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Non-relativistic Quark Model. Sugat V Shende KVI. 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
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Non-relativistic Quark Model Sugat V Shende KVI 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 constituent quarks with effective quark masses and a spatial configuration.
Y 1/3 Y=B+S 1 I3 -1/2 1/2 -2/3 u d 1/3 1/2 I3 s Mesons in quark Model The fundamental quark triplet : q qq combination gives : K0 K+ singlet {1} and octet {8} states JP = 0- q are spin ½ fermions, qq state total spin S = 0 or 1 8 0 1 - + J = L + S Parity = (-1)L+1 (-1)L arises from orbital motion 1 opposite intrinsic parities of q and q JP = 0-, 1- , 2+ K - K0
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 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[( - )]
In SU(3) In order to predict the nature of baryon multiplets, we should combine SU(3) flavour multiplets with the spin multiplets Symmetric combinations notation is (nSU(3),nSU(2)) where n is dimensionality
Y udd uud ddd udd uud uuu 1 uds Y dds uus dds uus uds 0 1 uds dss uss dss uss -1 0 sss -2 -1 I3 -1 -1/2 0 1/2 1 3/2 -3/2 -2 JP= (3/2)+ JP= (1/2)+ I3 -1 -1/2 0 1/2 1 3/2 -3/2 Baryons in quark Model Quark model successfully predicts a decuplet of (3/2)+ and octet of (1/2)+ baryons
The baryon mass relation : Y = U3 + ½ Q The mass of a particular U spin state |U,U3> is v vector s scalar <U,U3|H|U,U3> = <U,U3| H0 + Hv + Hs |U,U3> = m0 + mv + ms m0 mass arising from ‘very strong’ part of the interaction ms in a given U spin ms is same for all members mv is proportional to U3 I3 m - m m -m m - m 150 MeV U3 -
For (1/2)+ baryon octet: U=1 triplet is 0 a0+b0 n U3 -1 0 -1 I+ p(uud) n(udd) U- 0(uds) U- I+ +(uus) I+|I,I3> = [I(I+1)-I3(I3+1)] |I,I3+1> -(dds) 0(uds) U-|U,U3> = [U(U+1)-U3(U3-1)] |U,U3-1> 0(uss) +(dss) a = ½ b = ½3 (1/2)mn + 1/2 m = 1/4 m + 3/4 m accurate to about 1%.
Mass Difference between the multiplets Mass differences between multiplets spin-spin interaction (0) is the value of the wavefunction (r1,r2) at zero separation Example:
Quark model predictions for masses of (3/2)+ baryons Current and constituent masses of u, d and s quarks.
Isgur-Karl model 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 of central values of the masses quoted by PDG . PDG * or ** state *** state **** state
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 Hamiltonian : Where V is relative-position and momentum dependent potential Vstring -> string potential Vcoul -> color-Coulomb potential Vhyp -> hyperfine potential Vso(cm) ->spin-orbit potential for one-gluon exchange Vso(Tp) -> spin-orbit potential -> Thomas precession Thomas precession is the correction to the spin-orbit interaction
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 where cont is a constant parameter s(rij) is a running-coupling constant
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.
