AIMS-pp-08.pdf

(1)

The Quark Model

(2)

electron neutrino electron

down up

muon neutrino muon

strange charm

tau neutrino

tau

bo8om top

Increasing mass

+2/3

-‐1/3

-‐1

0 All

normal ma8er is made of these parCcles

The Building Blocks of Matter

(3)

Ground state

symmetric



=

0 ₂

1

2

1

2

3

2

1

2

1

2

1 ⊕

⊕

=

⊗

_SU(3)

ﬂavour

irreps

SU(3)_colourirreps! anCsymmetric!

must be symmetric

Baryons

Must be anC-‐ symmetric (Pauli

Principle)

ϕ_A = 1

6

(

rbg − brg + bgr

− gbr + grb − rgb

)

(4)

uuu

ddd

sss

uss+sus+ssu

3

dss+sds+ssd

3

dds+dsd+sdd 3

uus+usu+suu

3

uud+udu+duu

3

udd+dud+ddu 3

uds+usd+dsu+dus+sdu+sud 6

strangeness

0

-2

-3

i=3/2

i=1

i=1/2

i=0

-‐

1232 MeV

1384 MeV

1533 MeV

1672 MeV

J

=

3

2 i

3 Δ++ Δ+ Δ0 Δ− Σ+ Σ0 Σ−

Ξ* _Ξ*0

Ω−

(2(ud−du)s+(us−su)d−(ds−sd)u) / 12 ((us−su)d+(ds−sd)u) / 2

(ud−du)d 2

(ud−du)u

2

(ds−sd)d

2

(us−su)d

2

(ds−sd)s

2

(us−su)s 2

(2s(ud−du)+d(us−su)−u(ds−sd)) / 12

(d(us−su)+u(ds−sd)) / 2

d(ud−du) 2

u(ud−du) 2

d(ds−sd) 2

u(us−su) 2

s(ds−sd) 2

s(us−su) 2

⊕

strangeness

0

-2

939 MeV

1193 MeV

1318 MeV 1116 MeV

J

=

1

2 i

3 p n Σ+ Σ0 Σ−

Ξ Ξ0

(5)

Decuplet

E x a m p l e s

This is the only symmetric combination of spin with

χ

10

uuu

ddd

sss

uss+sus+ssu

3

dss+sds+ssd

3

dds+dsd+sdd 3

uus+usu+suu

3

uud+udu+duu

3

udd+dud+ddu 3

(6)

uuu

ddd

sss

uss+sus+ssu

3

dss+sds+ssd

3

dds+dsd+sdd 3

uus+usu+suu

3

uud+udu+duu

3

udd+dud+ddu 3

uds+usd+dsu+dus+sdu+sud 6

strangeness

0

-2

-3

i=3/2

i=1

i=1/2

i=0

-‐

1232 MeV

1384 MeV

1533 MeV

1672 MeV

J

=

3

2 i

3 Δ++ Δ+ Δ0 Δ− Σ+ Σ0 Σ−

Ξ* _Ξ*0

Ω−

We write down quark wavefunctions for the

J

=

3 / 2 states by taking products of

(7)

Octet

This is the only symmetric combination of spin with

χ

10 (2(ud−du)s+(us−su)d−(ds−sd)u) / 12

((us−su)d+(ds−sd)u) / 2 (ud−du)d

2

(ud−du)u

2

(ds−sd)d

2

(us−su)d

2

(ds−sd)s

2

(us−su)s 2

d(ud−du) 2

u(ud−du) 2

d(ds−sd) 2

u(us−su) 2

s(ds−sd) 2

s(us−su) 2

⊕

Can also write this as: 2

ψ

₁₂1/2

χ

₁₂8

+

2 ψ

₂₃1/2

χ

₂₃8

+

ψ

1/2₂₃

χ

₁₂8

+

ψ

₁₂1/2

χ

₂₃8

There are NO other symmetric combinations of spin with

χ

10

(8)

InterpretaCon?

Probability of ﬁnding a spin-‐down d-‐quark inside the proton is: p;1

2, 1

2 =N 2 1

2

(

↑↓ − ↓↑

)

↑ ⎡

⎣⎢

⎤ ⎦⎥

1

2

(

ud − du

)

u ⎡ ⎣⎢ ⎤ ⎦⎥ ⎧ ⎨

⎩ +2

1

2 ↑ ↑↓ − ↓↑

(

)

⎡

⎣⎢

⎤ ⎦⎥

1

2 u

(

ud − du

)

⎡

⎣⎢

⎤ ⎦⎥

1

2 ↑ ↑↓ − ↓↑

(

)

⎡

⎣⎢

⎤ ⎦⎥

1

2

(

ud − du

)

u ⎡

⎣⎢

⎤ ⎦⎥+

1

2

(

↑↓ − ↓↑

)

↑ ⎡

⎣⎢

⎤ ⎦⎥

1

2 u

(

ud − du

)

⎡ ⎣⎢ ⎤ ⎦⎥ ⎫ ⎬ ⎭

Normalize this state

⇒

p

;

1

2 ,

1

2 p

;

1

2 ,

1

2 =

1 ⇒

N

=

1

18 Expand it out:

p

;

1

2 ,

1

2 =

1

18 (

2 u

↑

d

↓

u

↑ −

u

↓

d

↑

u

↑ −

d

↑

u

↓

u

↑

+

2 d

↓

u

↑

u

↑

+

2 u

↑

u

↑

d

↓ −

u

↑

u

↓

d

↑

−

u

↑

d

↑

u

↓ −

d

↑

u

↑

u

↓ −

u

↓

u

↑

d

↑

)

(9)

(2(ud−du)s+(us−su)d−(ds−sd)u) / 12 ((us−su)d+(ds−sd)u) / 2

(ud−du)d 2

(ud−du)u

2

(ds−sd)d

2

(us−su)d

2

(ds−sd)s

2

(us−su)s 2

d(ud−du) 2

u(ud−du) 2

d(ds−sd) 2

u(us−su) 2

s(ds−sd) 2

s(us−su) 2

⊕

strangeness

0

-2

939 MeV

1193 MeV

1318 MeV 1116 MeV

J

=

1

2 i

3 p n Σ+ Σ0 Σ−

Ξ Ξ0

Λ

We write down quark wavefunctions for the

J

=

1 / 2 states by taking symmetric

(10)

Mesons

_•_{Bound state of a quark with an antiquark}

•  Pauli principle no longer an issue because particles are never

identical

Ground state

symmetric



=

0

1

2

1

2

1 ⊕

=

⊗

SU(3)ﬂavour

irreps must be colour neutral; guaranteed since we have an anCparCcle

s u s d u d

(

uu−dd

)

2 1 u s d u − d s −

(uu dd 2ss)

6 1

−

+ (uu+dd +ss)

3 1

8

χ

1

s u s d u d

(

uu−dd

)

2 1 u s d u − d s −

(uu dd 2ss)

6 1

−

+ (uu+dd +ss)

3 1

8

(11)

+

K 0

K

−

K _K0

+

π

−

π

0

π

8

η

0

η

Spin-0 Mesons

Theoretical

η

mesons are a bit different:

Experimental

(the octet and singlet get mixed together

because the strange quarks strongly interact and the quantum numbers are the same

(12)

+

*

K

0 *

K

0 *

K

−

*

K

+

ρ

−

ρ

0

ρ

8

φ

0

φ

Spin-1 Mesons

Theoretical

φ

mesons are a bit different:

Experimental

(13)

Can we do better?

M

_meson

=

sum of quark masses

+

colour dipole interaction energy

Quark Mass RelaCons

m

_s



150 MeV?

dipole-‐dipole interacCon

Yes! Quarks have

dipole moments

(14)

Electromagnetic Hyperfine splitting

Colour Hyperfine splitting

(15)

(16)

Colour factor

(17)

(18)

(19)

(20)

MagneCc Moments

Compute for each baryon wavefuncCon

(21)

For the proton:

Each state is orthogonal to the others, so do this term-‐by-‐term

First term contributes

All others are similar to one of these two, so get:

2nd term

contributes

(22)

Fit okay – but not fantastic