic scattering

experiments gave evidence that partons = quarks

Phys 450Spring 2003

Quark Model

One of the early successes of the quark model (Eightfold Way) was the prediction of the existence of the Ω-

before its discovery

Phys 450Spring 2003

A Little More (review) on Spin

Physics should be unchanged under symmetry operations

Rotations form a symmetry group So do infinitesimal rotations

The angular momentum operators are the generators of the infinitesimal rotation group

An infinitesimal rotation ε about z is U ψ(x,y,z) = ψ (R-1r) ~ ψ (x+εy,y-εx,z) = ψ(x,y,z) + ε(y∂ψ/∂x - x∂ψ/∂y) = (1 - iε(xpy – ypx))ψ = (1 – iεJ3)ψ

And the generators (angular momentum operators) satisfy commutation relations and have eigenvalues shown on the previous page

Phys 450Spring 2003

SU(2) Group (Jargon)

SU(2) group is the set of all traceless unitary 2x2 matrices (detU = 1)

U(2) group is the set of all unitary 2x2 matrices U†U = 1 U(θi) = exp(-iθiσi/2) σi are the Pauli matrices and Ji = σi/2

The generators of this group are the Ji The SU(2) algebra is just the algebra of the

generators Ji The lowest, nontrivial representation of the

group are the Pauli matrices The basis for this representation are the

column vectors

Phys 450Spring 2003

SU(2) Group Representations

Higher order representations (higher order spin states) can be built from the fundamental representation (by adding spin states via the CG coefficients) A composite system is described in terms

of the basis |jAjBJM> == |jAmA>|jBmB> The J’s and M’s follow the normal rules for

addition of angular momentum |jAjBJM> = ∑ CG(mAmB;JM>|jAjBmAmB> where

the CG are the Clebsch-Gordon coefficients we talked about earlier in the course

Phys 450Spring 2003

SU(2) Representations

The product of 2 irreducible representations of dimension 2jA+1

and 2jB+1 may be decomposed into the sum of irreducible representations of dimension 2J+1 where J = jA+jB, …, |jA+jB| Irreducible means …

What is he talking about???

Phys 450Spring 2003

SU(3) Group (Jargon)

SU(3) group is the set of all traceless unitary 3x3 matrices (detU = 1)

The generators of this group are the Fa

There are 32-1 = 8 generators Fa

They satisfy the algebra [Fa,Fb] = ifabcFc

fabc== structure constants The generators Fa = 1/2λa where λa are

the Gell-Mann matrices (see next page)

The basis for this representation are the column vectors

Phys 450Spring 2003

SU(3) Group

Note F3 and F8 are diagonal F3 == Isospin operator F8 == Hypercharge operator Later we’ll define Y = B+S and Experimentally we find Q = I3 + Y/2

Phys 450Spring 2003

SU(3) Represenations

Combining 2 SU(3) objects 3 x 3 = 6 + 3

It’s a 3 because in Y, I3 space the u, d, s triangle looks like the ud, us, ds triangle

Phys 450Spring 2003

SU(3) Representations

Combining 3 SU(3) objects 3 x 3 x 3 = 3 x (6 + 3) = 10 + 8 + 8 + 1 Note the 8’s! Note the symmetry is S, MS, MA, A The mixed symmetry representations

are given on the next page

Phys 450Spring 2003

Quark Model

Hopefully you’ve caught on to what we’ve done

Let u, d, s be the SU(3) basis states Define isospin Ii = λi/2

Define hypercharge Y = λ8/√3 = B+S Since λ3 and λ8 are diagonal, I3 and Y are

conserved and represent additive quantum numbers

Note I2, S, Q = I3 + Y/2 are also diagonal and hence are conserved and represent additive quantum numbers

Phys 450Spring 2003

Quark Model

u d s

I 1/2 1/2 0

I3 1/2 -1/2 0

Y 1/3 1/3 -2/3

Q 2/3 -1/3 -1/3

B 1/3 1/3 1/3

S 0 0 -1

Spin 1/2 1/2 1/2

P + + +

u d s

I ½ 1/2 0

I3 -1/2 ½ 0

Y -1/3 -1/3 2/3

Q -2/3 1/3 1/3

B -1/3 -1/3 -1/3

S 0 0 1

Spin 1/2 1/2 1/2

P - - -

Phys 450Spring 2003

Quark Model

A convenient way to display the multiplet is to show its elements on a weight diagram in Y-I3 space

Note that the combinations ud, us, ds would appear in the same triangle as s, d, u

Phys 450Spring 2003

Mesons

3 x 3 = 8 + 1 One can determine the multiplet by explicit

calculation of the representation or by the following trick