Chapter 5 Quark Model
Chapter 5 5.1 Introduction 5.2 Quark Model 5.3 Meson and Baryon wave function 5.4 Magnetic moment and masses of baryons 5.5 Interactive Exercise
Non Relativistic Quark model Quark Model was proposed by Gell-Mann and Zweig in 1964. Quarks are strongly interacting particles. Initially there were three flavors of quarks, up, down and strange. Quarks are absolutely confined within baryons and mesons, this phenomenon is called as Quark Confinement. Quarks are spin half particles, means they are fermions. Strong interactions between quarks is flavor independent. Hadrons always participate in strong interaction. Hadron family contains both fermions and bosons. Baryons which have half integral spin are fermions. Mesons with integral spin are bosons. Later three more flavors of quarks were discovered, charm, beauty and truth. All quarks are assigned baryon number of 1/3. Up, charm and top quark have an electric charge of +2/3. Electric charge of down,strange and bottom quark is -1/3. Strong forces on u & d quarks are same. Electromagnetic forces are different because quarks have different electric charges.
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Introduction Eightfold Way Murray Gell-Mann introduced the Eightfold Way in 1961. Eightfold Way arranged the baryons and mesons into weird geometrical patterns, according to their charge and strangeness. The eight lightest baryons fit into hexagonal array with two particles at the center. This group is called baryon octet. n
S = -1
S = -2
Fig 1: baryon octet
Q = -1
Particles of like charge lie along the downward sloping diagonal lines Q = +1 for proton and Σ+ (sigma plus). Q = 0 for the neutron, the Λ (lambda), the Σ0 (sigma zero) and the Ξ0 (cascade zero). Q = -1 for Σ(sigma minus) and Ξ- (cascade minus). Horizontal lines associate particles of like strangeness: S = 0 for proton and neutron. S = -1 for sigma’s and lambda. S = -2 for two Ξ’s. Dayalbagh Educational Institute
The eight lightest mesons fill a similar hexagonal pattern, forming the pseudoscalar meson nonet. K0
Fig 2: Pseudoscalar mesons
Q=0 Q = -1
Once again, diagonal lines determine charge and horizontal lines determine strangeness but this time top line has strangeness, S = 1, middle line S = 0 and bottom line S = -1, this is again a historical accident. Gell-Mann could have designed S = 1 to proton and neutron. S = 0 to the Σ’s and Λ. S = -1 to Ξ’s. In 1953 there were no reasons to assign proton and neutron strangeness 1. Thus they were assigned strangeness 0. In 1961 a new term hypercharge was introduced which is represented by Y and Y = B + S, where B is baryon number and S is strangeness. Baryon number for meson is zero and for baryon is 1, thus Y = S for mesons and Y = S + 1 for baryons. Dayalbagh Educational Institute
Hexagons were not the only allowed figure, there was a triangular array, incorporating 10 heavier baryons which is called baryon decuplet. Δ0
S=0 Q=2 Σ*-
S = -1
S = -2
S = -3
Fig 3: Baryon Decuplet
Q = -1 As Gell-Mann was fitting these particles into decuplet, an absolutely lovely thing happened. Nine of the particles were known experimentally, but at that time the tenth particle at the very bottom of triangle with a charge of -1 and strangeness -3 was missing. No particle with these properties had been detected in the laboratory. Dayalbagh Educational Institute
Gell-Mann boldly predicted that such particle would be found and told the experimentalists exactly how to produce it. He calculated its mass and its lifetime and in 1964 the omega minus particle was discovered. Since the discovery of omega minus (立-) the Eightfold Way was not taken seriously. Over next ten years, every new hadron found a place in one of the eightfold way supermultiplets. In addition to baryon octet and decuplet, there exist antibaryon octet or decuplet with opposite charge and opposite strangeness. Eightfold way did more than merely classifying the hadrons and its real importance lies in the organizational structures it provided.
Non Relativistic Quark Model A non relativistic quark model is proposed for baryons, according to which any two quarks are assumed to interact with each other through primary-wave forces. Such forces are capable of producing strong binding in a three-quark system in a spatially antisymmetric state of angular-momentum. Thus it makes the model compatible with an extension of the representation of SU6. If the strength of the quark-quark force is adjusted to fit some central baryon mass (m0), the model predicts a 2-quark bound state at a mass ~1/2(M+m 0), where M is the central mass of a quark. Dayalbagh Educational Institute
The validity of the non relativistic description is shown to depend on the smallness of a certain "inverse range parameter" Î˛ compared with the quark mass M, and this condition is fully compatible with the present experimental knowledge of baryon sizes, as measured by the charge radius of the proton. Further, using an SU(3) invariant interaction, an "equal interval rule" for the baryon masses follows dynamically from the assumption of a mass difference between the singlet and doublet quarks, under the same condition, Î˛<<M, as above. It is argued that a p-wave quark interaction, which leads to the formation of antisymmetric spatial states more easily than formation of symmetric states, gives a "saturated system" at the 3-quark level. This reduces considerably the (undesirable) prospects of very strong binding of a larger number of quarks, compared to the situation with s-wave forces (which facilitate the formation of symmetric states in multiquark systems with stronger binding as the number of quarks is increased). By ruling out the generally stronger s-wave forces as the main bond between two quarks, the model leaves scope for their action in quark anti-quark systems, which should require stronger binding in order to generate the (less massive) mesons.
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1.3 Chiral quark model Chiral quark model was originated by Weinberg and deeloped by Manohar and Georgi  to improve Non Relativistic Quark Model. Quantum Chromodynamics is the theory of strong interaction in which quarks and gluons interacts with a strength αs called coupling constant which varies with energy. QCD (Quantum Chromodynamics) breaks at low energy. Non Relativistic quark model is used to describe properties of hadrons in low energy region. Non Relativistic quark model gives good results for masses of hadrons, but moderate results for magnetic moments and fails to describe spin structure of baryons. Chiral Quark Model describes properties of hadron at low energies and incorporates spontaneous chiral symmetry breaking theory. χQM (Chiral Quark Model) gives an idea that there is set of internal bosons, which couple to quarks in interior of hadrons particularly the baryons but not at so small diatance that QCD is applicable. QCD has features of confinement and chiral symmetry breaking. Confinement means coupling constant, αs increases at low momentum transfer and long distance and chiral symmetry breaking means if mass of quark tends to zero then the QCD Lagrangian is invariant under the independent SU(3) transformations of the left-handed and right-handed light quark fields. Dayalbagh Educational Institute
Chiral symmetry is observed in two modes, Wigner mode and Goldstone mode. One of the basic ideas of the Ď‡QM is that the non-perturbative QCD phenomenon of chiral symmetry breaking will take place at an energy scale much higher than that of QCD confinement.
Confining and Covariant Diquark-Quark Model Baryons are treated as bound states of scalar or axialvector diquarks and a constituent quark which interacts through quark exchange. Fully four-dimensional wave functions is obtained for both octet and decuplet baryons as solutions of the corresponding Bethe-Salpeter equation. Applications currently under investigation electromagnetic and strong form factors strangeness production processes.
Constituent Quark Model Static properties of hadrons have been introduced in various models. The static properties of hadrons (charge radius, magnetic moment, etc.) are useful for understanding the quark structure of hadron. In this model the hypercentral constituent quark and isospin dependent potentials were introduced. Here constituent quarks interact with each other via a potential in which the three body force effect and standard two-body potential contributions was taken into account. Dayalbagh Educational Institute
The main focus of this model was the interquark potential, which contains a spin-independent and spin dependent terms characterized by the presence of a long range part giving rise to confinement. The standard hyperfine interaction is used in order to reproduce the splitting within the SU (6) multiplets. To introduce the isospin nonconfining potential we have the chiral Constituent Quark Model (CQM). The nonconfining part of the potential is provided by the interaction with the Goldstone bosons, giving rise to a spin- and isospin dependent part. An isospin dependence of the quark potential can be obtained by means of quark exchange.
Relativistic quark model The charge radius 〈r2〉 and magnetic moment 〈μ〉 of baryons have been formulated as expectation values with respect to Salpeter amplitudes. Resulting operators allow for a sensible physical interpretation and generalize the well known nonrelativistic expressions. The method is applicable under the assumption of free quark propagators and instantaneous interaction kernels. The empirical radii and magnetic moments is satisfactorily described. There are many allowed quasi-two-body decaying cascades. Decay widths are not correctly reproduced quantitatively. Dayalbagh Educational Institute
Introduction- Wave Function