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Wolfgang Bentz Department of Physics, School of Science Tokai University 10 1514 15 10 m 10 10 m 10 m 1 fm
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 color charge (quark) in the vacuum:
The physical_ quark with color c has a cloud of gluons (g) and ( qq ) 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 cp23 cR 3 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
[Note: The product of two Lorentz 4-vectors a (a 0 , 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: G p ( Q 2 ) = d 3 r e i q r p ( r ) , where M z 3 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 z is the z-component 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 2 p 2p n 2n 1 G E ( Q ) = G M ( Q ) / = G M ( Q ) / = 2 2 2 , ( = 0 . 8 4 G e V ) ( 1 + Q / ) [Here p = 2.79, n = -1.91 are the proton and neutron 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)
Most recent data at high Q 2 show deviations from the dipole form: Electric form factor of proton decreases faster than the magnetic one!
(from ref. 5) We have performed model calculations of the form factor for a free proton and also for a bound proton:
Electron scattering on proton in the quark model
1 .2 D ip o le p a ra m e triza tio n Electric form factor of D e n sity = 0 .0 fm -3 D e n sity = 0 .1 6 fm -3 proton:
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n 0 .8
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Experiment
r
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f Free proton
m r 0 .4
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F Bound proton
T. Horikawa, W. Bentz, 0 0 .5 1 1 .5 2 Nucl. Phys. A 762 (2005) 102. 2 M o m e n tu m tra n sfe r [G e V ] Concept of scaling: Generally, we say that “scaling” is valid in electron-proton scattering if the ratio (defined earlier) ( Q 22 = q > 0 ) measured cross section F ( Q 2 , x ) ( x = Q 2 / 2 p q ) Mott cross section (q…momentum transfer) is (almost) independent of Q 2 . Scaling can hold only in a certain range of . Physically, it indicates that the electron scatters elastically from some particle, the size of which is small compared to a typical scale / Q , which can be resolved in the scattering process. For example, in elastic e-p scattering, “scaling’’ holds if Q is very small. ( Q c << c / 0. 8 f m 0. 2 5 G e V . ) In this case, the form factor can be approximated by its value at Q=0, and the ratio F( , x) reduces to the kinematical constraint ( x - 1 ) . F( Q 2,x) ( x - 1 ) (Elastic e-p scattering at very low .) x 1 Inelastic electron-proton scattering at high energies： Deep inelastic scattering (DIS) The proton breaks up, and many new hadrons are formed. Hadrons with total momentum p’
Kinematical constraint for inelastic scattering: q 222 ( p q ) p ' > M N 222 M NN q 2 p q > M p Q 2 x < 1 ( Q 22 q 0 ) electron proton 2 p q If the outgoing hadrons are not distinguished in the experiment (that is, if only the final electron is measured), this process is called “inclusive” inelastic scattering. In this case, the ratio measured cross section F ( Q 2 , x ) Mott cross section
is called “structure function’’. [Note: Actually, for the proton there are two structure functions: F ( Q 2 , x ) and F ( Q 2 , x ) ] 1 2 The experimental data clearly show a “scaling” behavior: The structure functions are almost independent of Q 2 over a wide range of , as long as Q 22 > 2 G e V . As function of Q 2 for several x :
As function of for several :
(from ref. 6) This scaling arises from the elastic scattering of the electrons on the quarks (or “partons’’) within the proton Feynman’s “parton model” . Hadrons form by recombination of quarks
One of the quarks (momentum k) scatters elastically with the electron, q k+q and receives a large momentum q. k
electron proton (momentum p) Kinematical constraint imposed by this “quasielastic” process: 22 2 222 ( k q ) k M q M qq q 2 k q M Q 2 1 ( Q 22 q 0 ) ( M = quark mass) 2 k q q If z<1 denotes the momentum fraction of a quark in the proton, then the quark momentum is k=pz , and Q 22 Q x = = z = z 2 p q 2 k q The momentum fraction of the quark interacting with the electron is equal to the Bjorken variable !!
By studying the deep inelastic scattering, we can obtain information on the momentum distribution of quarks inside the proton !!
If the quark is a point particle, its form factor is a constant equal 2 to its charge: G qq ( Q ) = e . Then, in the parton model, we obtain the following expression for the structure function of the proton: 1 F ( Q 22 , xx ) = d z G 2 ( Q ) ( - z ) q ( z ) q q 0 probability that quark point-quark: kinematical 2 (flavor q) has momentum G ( Q ) = e constraint qq fraction z Finally we obtain:
2 2 independent of Q 2 F ( Q , xx ) = e q q ( ) q If the proton consists of 3 quarks (uud), one naively expects that one quark (from ref. 6) carries 1/3 of the total momentum.
We expect that the momentum distributions q(x) [=u(x) or d(x)] have a peak around x =1/3. But there is no peak in the data !?
If we consider the difference between proton and neutron pn structure functions F 22 ( xx ) F ( ) , we see a peak:
(from ref. 1)
Interpretation: There are 2 kinds of quarks in the nucleon: (i) three “valence quarks’’ , which are always present, and (ii) many “sea quarks’’, which are appear and disappear in pairs “at random’’ due to vacuum fluctuations. Each valence quark carries about 1/3 of the total momentum, and the sea quarks carry only very small momenta. 2 F 2 ( x ) = xx e q q ( ) q sea quarks
valence quarks
sea x quarks 1/3 valence (see ref. 4, p. 23) quarks
For the proton the valence quarks are uud, and for the neutron udd. The sea quarks should be about the same in proton and neutron. (Recent results have shown that this is only approximately true!) pn In the difference F 22 ( xx ) F ( ) , only the contribution from valence quarks remains, and a peak is seen at x 1 / 3 . Spin dependence of parton distributions: Experiments using polarized beams have also measured the spin dependence of the DIS cross section, i.e., the difference
q - q
p p electron proton electron proton
Here and denote the spin directions parallel and opposite to the momentum. (Longitudinal polarization.) In the parton model, the quark momentum distributions can be written as q ( xxx ) = q ( ) + q ( ) , where q ( x ) is the probability that a quark, with flavor q and spin parallel to the total proton spin, has a momentum fraction x . From the unpolarized experiments, we obtain only the combination q ( xxx ) = q ( ) + q ( ) , but from the polarized experiments we obtain also
q ( x ) q ( x ) q ( x ) . By integrating this over x , we obtain the contribution of the quark with flavor q to the proton spin: 1 q d xx q ( ) 0 Naively, we expect that the proton spin is 100% due to the spin of the quarks, i.e., the naive expectation is ? u + d + s = 1 . [Note: The strange quark contribution s is only to the sea quarks.] Experiment: Very precise experimental data for polarized and unpolarized structure functions have been obtained at the HERA collider (28 GeV positrons on 820 GeV protons), located at the DESY laboratory in Hamburg, Germany.
DESY facilities with Inside the tunnel of HERA HERA and PETRA (from ref. 7) g (?) Parton distributions unpolarized q ( xxx ) = q ( ) + q ( ) obtained by analysis of unpolarized DIS experiments at Q 22 = 4 GeV . Shown are the valence (v) and sea dv u v (s) quark momentum distributions in the proton, and also the gluon (g) d s momentum distribution as functions us of the momentum fraction x .
g (??) polarized Parton distributions q ( x ) q ( x ) q ( x ) u v obtained by analysis of polarized DIS experiments at . The gluon contribution
us ds is not well known. dv (from ref. 8) There are many surprises, for example: (i) Gluon contributions seem to be large. (ii) The quark spins give the following contributions to the spin of the proton (valence and sea quark contributions are added up): u = 0 .8 2 0 . 0 2 (82% of the proton spin) d = 0 . 4 3 0 . 0 2 (- 43% of the proton spin) s = 0 . 1 0 0 . 0 2 ( - 10% of the proton spin) Total spin sum: u + d + s = 0 . 2 9 0 . 0 6 (only 29% of the proton spin) “Proton spin crisis”: Less than 1/3 of the proton spin is due to the spin of the quarks. The rest must come from the orbital angular momentum of the quarks, or from the gluons, or both. These facts are difficult to understand in simple quark models, where there are no gluons, and the proton spin is almost entirely due to the spin of the quarks. “Proton spin crisis”: What is the origin of the proton spin?
(from ref. 4, p. 33)
In order to solve this puzzle: Experiments have now started at RHIC (Relativistic Heavy Ion Collider at Brookhaven, near New York, U.S.A.) By deep inelastic polarized proton-proton scattering experiments (E=250 GeV) one can measure the function g ( x ) , that is, one can determine the contribution of the gluons to the spin of the proton. RHIC (Relativistic Heavy Ion Collider) at Brookhaven, U.S.A
(from ref. 9) Polarized p-p collision experiments will be performed in the RHIC-Spin experiment
“Gluon fusion processes’’ will give information on the proton contribution of gluons to the spin of the proton.
(from ref. 10) proton As a last topic of this lecture: Deep inelastic scattering of electrons from NUCLEI:
Many hadrons (not observed) are formed after the scattering
q (unpolarized scattering)
p electron nucleus, for example calcium Using again the parton model, one can extract the momentum distributions of quarks inside a BOUND proton. By comparison with the case of a single proton, one can see whether a proton bound in the nucleus is different from a free proton or not. Results of experiments at CERN (Swiss, Europe) and SLAC (Stanford, U.S.A.): The figure below shows the ratio (momentum distribution of quarks in a bound proton) (momentum distribution of quarks in a free proton)
1) For intermediate momentum fractions in the region x 0.5 - 0.8 the ratio is smaller than 1. 2) For small momentum fractions in the region x 0.1 - 0.2 the ratio is a bit larger than 1. This result is called the “EMC effect’’. (EMC means: European Muon Collaboration)
Calculation: I. Cloet, W. Bentz, A.W. Thomas, Phys. Rev. Lett. 95 (2005) 052302 The EMC effect shows that the average momentum of the quarks in a bound nucleon is smaller than in a free nucleon.
Physical interpretation:
The nucleon bound in the nucleus is somewhat larger than a free nucleon. (“Swelling’’ of the nucleon.) Therefore, the quarks in the bound nucleon are confined to a larger region of space. Due to the uncertainty principle, their average momentum becomes smaller.
This EMC effect is very important, because it provides a connection between nuclear physics and elementary particle (quark) physics.
The study of the properties of a nucleon bound in the nucleus (“medium modification of nucleon properties’’) is a very important and active field of recent research. Study the spin-dependence of electron-nucleus scattering “What is the contribution of quark spin to the spin of the nucleus?”
Spin of Spin of electron electrons Spin of nucleus Spin of nucleus There are no experimental data yet! We have made theoretical predictions for the “polarized EMC ratio”: (momentum distribution of quarks in a bound proton) (momentum distribution of quarks in a free proton) where the quark spin is parallel to the proton spin.
Calculation: I. Cloet, W. Bentz, A.W. Thomas, Phys. Rev. Lett. 95 (2005) 052302 This theoretical predictions shows: (quark spin in a bound proton) = 0.8 (quark spin in a free proton) In the nucleus, the quarks carry smaller spin and larger orbital angular momentum! This prediction will be confirmed at ongoing experiments at Jefferson Laboratories (JLab, US).
Spin of the quarks depends on the environment! (Figure from JLab’s home page) Summary of this lecture
Quarks are the building blocks of matter. From accelerator experiments, we know many things about quarks in the nucleon and in the nucleus: Charge distributions, momentum distributions, spin distributions, etc.
By comparison with theoretical calculations, we can study the interactions between quarks and the interactions between nucleons. References: 1) F. Halzen, A.D. Martin: Quarks and Leptons (Wiley, 1984) 2) L. Schiff: Quantum Mechanics (McGraw-Hill, 1968) 3) A.W. Thomas, W. Weise: The Structure of the Nucleon (Wiley, 2001) 4) Committee on Nuclear Physics, National Research Council: Nuclear Physics (National Academy Press, 1999). http://www.nas.edu/books/030962764/html/index.html 5) URL of the Jefferson laboratory: http://www.jlab.org 6) R.K. Ellis, W.J. Stirling, B.R. Webber: QCD and Collider Physics (Cambridge, 1996) 7) URL of DESY laboratories: http://www.desy.de 8) M. Glueck, E. Reya, A. Vogt: Eur. Phys. J. C5 (1998) 461. 9) URL of the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory (BNL): http://www.bnl.gov/rhic 10) URL of the RIKEN-BNL Research Center: http://www.rarf.riken.go.jp/rarf/riken 11) D.F. Geesaman, K. Saito, A.W. Thomas, Annual Review of Nuclear and Particle Science Vol. 45 (1995), p. 337.