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Lecture 3: the Structure of the Nucleon

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Lecture 3: the Structure of the Nucleon Ｒｅｃｅｎｔ Ｔｏｐｉｃｓ ｉｎ Ｑｕａｒｋ Ｐｈｙｓｉｃｓ Wolfgang Bentz Department of Physics, School of Science Tokai University 10 15 14 15 10 m 10 10 m 10 m 1 fm electron proton nucleus neutron proton neutron u-quark d-quark atomic nucleon thing atom nucleus almost empty, densely packed, densely packed, electromagnetic strong strong interaction interaction interaction Some historical remarks: In the early days of nuclear physics, protons and neutrons were considered as elementary particles. However, (i) In the 1930’s (O. Stern): Magnetic moment of proton is not the same (in magnitude) as the one of the electron, but almost three times larger Proton is not a Dirac particle. (ii) In the 1950’s (R. Hofstadter): Cross section for electron-proton scattering differs from that of point particles Proton has a size! (iii) In the 1960’s (M. Gell-Mann, G. Zweig): Like in a puzzle game, strongly interacting particles (“hadrons’’), like proton and neutron, can be made up of elementary fermions (“quarks’’: u,d,s) and their antiparticles (“antiquarks’’). How does this “puzzle’’ work? a) Consider hadrons with integer spin (mesons): There are 9 mesons with spin zero, negative parity, and mass below 1 GeV: - ( , 0 , ) , (KKKK , 00 , , ) , , ' Can be- explained by assuming quark-antiquark bound states: ( qq 1 , 2 ) with q i u , d , s . 9 different combinations. b) Consider hadrons with half integer spin (baryons): There are 10 baryons with spin 3/2, positive parity, and mass below 2 GeV: ( ++ , + , 0 , ) , ( +* , 0* , * ) , ( 0* , * ) , . Can be explained by assuming 3-quark bound states: ( q 1 , q 2 , q 3 ) with 10 different combinations [Note: For the baryons with spin 1/2 and positive parity, there are only 8 members: (p, n) , ( +0 , , ) , 0 , ( 0 , ) . This is because (uuu), (ddd) and (sss) cannot form spin 1/2, and there are 2 independent (uds) states with spin 1/2.] iv) In the 1970’s (Stanford Linear Accelerator – SLAC): In deep inelastic electron-proton scattering, the phenomenon of “Bjorken scaling’’ was discovered, which confirmed the quark structure of the proton. v) Nowadays we know that there are 6 types (flavors) of quarks. For nuclear physics, the most important ones are u, d, s. We also know that quarks have an additional quantum number, called color (Greenberg, Han and Nambu): Without this ++ additional quantum number, the spin 3/2 state ( u u u ) would be totally symmetric, in contradiction to the Pauli principle. Hadronic states must be totally antisymmetric in color (“white’’). A single quark has never be observed in isolation Color confinement was postulated: A single quark can never be isolated from a_ physical hadron. For the case of a meson ( qq bound state), this means that the potential energy between the quark and antiquark increases with increasing distance： V (r) a s r (Confinement) Moreover, the analysis of high energy processes (which probe short distances (r) between the quarks) shows that for small distances perturbation theory is applicable V(r) becomes weaker at small r (Asymptotic freedom) V(r) confinement r r q - Coulomb-like attraction ( / r ( < 1 ) ) at small r Qualitative picture of confinement and asymptotic freedom in terms of “screening’’: a) Screening of an electric charge (electron) in the vacuum: Because of self interactions, a physical electron has a cloud of virtual photons ( ) and electron-positron pairs ( ee ) around it. The has no charge. Due to the Coulomb force, the virtual positrons tend to surround the electron, while the virtual electrons tend to spread in space “Screening’’. virtual pair + virtual + + + + e + + test charge + A test charge will see a smaller electron charge at large distances than at small distances. The test charge feels a weaker interaction at larger r Interaction becomes weaker as r increases. _ b) Antiscreening of a colorqq charge (quark) in the vacuum: The physical quark with color c has a cloud of gluons (g) and ( ) pairs around it. But the gluons have color! The theory shows that the gluons carry away color charges other than the quark color charge c “Antiscreening’’. virtual pair virtual g gluon self interaction but also: (see ref. 1) r r r test color charge r r quark A test color charge will see a larger r r r quark color charge at large distances than at small distances. The test charge feels a stronger interaction at larger r Interaction becomes stronger as r increases. The most important difference between quantum electrodynamics (QED) and quantum chromodynamics (QCD) is that the “gauge particle’’ in QED has no electric charge, but in QCD it has does have color charge. We say: QED is an Abelian gauge theory, and QCD is a non-Abelian gauge theory. This difference leads to color confinement and asymptotic freedom in QCD. Asymptotic freedom can be derived rigorously by using the “renormalization group equations’’ of QCD. However, an exact proof of confinement is still missing. Numerical supports for confinement come from Lattice Gauge calculations. A schematic model of the nucleon, which takes into account confinement and asymptotic freedom, is the bag model: V(r) V(r)=0 V(r)= RR (square well potential) r R pressure B 3 massless quarks move freely inside a cavity (“bag’’) of radius R. The pressure B (from outside) is introduced in order to stabilize the Fermi pressure of the 3 quarks. The bag radius R is determined by the condition MR N / = 0 , where M N is the nucleon mass. The nucleon mass in the bag model is: 4 M c23 3 p c + R B Nq 3 Here p q is the momentum of a quark inside the bag, which is determined from the boundary condition of the wave function as Elastic electron-proton (e-p) scattering p+q Kinematical constraint for elastic scattering: 22 2 q ( p q ) p M N 222 M NN q 2 p q M Q 2 p x 1 ( Q 22 q 0 ) electron proton 2 p q 0 [Note: The product of two Lorentz 4-vectors a (a , a ) 00 and b (b 0 , b ) is defined as a b a b a b . q is the momentum transfer, and for electron scattering we have q 2 ( q 0 ) 2 ( q ) 2 0 . To make the formulae simpler, we will use “natural units” from here: = c = 1 ] The variable x defined above is called the “Bjorken variable’’. For elastic scattering, x = 1. In the experiments, the (differential) “cross section’’ is measured. (For the definition of cross section, see any text book on mechanics or quantum mechanics, for example ref. 2.) From the data, one can extract the “form factor’’ of the proton p 2 from the ratio of the measured cross section to the G E ( Q ) cross section for a point-like proton (“Mott cross section’’): measured cross section F ( Q 2 , x ) ( x 1 ) . Mott cross section Effect of proton size kinematical constraint Note: Actually, here we refer to the “electric form factor’’ . For a spin 1/2 particle like the proton, there is also a second p 2 form factor – the “magnetic form factor’’ G M ( Q ). What is the physical meaning of ? It is the Fourier transform of the proton’s charge density : = d 3 r e i q r p ( r ) Note that for a point-particle the charge density is a delta – function, and the form factor is a constant : p o i n t p o i n t 2 ( r ) = ( r ) G E ( Q ) = 1 But the proton consists of three quarks, and therefore it has a finite 3 pi i extension: ( r ) = p | e ( r r i ) | p , where e is the i=1 charge of the i-th quark. Similarly, the magnetic form factor of the proton is the Fourier transform of the magnetic moment density inside the proton: = p ( r ) , where z G p ( Q 2 ) 3 M i p i iie z ( r ) = p | m zi ( r r ) | p , where m zz = 2 s i=1 2 m is the magnetic moment of the i-th quark. i s 3 i q r [ z is the z-component d r e of the spin operator of the i-th quark.] The next slide shows the experimental data. electric proton magnetic proton electric neutron magnetic neutron (from ref. 3) These experimental data show that the proton form factors and the neutron magnetic form factor have a “dipole form:” p 2p n 2n 1 = G M ( Q ) / = G M ( Q ) / = 2 2 2 , ( = 0 . 8 4 G e V ) ( 1 + Q / ) p n [Herep 2 = 2.79, = -1.91 are the proton and neutron G E ( Q ) magnetic moments.] This corresponds to an exponential form of the proton’s electric charge and magnetization density, as e x p ( - r ) . The experimental data give the following charge distributions (multiplied by r 2 ) inside proton and neutron: neutron proton (from ref. 4, p. 34) Blue: possible values Yellow: probable values + charge - charge in center outside The “root-mean-square” radii r { r 2 } 1/2 { [ ( r 2 d r ) r 2 ( r )] / [ ( r 2 d r ) ( r )] } 1/2 00 for the proton electric and magnetic distributions are both equal to 0.86 fm . The study of nucleon form factors at high Q 2 is a very active field of research both experimentally and theoretically. The most precise data are now taken at the Jefferson Laboratory, Virginia, U.S. Continuous Electron Beam Accelerator Facility (CEBAF) at the Jefferson Laboratory (Jlab). (from ref. 5) Parts of the linear electron accelerator (energy 5 GeV) at Jlab: (from ref. 5) Q 2 Most recent data at high show deviations from the dipole form: Electric form factor of proton decreases faster than the magnetic one! (from ref.
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