Quark Model. Quark Model

(1)

Quark Model

Hadrons Isospin, Strangeness Quark Model 3 Flavours u, d, s Mesons Pseudoscalar

and vector mesons

Baryons Decuplet, octet Hadron Masses Spin-spin coupling Heavy Quarks Charm, bottom, Heavy quark Mesons Top quark

Motivation for Quark Model

Particle “Zoo” proliferates

“ … the finder of a new particle used to be rewarded by a Nobel prize, but such a discovery ought to be punished by a $10000 fine” Lamb, 1955

Outline

(2)

Isospin

Nucleons

Proton and neutron have almost equal mass Strong nuclear force is charge independent

V_pp≈ V_pn ≈ V_nn

Isospin

p and n form part of single entity with

isospin ½ analogous to ↑ and ↓ of spin ½ Isospin I is conserved in strong interactions Addition by rules of angular momentum

Isospin Multiplets

Useful for classification of hadrons, see slide 1 2I+1 states in a isospin muliplet |I, I₃ >

Quark Model

Gives natural explanation for Isospin

n_i number of i quarks Isospin works well

Masses of u and d quark are almost equal

(

nu nd nd nu

)

I = ₂1 − + − 3

(3)

Isospin

Conservation

Conservation Law

Isospin I is conserved in strong interactions

Allows to calculate ratios of cross sections and

branching fractions in strong interactions

Delta(1232) Resonance

Cross sections

In agreement with

I=3/2 Isospin prediction

n p p p p p 0 0 0 π π π π π π → ∆ → → ∆ → → ∆ → − − − + + + +

(

)

(

)

(

)

23mb 1x 3x mb 70 all 9x mb 200 0 0 ≈ ≈ → ∆ → ≈ ≈ → ∆ → ≈ ≈ → ∆ → − − − + + + + p p p p p π π σ π σ π π σ Production Isospin addition Matrix element depends on I, not I₃ 2 1 2 1 3 1 2 1 2 3 3 2 2 1 2 1 0 2 1 2 1 3 2 2 1 2 3 3 1 2 1 2 1 2 3 2 3 2 1 2 1 , , , 0 , 1 : , , , 1 , 1 : , , 1 , 1 : − + − = − − − − = − = − + n p p π π π

(

)

(

)

(

)

3 1 2 3 3 2 0 0 1 3 2 3 3 1 0 3 M M n p M M M p p M M p p M − = → ∆ → + = → ∆ → = → ∆ → − − − + + + + π π π π π π Mass 1232 MeV Width 120 MeV 2 1 1 2 1 1 2 3 3 2 3 3 H M H M = = 2 M ∝ σ

(4)

Strangeness

Strange Particles

Discovered in 1947 Rochester and Butler

V, “fork”, and K, “kink”

Production of V(K0_,_Λ_{) and K}± via strong interaction,

weak decay

Associated Production

Strange particles produced in pairs Pais

Strangeness S

Additive quantum number Gell-Mann Nishijima Conserved in strong and electromagnetic interactions

Violated in weak decays Non-zero for Kaons

and hyperons

Naturally explained in quark model

(

)

s p s K s O K p K 10 10 0 23 0 10 63 . 2 10 89 . 0 10 0 − Λ − − − + − − × = → Λ × = → = Λ → τ π τ π π τ π Ξ − = Σ Λ − = = ∆ = − + : 2 ... , , , , : 1 , : 1 ... , , , , : 0 0 0 S K K S K K S n p S π s s n n S = −

(5)

Nuclear and Particle Physics Franz Muheim 5

Quark Model

3 Quark Flavours u, d, s

1964 - introduced by Gell-Mann & Zweig

Charge, Isospin and Strangeness

Additive quark quantum numbers are related Q = I₃ + ½(S + B) not all independent

Gell-Mann Nishijima predates quark model valid also for hadrons

Baryon number B quarks B = +1/3 anti-quarks B = -1/3

Hypercharge Y = S + B is useful quantum number Quark model gives natural explanation

for Isospin and Strangeness

Quark

Charge

Q [e] Isospin|I, I₃ > Strange-ness S

up (u) +2/3 |½, +½ › |½, -½ › |0,0› 0 down (d) -1/3 0 strange (s) -1/3 -1 Gell-Mann Zweig

3 Quark Flavours u, d, s

(6)

Mesons

Bound States

Zero net colour charge

Zero net baryon number B = +1/3 +(-1/3) = 0

Angular Momentum L

For lightest mesons

Ground state

L = 0 between quarks

Parity P

Intrinsic quantum number of quarks and leptons P=+1 for fermions P=-1 for anti-fermions

Total Angular Momentum J

include quark spins

S = 0 spins anti-aligned ↑↓ or ↓↑

ÎJP _{= 0}- _{Pseudo-scalar mesons}

S = 1 spins aligned ↑↑ or ↓↓

ÎJP _{= 1}- _{Vector mesons}

Quark flavours

non-zero flavour states

zero net flavour states have identical additive quantum numbers

Physical states are mixtures

( )

( )( )( )

1 1 1 1 for 0 1 = − = − − + = − = L P P q q P L L q q q q S L Jr = r + s s d d u u d s u s s d u d s u d u , , , , , , , q q q q

(7)

Mesons

Pseudoscalar Mesons J

P

= 0

-Vector Mesons J

P

= 1

-Isospin I₃ Strangeness S Strangeness S Isospin I₃ Kaons: K+_{, K}0_{, anti-K}0_{, K} -Pions: π+_,_π0_,_π -Etas: η, η’ Kstar: K*+_{, K}*0_{, anti-K}*0_{, K} *-rho: ρ+_,_ρ0_,_ρ -omega/phi: ω, φ

(8)

Baryon

Decuplet

Baryon Wavefunction

Ψ(total) = Ψ(space) Ψ(spin) Ψ(flavour) Ψ(colour)

Space symmetric - L = 0

Flavour symmetric, e.g. uuu, (udu+duu+uud)/√3

Spin symmetric

all 3 quarks aligned → S = 3/2 Colour antisymmetric

Total antisymmetric - obeys Pauli Exclusion Principle

Baryon Decuplet J

P

= 3/2

+

Quark model predicted unobserved state Ω- _(sss)

Strangeness S Isospin <Mass> Delta 1232 MeV Sigma* 1385 MeV Cascade* 1533 MeV Omega- _{1672 MeV} uuu

(9)

Baryon Octet

Baryon Wavefunction

Ψ(space) symmetric (L = 0) Ψ(colour) antisymmetric

Mixed symmetric Ψ(spin, flavour)

Flavour mixed symmetric: e.g. (ud - du) u/√2

Spin as flavour: e.g. (↑↓ - ↑↓) ↑/√2

Spin-flavour e.g. (u↑d↓ - d↑u↓ - u↓d↑ + d↓u↑) u↑/√6

Symmetrisation by cyclic permutations

Ψ(proton, s=+½) = ( 2u↑u↑d↓ - u↑u↓d↑- u↓u↑d↑

+2d↓u↑u↑ - d↑u↑u↓- d↑u↓u↑

+2u↑d↑u↓ - u↑d↓u↑- u↓d↑u↑) /√18

Baryon Octet J

P

= ½

+

Lightest baryons stable or long-lived

Antibaryons

(

p, n,...

)

also form Octet and Decuplet

Strangeness S Isospin <Mass> p,n 938.9 MeV Sigma 1193 MeV Lambda 1116 MeV Cascade 1318 MeV (Xi)

(10)

Discovery of

Ω

-

-Ω

-

(sss) Hyperon

Hyperon - baryon with at least one s quark

Quark model predicted existence and mass Missing member of baryon decuplet JP _{= 3/2}+

discovered 1964 at Brookhaven K- _beam _onto_hydrogen _target

Bubble Chamber detector

p e e e e K K p K − − + − + + − − − + + Λ + Ξ + + Ω → + π γ γ π π a a a a a a 0 0 0 0 .

(11)

Hadron

Masses

Quark Masses

u, d & s quark masses light at short distance

q2 _{> 1 GeV}2 _m

u < md ~ 5 MeV ms ~ 100 MeV Constituent mass is relevant for quark model

q2 _{< 1 GeV}2 _m

u = md ~ 300 MeV ms ~ 500 MeV

Meson Masses

m(K) > m(π) due to m_s> m_u, m_d

m(ρ) > m(π) same quark content e.g. ρ+_,_π+_{: (u-dbar)}

Mass difference is due to quark spins

Chromomagnetic Mass Splitting

Spin-spin coupling of quarks S₁ = S₂ = 1/2

analogous to hyperfine splitting in el. mag. interaction

Meson Masses

m_u = m_d = 310 MeV

m_s = 483 MeV

A = (2m_u)2 _{· 160 MeV} Excellent agreement

What about eta(‘)? ( )

(

)

( ) ⎪ ⎩ ⎪ ⎨ ⎧ = − = − = = − = + − + − + = − − = ⋅ ⋅ + + = ⋅ ∝ ∆ 0 4 3 4 3 0 1 4 1 4 3 1 ) 1 ( ) 1 ( ) 1 ( 2 1 2 1 2 2 1 1 2 2 2 1 2 2 1 2 1 2 1 2 1 2 1 2 1 S S S S S S S S S S S S S S m m S S A m m q q m m m S S E r r r r r r r r r α Mass [MeV] ω 780 782 K* 896 894

Meson Prediction Experiment

π 140 138 K 484 496 ρ 780 770 φ 1032 1019 ( ) 2 1 2 1 2 1 m m S S A m m q q m r r ⋅ + + =

(12)

Heavy Quarks

Charm and bottom quarks

Charmonium (c-cbar) --- see QCD lecture

1977 Discovery of Upsilon States

Interpretation is Bottomonium (b-bar)

Spectroscopy

Charmonium and Upsilon m_c ~ 1.1 … 1.4 GeV m_b ~ 4.1 … 4.5 GeV

Heavy-light Mesons and Baryons

Charmed (c-quark) hadrons

Bottom-quark hadrons

Top quark

Decays before forming bound states

m_t ~ 174 GeV discovered in 1995 at Fermilab cud J s c D d c D u c D J s c D d c D u c D J c P s P s P = Λ = = = = = = = = = + − + + − + + − 2 1 , , , 1 , , , 0 * * 0 * 0 bud J b s B b d B b u B J b s B b d B b u B J b P s P s P = Λ = = = = = = = = = − + − + − 0 0 * 0 * * 0 0 2 1 , , , 1 , , , 0