Nuclear and Particle Physics Franz Muheim 1

Quark ModelQuark Model

HadronsIsospin, Strangeness

Quark Model3 Flavours u, d, s

MesonsPseudoscalarand vector mesons

BaryonsDecuplet, octet

Hadron MassesSpin-spin coupling

Heavy QuarksCharm, bottom, Heavy quark MesonsTop quark

Motivation for Quark ModelParticle “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

OutlineOutlineHadrons known in 1960

Nuclear and Particle Physics Franz Muheim 2

IsospinIsospin

NucleonsProton and neutron have almost equal massStrong nuclear force is charge independent

Vpp≈ Vpn ≈ Vnn

Isospinp and n form part of single entity withisospin ½ analogous to ↑ and ↓ of spin ½Isospin I is conserved in strong interactionsAddition by rules of angular momentum

Isospin MultipletsUseful for classification of hadrons, see slide 12I+1 states in a isospin muliplet |I, I3 >

Quark Model Gives natural explanation for Isospin

ni number of i quarksIsospin works wellMasses of u and d quark are almost equal

( )uddu nnnnI −+−= 21

3

Nuclear and Particle Physics Franz Muheim 3

IsospinIsospin ConservationConservation

Conservation LawIsospin I is conserved in strong interactionsAllows to calculate ratios of cross sections and branching fractions in strong interactions

Delta(1232) Resonance

Cross sections

In agreement withI=3/2 Isospin prediction

nppppp

00

0

ππ

ππ

ππ

→∆→

→∆→

→∆→

−

−−

++++

( )( )

( ) 1xmb 233xmb 70all9x mb 200

0

0

≈≈→∆→

≈≈→∆→

≈≈→∆→

−−

−

++++

ppp

pp

ππσ

πσ

ππσ

Production

Isospin addition

Matrix elementdepends on I, not I3

21

21

31

21

23

32

21

210

21

21

32

21

23

31

21

21

23

23

21

21

,,,0,1:

,,,1,1:

,,1,1:

−+−=−

−−−=−

=−

+

n

p

p

π

π

π

( )( )( ) 13

233

200

132

3310

3

MMnpM

MMppM

MppM

−=→∆→

+=→∆→

=→∆→

−

−−

++++

ππ

ππ

ππ

Mass 1232 MeVWidth 120 MeV

21

121

1

23

323

3

HM

HM

=

=

2M∝σ

Nuclear and Particle Physics Franz Muheim 4

StrangenessStrangeness

Strange ParticlesDiscovered in 1947 Rochester and ButlerV, “fork”, and K, “kink”

Production of V(K0, Λ) and K±

via strong interaction, weak decay

Associated ProductionStrange particles produced in pairs Pais

Strangeness SAdditive quantum number Gell-Mann NishijimaConserved in strong and electromagnetic interactionsViolated in weak decaysNon-zero for Kaonsand hyperonsNaturally explained in quark model

( )

sp

sKsOKp

K

10

100

230

1063.2

1089.010

0

−Λ

−

−−+

−−

×=→Λ

×=→

=Λ→

τπ

τππ

τπ

Ξ−=ΣΛ−=

=∆=−

+

:2...,,,,:1,:1...,,,,:0

0

0

SKKSKKSnpS π

ss nnS −=

Nuclear and Particle Physics Franz Muheim 5

Quark ModelQuark Model

3 Quark Flavours u, d, s

1964 - introduced by Gell-Mann & Zweig

Charge, Isospin and StrangenessAdditive quark quantum numbers are relatedQ = I3 + ½(S + B) not all independent

Gell-Mann Nishijima predates quark modelvalid also for hadrons

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

Hypercharge Y = S + B is useful quantum numberQuark model gives natural explanationfor Isospin and Strangeness

Quark ChargeQ [e]

Isospin|I, I3 >

Strange-ness S

up (u) +2/3 |½, +½ ›

|½, -½ ›

|0,0›

0down (d) -1/3 0

strange (s) -1/3 -1

Gell-Mann

Zweig

3 Quark Flavours u, d, s3 Quark Flavours u, d, s

Nuclear and Particle Physics Franz Muheim 6

MesonsMesons

Bound StatesZero net colour chargeZero net baryon number B = +1/3 +(-1/3) = 0

Angular Momentum LFor lightest mesons Ground stateL = 0 between quarks

Parity PIntrinsic quantum number of quarks and leptonsP=+1 for fermions P=-1 for anti-fermions

Total Angular Momentum Jinclude quark spins

S = 0 spins anti-aligned ↑↓ or ↓↑JP = 0- Pseudo-scalar mesons

S = 1 spins aligned ↑↑ or ↓↓JP = 1- Vector mesons

Quark flavoursnon-zero flavour stateszero net flavour states

have identical additive quantum numbersPhysical states are mixtures

( ) ( )( )( )( ) 0for 1111

1

=−=−−+=

−=

L

PPqqPL

Lqq

qq

SLJ +=rr

ssdduudsussdudsudu

,,,,,,,

qq

qq

Nuclear and Particle Physics Franz Muheim 7

MesonsMesons

Pseudoscalar Mesons JP = 0-

Vector Mesons JP = 1-

Isospin I3

Stra

ngen

ess

SSt

rang

enes

s S

Isospin I3

Kaons:K+, K0, anti-K0, K-

Pions: π+, π0, π-

Etas: η, η’

Kstar: K*+, K*0, anti-K*0, K*-

rho: ρ+, ρ0, ρ-

omega/phi: ω, φ

Nuclear and Particle Physics Franz Muheim 8

Baryon Baryon DecupletDecuplet

Baryon WavefunctionΨ(total) = Ψ(space) Ψ(spin) Ψ(flavour) Ψ(colour)Space symmetric - L = 0Flavour symmetric, e.g. uuu, (udu+duu+uud)/√3Spin symmetric

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

Total antisymmetric - obeys Pauli Exclusion Principle

Baryon Decuplet JP = 3/2+

Quark model predicted unobserved state Ω- (sss)

Stra

ngen

ess

S

Isospin

<Mass>Delta 1232 MeV

Sigma* 1385 MeV

Cascade* 1533 MeV

Omega- 1672 MeV

uuu

Nuclear and Particle Physics Franz Muheim 9

Baryon OctetBaryon Octet

Baryon WavefunctionΨ(space) symmetric (L = 0) Ψ(colour) antisymmetricMixed symmetric Ψ(spin, flavour) Flavour mixed symmetric: e.g. (ud - du) u/√2Spin as flavour: e.g. (↑↓ - ↑↓) ↑/√2Spin-flavour e.g. (u↑d↓ - d↑u↓ - u↓d↑ + d↓u↑) u↑/√6Symmetrisation by cyclic permutationsΨ(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 JP = ½+

Lightest baryons stable or long-livedAntibaryons also form Octet and Decuplet( )...,, np

Stra

ngen

ess

S

Isospin

<Mass>p,n 938.9 MeV

Sigma 1193 MeVLambda 1116 MeV

Cascade 1318 MeV(Xi)

Nuclear and Particle Physics Franz Muheim 10

Discovery of Discovery of ΩΩ--

Ω- (sss) HyperonHyperon - baryon with at least one s quarkQuark model predicted existence and massMissing member of baryon decuplet JP = 3/2+

discovered 1964 at BrookhavenK- beam onto hydrogen targetBubble Chamber detector

pee

ee

KKpK

−

−+

−+

+

−−−

++Λ

+Ξ

++Ω→+

π

γγπ

π

a

a

a

a

a

a00

0

0.

Nuclear and Particle Physics Franz Muheim 11

HadronHadron MassesMasses

Quark Massesu, d & s quark masses light at short distanceq2 > 1 GeV2 mu < md ~ 5 MeV ms ~ 100 MeVConstituent mass is relevant for quark modelq2 < 1 GeV2 mu = md ~ 300 MeV ms ~ 500 MeV

Meson Massesm(K) > m(π) due to ms > mu, mdm(ρ) > m(π) same quark content e.g. ρ+, π+: (u-dbar)Mass difference is due to quark spins

Chromomagnetic Mass SplittingSpin-spin coupling of quarks S1 = S2 = 1/2analogous to hyperfine splitting in el. mag. interaction

Meson Massesmu = md = 310 MeVms = 483 MeVA = (2mu)2 · 160 MeVExcellent agreementWhat about eta(‘)?

( )

( ) ( )

⎪⎩

⎪⎨

⎧

=−=−

==−=

+−+−+=−−=⋅

⋅++=

⋅∝∆

043

430

141

431

)1()1()1(21

21

22112

22

12

21

21

2121

21

21S

S

S

SSSSSSSSSSS

mmSSAmmqqm

mmSSE

rrrrr

rrrr

α

Mass [MeV]

ω 780 782K* 896 894

Meson Prediction Experimentπ 140 138K 484 496ρ 780 770

φ 1032 1019

( )21

2121 mm

SSAmmqqmrr

⋅++=

Nuclear and Particle Physics Franz Muheim 12

Heavy QuarksHeavy Quarks

Charm and bottom quarksCharmonium (c-cbar) --- see QCD lecture1977 Discovery of Upsilon StatesInterpretation isBottomonium (b-bar)

SpectroscopyCharmoniumand Upsilonmc ~ 1.1 … 1.4 GeVmb ~ 4.1 … 4.5 GeV

Heavy-light Mesons and Baryons Charmed (c-quark) hadrons

Bottom-quark hadrons

Top quarkDecays before forming bound statesmt ~ 174 GeV discovered in 1995 at Fermilab

cudJ

scDdcDucDJ

scDdcDucDJ

cP

sP

sP

=Λ=

====

====

+−

++−

++−

21

,,,1

,,,0**0*

0

budJ

bsBbdBbuBJ

bsBbdBbuBJ

bP

sP

sP

=Λ=

====

====

−

+−

+−

0

0*0**

00

21

,,,1

,,,0