Non-relativistic Quark Model

Sugat V ShendeKVI

Student Seminar on Subatomic Physics

Date: 26 Oct 2005

Contents

• Quark Model

• Mesons in quark Model

• Baryons in quark Model

• Baryon mass relation

• Isgur –Karl Model

• Relativized quark Model

•Literature : 1) Nuclear and Particle Physics by Burcham and Jobes 2) S. Capstick, W. Roberts, Prog. Part. Nucl. Phys. 45,(2000) S241 - S331.

Quark Model

The baryon consists of valence quarks, sea quarks and gluons.

The quark models assumes 3 constituentquarks with effective quark masses anda spatial configuration.

Mesons in quark Model

The fundamental quark triplet :

I3

Y

1/3

-2/3

1/2-1/2

Y=B+S

I3

1/3

1

1/2

su

ds us

sd

ud

s

q

qq combination gives :

singlet {1} and octet {8} states K0

JP = 0- K+

- +

K - K0

0

1

8

q are spin ½ fermions, qq state total spin S = 0 or 1

J = L + S Parity = (-1)L+1

(-1)L arises from orbital motion1 opposite intrinsic parities of q and q

JP = 0-, 1- , 2+

Baryons in quark Model

Baryons qqq state

The wavefunction: = (space) (flavour) (spin) (colour) must be antisymmetric

Each quark flavour comes in three colours, Red, Green and Blue

= (1/6){ |RGB> + |GBR> + |BRG> - |GRB> - |BGR> - |RBG>}

is antisymmetric in the exchange of any two quark colours

Baryons in quark Model

AS MMS 224222

S4

In SU(2) the direct product of three spin doublets

ms (s,ms)

+3/2 ()+1/2 1/3[( + ) + ()]-1/2 1/3[( + ) + () ]-3/2 ()

ms (s,ms)

+1/2 1/6[( + ) - 2()]-1/2 1/6[( + ) - 2() ]

ms (s,ms)

+1/2 1/2[( - )]-1/2 1/2[( - )]AM2

SM2

3363)36(3333

)224()18810(ASSS MMSAMMS

)2,8()4,10(

In order to predict the nature of baryon multiplets,we should combine SU(3) flavour multiplets with the spin multiplets

In SU(3)

Symmetric combinations

notation is (nSU(3),nSU(2)) where n is dimensionality

AMMS AS18810

Baryons in quark Model

JP = (3/2)+

0 1/2 1 3/2-1/2-1-3/2

1

0

-1

-2

Y

I3

uuuuududdddd

uusudsdds

ussdss

sss

JP = (1/2)+

0 1/2 1 3/2-1/2-1-3/2

1

0

-1

-2

Y

I3

uududd

uusuds

dds

dss ussuds

Quark model successfully predicts a decuplet of (3/2)+

and octet of (1/2)+ baryons

The baryon mass relation :

m - m m -m m - m 150 MeV

The mass of a particular U spin state |U,U3> is

<U,U3|H|U,U3> = <U,U3| H0 + Hv + H

s |U,U3> = m0 + mv + ms

v vectors scalar

m0 mass arising from ‘very strong’ part of the interactionms in a given U spin ms is same for all membersmv is proportional to U3

Y = U3 + ½ Q

U3

I3

-

For (1/2)+ baryon octet:

(1/2)mn + 1/2 m = 1/4 m + 3/4 m

accurate to about 1%.

p(uud)n(udd)

+(uus)

0(uds)

-(dds)

+(dss)0(uss)

0(uds)

I+

I+

U-U-

I+|I,I3> = [I(I+1)-I3(I3+1)] |I,I3+1>

U-|U,U3> = [U(U+1)-U3(U3-1)] |U,U3-1>

U=1 triplet is 0 a0+b0 nU3 -1 0 -1

a = ½ b = ½ 3

Mass Difference between the multiplets

21

212)0(

3

8

mm

ssEhfs

21

212121 )(

mm

ssammqqm

ji

ji

ji mm

ssammmqqqm

321321 )(

Mass differences between multiplets spin-spin interaction

(0) is the value of the wavefunction (r1,r2) at zero separation

Example:

24

33

uuN m

amm

Current and constituent masses of u, d and s quarks.

Quark model predictions for masses of (3/2)+ baryons

Isgur-Karl model

i ji

ijhyp

ij

i

ii HV

m

pmH )()

2(

2

ijsijqqqij rbrCV 3/2

ji

ij

ijjiji

ijijji

ji

sijhyp SS

r

rSrS

rrSS

mmH

233 ))((31

)(3

8

3

2

Spin independent potential

Vij is written as harmonic-oscillator potential Kr2ij/2 +

anharmonicity

Anharmonicity :

anharmonic perturbation is assumed to be a sum of two body forces U = i<j Uij

it is flavor independent and spin-scalar and symmetric.

the anharmonicity is treated as a diagonal perturbation on the energies on the states and so it is not allowed to cause mixing between the N=0 and N=2 band states. It causes splitting between the N=1 band states only when the quark masses are unequal.

The anharmonic and hyperfine perturbations applied to the positive-parity excited non strange baryons. Isgur-Karl model shown as bars, the range ofcentral values of the masses quoted by PDG .

* or ** state

*** state

**** state

PDG

Criticism of the non-relativistic quark model

• This model is non-relativistic.

•The quarks used are the constituent quarks which have masses of several 100MeV and are extended objects.

• At higher energies the full QCD structure of the nucleon become noticeable and the model cease to be applicable.

• In the form of the hyperfine interaction spin-orbit interaction should have been included.

Relativized quark Model

i

ii VmpH 22

)()(0/

lim TpsocmsohypCoulstringmp

VVVVVVii

Hamiltonian :

Where V is relative-position and momentum dependent potential

Vstring -> string potentialVcoul -> color-Coulomb potentialVhyp -> hyperfine potentialVso(cm) ->spin-orbit potential for one-gluon exchangeVso(Tp) -> spin-orbit potential -> Thomas precession

Thomas precession is the correction to the spin-orbit interaction

cont

ijij

cont

ji

jirij

ji

jiijs

ji

jiijcont EE

mme

mm

SSr

EE

mmV

2

1

2/3

32

1

22

3

2

3

8

where cont is a constant parameter s(rij) is a running-coupling constant

Extra terms included: 1) the inter-quark coordinate rij is smeared out suggested by relativistic kinematics 2) the momentum dependence away from the p/m -> 0 limit is parametrized by introducing factors which replace the quark masses mi in the nonrelativistic model by roughly Ei.

smearing function

Mass predictions and N Decay amplitudes for nucleon resonances

Conclusion :-

• Quark model assumes the baryon consists of 3 constituent quarks.

• Quark model successfully predicts a decuplet of (3/2)+ and octet of (1/2)+ baryons

• It predict the masses of (3/2)+ baryons correctly.