Non-relativistic Quark Model
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Transcript of Non-relativistic Quark Model
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.