Ｒｅｃｅｎｔ Ｔｏｐｉｃｓ ｉｎ Ｑｕａｒｋ Ｐｈｙｓｉｃｓ

Wolfgang Bentz

Department of Physics, School of Science

Tokai University

proton

neutron

nucleus

electron

proton

neutron u-quark

1010 m

d-quark

15 1410 10 m 1510 m

thing atom atomic

nucleus nucleon

almost empty, densely packed, densely packed,

electromagnetic

interaction

strong

interaction

strong

interaction

1 fm

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( , , ) , 0 0( , , , ) , K K K K , '

-

Can be explained by assuming quark-antiquark bound states:

1 , 2( ) q q- with i u , d , s .q

b) Consider hadrons with half integer spin (baryons):

There are 10 baryons with spin 3/2, positive parity, and mass

below 2 GeV:

9 different combinations.

++ + 0( , , , ) , +* 0* *( , , ) ,

0* *( , ) , .

Can be explained by assuming 3-quark bound states:

1 2 3( q , q , q ) i u , d , s .q with 10 different combinations

[Note: For the baryons with spin 1/2 and positive parity, there

are only 8 members: + 0( , , ) , 0( , ) . (p, n) ,

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

would be totally symmetric, in contradiction to the Pauli principle.

Hadronic states must be totally antisymmetric in color (“white’’).

++ ( u u u )

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 ( bound state), this means that the

potential energy between the quark and antiquark increases with

increasing distance：

_

q q

r

q - q

V (r) a s r

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)

r

V(r)

confinement

Coulomb-like attraction

( ) at small r / r ( < 1 )

(Confinement)

Because of self interactions, a physical electron has a cloud

of virtual photons ( ) and electron-positron pairs ( ) 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’’.

Qualitative picture of confinement and asymptotic freedom in terms

of “screening’’:

a) Screening of an electric charge (electron) in the vacuum:

e e

virtual virtual pair

+ +

+ + +

+ +

+ 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

( ) 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’’.

_

q q

virtual g virtual pair gluon self interaction

but also:

(see ref. 1)

r r

r r

r r

r

r

quark

test color charge

A test color charge will see a larger

quark color charge at large distances

than at small distances.

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.

The test charge feels a stronger interaction at larger r

Interaction becomes stronger as r increases.

A schematic model of the nucleon, which takes into account

confinement and asymptotic freedom, is the bag model:

r

R

V(r)

V(r)=0 V(r)= 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 , where is the nucleon

mass.

/ = 0NM R NM

(square well potential)

2 3

4 3 +

3N qM c p c R B

The nucleon mass in the bag model is:

Here is the momentum of a quark inside the bag, which is

determined from the boundary condition of the wave function as

qp

Elastic electron-proton (e-p) scattering

q

p

p+q

electron proton

Kinematical constraint for elastic scattering:

22 2

N( p q ) p M 2 22

N NM q 2 p q M 2

2 2Q 1 ( Q q 0 )

2 p qx

[Note: The product of two Lorentz 4-vectors 0a (a , a )

0b (b , b )

and is defined as 0 0a b a b a b

q is the momentum transfer, and for electron scattering we have

.

2 0 2 2q ( q ) ( q ) 0 .

The variable x defined above is called the “Bjorken variable’’.

For elastic scattering, x = 1.

To make the formulae simpler, we will use “natural units” from here: ] = = 1c

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

EG ( Q ) from the ratio of the measured cross section to the

cross section for a point-like proton (“Mott cross section’’):

2F ( Q , ) x measured cross section

Mott cross section

p 2

EG ( Q ) ( 1 ) .x

p 2

EG ( Q )What is the physical meaning of ?

It is the Fourier transform of the proton’s charge density :

p 2

EG ( Q )p ( r )

3 i q r d r e

=

p ( r )

Note: Actually, here we refer to the “electric form factor’’ . p 2

EG ( Q )For a spin 1/2 particle like the proton, there is also a second

form factor – the “magnetic form factor’’ . p 2

MG ( Q )

Effect of proton size kinematical constraint

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 2

EG ( Q ) = 1p o i n t ( r ) = ( r )

But the proton consists of three quarks, and therefore it has a finite

3i

z i

i=1

= p | m ( r r ) | p ,

extension: where is the ie

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 2

MG ( Q )3 i q r d r e

p

z ( r ) ,

=

p

z ( r )

3p i

i

i=1

( r ) = p | e ( r r ) | p ,

where

where i

i i

z z

e m = 2 s

2 m

is the magnetic moment of the i-th quark. i

zs [ is the z-component of the spin operator of the i-th quark.]

The next slide shows the experimental data.

electric proton

electric neutron

magnetic proton

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

EG ( Q )p 2 p

MG ( Q ) / n 2 n

MG ( Q ) / 2 2 2

1 ,

( 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 ) inside proton and neutron:

neutron proton

2r

+ charge

in center - charge

outside

Blue: possible values

Yellow: probable values

( = 0 . 8 4 G e V )

(from ref. 4, p. 34)

The study of nucleon form factors at high is a very active field

of research both experimentally and theoretically.

2Q

The “root-mean-square” radii

2 1/2 2 2 2 1/2

0 0

r { r } { [ ( r d r ) r ( r )] / [ ( r d r ) ( r )] }

for the proton electric and magnetic distributions are both equal

to 0.86 fm .

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:

Most recent data at high show deviations

from the dipole form: Electric form factor of

proton decreases faster than the magnetic one!

2Q

(from ref. 5)

(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

Electric form factor of

proton:

Experiment

Free proton

Bound proton

T. Horikawa, W. Bentz,

Nucl. Phys. A 762 (2005) 102.

M o m e n tu m tra n sfe r [G e V ]2

Fo

rm f

acto

r (P

roto

n)

D ip o le p a ra m e tr iza t io n D e n sity = 0 .0 fm D e n sity = 0 .1 6 fm

-3 -3

0 0 .5 1 1 .5 2

0 .4

0 .8

1 .2

Concept of scaling:

2F ( Q , ) x measured cross section

Mott cross section

Generally, we say that “scaling” is valid in electron-proton

scattering if the ratio (defined earlier)

is (almost) independent of . 2Q

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 ,

which can be resolved in the scattering process.

2Q

/ Q

For example, in elastic e-p scattering, “scaling’’ holds if Q is very

small. ( ) 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 .

Q c << c / 0. 8 f m 0. 2 5 G e V . 2Q

( - 1 )x

(q…momentum transfer)

2 2

2

( Q = q > 0 )

( = Q / 2 p q )x

2Q F( ,x)

x 1

( x - 1 )(Elastic e-p scattering at very

low .) 2Q

Inelastic electron-proton scattering at high energies：

Deep inelastic scattering (DIS)

p’

q

p

electron proton

Hadrons with total

momentum

Kinematical constraint for inelastic scattering:

22 2

N( p q ) p ' > M 2 22

N NM q 2 p q > M 2

2 2Q < 1 ( Q q 0 )

2 p qx

The proton breaks up, and many new hadrons are formed.

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.

2F ( Q , ) x measured cross section

Mott cross section

In this case, the ratio

is called “structure function’’. [Note: Actually, for the proton

there are two structure functions: 2

1F ( Q , x ) 2

2F ( Q , x )

2Q The experimental data clearly show a “scaling” behavior:

The structure functions are almost independent of

over a wide range of , as long as . 2Q 2 2Q > 2 G e V

and ]

As function of for several : 2Q

As function of for several : 2Q

(from ref. 6)

x

x

This scaling arises from the elastic scattering of the electrons

on the quarks (or “partons’’) within the proton

Feynman’s “parton model” .

electron proton (momentum p)

k

k+q

Hadrons form by

recombination of quarks

q

One of the quarks (momentum k)

scatters elastically with the electron,

and receives a large momentum q.

Kinematical constraint imposed by this “quasielastic” process: 22 2

q( k q ) k M 2 22

q qM q 2 k q M 2

2 2Q 1 ( Q q 0 )2 k q

qM ( = quark mass)

If z<1 denotes the momentum fraction of a quark in the proton, then

the quark momentum is k=pz , and 2 2Q Q

= = z = z 2 p q 2 k q

x

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

to its charge: . Then, in the parton model, we obtain

the following expression for the structure function of the proton:

2

q qG ( Q ) = e

122 2

q

q 0

F ( Q , ) = d z G ( Q ) ( - z ) q ( z ) x x

point-quark:

2

q qG ( Q ) = e kinematical

constraint

probability that quark

(flavor q) has momentum

fraction z

22

q

q

F ( Q , ) = e q ( )x x

Finally we obtain:

independent of 2Q

If the proton consists of 3 quarks (uud),

one naively expects that one quark

carries 1/3 of the total momentum.

We expect that the momentum

distributions q(x) [=u(x) or d(x)]

have a peak around =1/3.

But there is no peak in the data !?

x

(from ref. 6)

If we consider the difference between proton and neutron

structure functions we see a peak: p n

2 2F ( ) F ( ) , x x

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.

(from ref. 1)

2F ( )x

x1/3

sea quarks

valence quarks

valence

quarks

sea

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!)

p n

2 2F ( ) F ( ) , x xIn the difference only the contribution

from valence quarks remains, and a peak is seen at 1 / 3 .x

(see ref. 4, p. 23)

2

q

q

= e q ( )x x

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

p electron proton

q

p

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 ( ) = q ( ) + q ( ) ,x x x

where is the probability that a quark, with flavor q and spin

parallel to the total proton spin, has a momentum fraction .

q ( ) x

x

From the unpolarized experiments, we obtain only the combination

q ( ) = q ( ) + q ( ) ,x x x

but from the polarized experiments we obtain

also

q ( ) q ( ) q ( ) .x x x

xBy integrating this over , we obtain the contribution of the

quark with flavor q to the proton spin: 1

0

q d q ( ) x x

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 is only to the

sea quarks.]

s

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

HERA and PETRA Inside the tunnel of HERA

(from ref. 7)

Parton distributions

q ( ) = q ( ) + q ( ) x x x

obtained by analysis of unpolarized

DIS experiments at .

Shown are the valence (v) and sea

(s) quark momentum distributions

in the proton, and also the gluon (g)

momentum distribution as functions

of the momentum fraction .

2 2Q = 4 GeV

x

vuvd

susd

g

vu

vdsu sd

g

(?)

(??) Parton distributions

q ( ) q ( ) q ( ) x x x

obtained by analysis of polarized

DIS experiments at .

The gluon contribution

is not well known.

2 2Q = 4 GeV g

(from ref. 8)

unpolarized 2 2Q = 4 GeV

2 2Q = 4 GeV

polarized

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

d = 0 . 4 3 0 . 0 2

s = 0 . 1 0 0 . 0 2

(82% of the proton spin)

(- 43% of the proton spin)

( - 10% of the proton spin)

u + d + s = 0 . 2 9 0 . 0 6

Total spin sum:

(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)

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 , that is,

one can determine the contribution of the gluons to the spin of

the proton.

g ( )x

In order to solve this puzzle:

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

contribution of gluons to the

spin of the proton.

(from ref. 10)

proton

proton

As a last topic of this lecture:

Deep inelastic scattering of electrons from NUCLEI:

q

p

electron nucleus, for example calcium

Many hadrons (not observed) are formed after the scattering

Using again the parton model, one can extract the momentum

distributions of quarks inside a BOUND proton.

(unpolarized scattering)

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)

0.1 - 0.2x

2) For small momentum

fractions in the region

the ratio is a bit larger than 1.

0.5 - 0.8x

1) For intermediate momentum

fractions in the region

the ratio is smaller 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

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.

The EMC effect shows that the average momentum of the quarks

in a bound nucleon is smaller than in a free nucleon.

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?”

s

Spin of

electron Spin of nucleus

Spin of

electron 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)

(quark spin in a free proton) = 0.8

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

From accelerator experiments, we know many things

about quarks in the nucleon and in the nucleus:

Charge distributions, momentum distributions,

spin distributions, etc.

Quarks are the building blocks of matter.

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.