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Ｒｅｃｅｎｔ Ｔｏｐｉｃｓ ｉｎ Ｑｕａｒｋ
Ｐｈｙｓｉｃｓ
Wolfgang Bentz
Department of Physics, School of Science
Tokai University
10
10
10 15 10 14 m
m
electron
10 15 m  1 fm
proton
nucleus
neutron
proton
neutron
thing
atom
atomic
nucleus
almost empty,
densely packed,
electromagnetic
interaction
strong
interaction
u-quark
d-quark
nucleon
densely packed,
strong
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
( ,  ,  ) , ( K , K , K , K ) ,  ,  '



0

0
Can be explained by assuming quark-antiquark bound states:
( q 1 , q-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 q i  u , d , s . 10 different combinations
[Note: For the baryons with spin 1/2 and positive parity, there
are only 8 members:
0
+
0

0

(p, n) , (  ,  ,  ) ,  , (  ,  ) .
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
++

 (uuu)
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’’).
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 ( q q 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
q
q-
r
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



e
e
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’’.
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
_
( q q) 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
gluon self interaction
virtual g
but also:
(see ref. 1)
r
r
r
r
test color charge
r
A test color charge will see a larger
r
r
quark color charge at large distances
r
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.
quark
A schematic model of the nucleon, which takes into account
confinement and asymptotic freedom, is the bag model:
V(r)
RR
V(r)=
V(r)=0

(square well potential)
r
pressure B
R
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  M N /  R = 0 , where M N is the nucleon
mass.
The nucleon mass in the bag model is:
4 3
M N c  3 pq c +
R B
3
Here p q is the momentum of a quark inside the bag, which is
2
determined from the boundary condition of the wave function as
Elastic electron-proton (e-p) scattering
p+q
Kinematical constraint for elastic scattering:
(p  q)  p
2
q
2
 MN
2
MN  q 2  2p  q  M N
Q2
x
1
(Q2   q2  0)
2p  q
2
p
electron
proton
[Note: The product of two Lorentz 4-vectors a

2

 (a
0

, a )


and b  (b , 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 ]

0
0
0
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
G E ( Q 2 ) from the ratio of the measured cross section to the
cross section for a point-like proton (“Mott cross section’’):
measured cross section
p
2
2

G
(
Q
) (x  1).
F(Q ,x ) 
E
Mott cross section
Effect of proton size
kinematical constraint
p
2
Note: Actually, here we refer to the “electric form factor’’ G E ( Q . )
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 ).
p
What is the physical meaning of G E ( Q 2 ) ?

p
It is the Fourier transform of the proton’s charge density  ( r ) :
p
2
GE (Q ) =
d

3
r e

i q  r

 (r)
p
Note that for a point-particle the charge density is a delta – function,
and the form factor is a constant :

point


(r)= (r)
GE
point
(Q 2 )=1
But the proton consists of three quarks, and therefore it has a finite

extension:  ( r ) =  p |
charge of the i-th quark.
p
3


i
where
is the
e

(
r

r
)
|
p

,
e

i
i
i=1
Similarly, the magnetic form factor of the proton is the Fourier
transform of the magnetic moment density inside the proton:
p
d
= p |  m
GM (Q2 ) =

z (r)
p

3
r e
3
i=1
i
z

i q  r



 z ( r ),
p
where
 ( r  r i ) | p  , where m z
i
ei
i
=
2s z
2m
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:”
1
p
p
n
2
2
p
2
n
, ( =0.84GeV)
G E ( Q ) =G M ( Q ) /  = G M ( Q ) /  =
2
2 2
(1+Q / )
n
p
[Here 
= 2.79,  = -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
in center
- charge
outside
The “root-mean-square” radii


0
0
 r   {  r 2  }1/2  { [  ( r 2 d r ) r 2  ( r )] / [ ( r 2 d r )  ( r )] }1/2
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
Electric form factor of
proton:
Form factor (Proton)
D ip o le p a r a m e t r iza t io n
D e n s it y = 0 .0 f m - 3
D e n s it y = 0 .1 6 f m - 3
0 .8
Experiment
Free proton
Bound proton
0 .4
0
0 .5
1
1 .5
2
M o m e n t u m t r a n sf e r [ G e V ]
2
T. Horikawa, W. Bentz,
Nucl. Phys. A 762 (2005) 102.
Concept of scaling:
Generally, we say that “scaling” is valid in electron-proton
scattering if the ratio (defined earlier)
F(Q ,x ) 
2
measured cross section
Mott cross section
(Q2 =  q 2 >0)
( x = Q2 / 2 p  q )
(q…momentum transfer)
is (almost) independent of Q 2 .
Scaling can hold only in a certain range of Q 2 . 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( Q 2, x)
reduces to the kinematical constraint  ( x - 1 ) .
2
F( Q ,x)
 (x-1)
(Elastic e-p scattering at very
2
low Q .)
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:
(p  q) 2  p'2 >M N
q
M N  q 2  2p  q>M N
Q2
x
<1
(Q2   q2  0)
2p  q
2
p
electron
proton
2
2
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
2
F(Q ,x ) 
Mott cross section
is called “structure function’’. [Note: Actually, for the proton
there are two structure functions: F 1 ( Q 2 , x ) and F 2 ( Q 2 , x ) ]
The experimental data clearly show a “scaling” behavior:
The structure functions are almost independent of Q 2
over a wide range of Q 2 , as long as Q 2 > 2 G e V .2
As function of
Q 2 for several x
:
As function of
(from ref. 6)
x
for several
Q2
:
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,
and receives a large momentum q.
k+q
q
k
electron
proton (momentum p)
Kinematical constraint imposed by this “quasielastic” process:
Mq  q 2  2k  q  Mq
(k  q)  k  Mq
Q2
1
(Q2   q2  0)
2k  q
2
2
2
2
2
( M q = quark mass)
If z<1 denotes the momentum fraction of a quark in the proton, then
the quark momentum is k=pz , and
2
2
Q
Q
x=
=z
=z
2p  q
2k  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 q ( Q ) = e q . Then, in the parton model, we obtain
the following expression for the structure function of the proton:
1
F(Q 2 ,x )=   dz G q (Q 2 )   (x-z) q(z)
2
q
0
point-quark:
Gq (Q2 )=eq
kinematical
constraint
probability that quark
(flavor q) has momentum
fraction z
Finally we obtain:
F(Q ,x )= eq q(x )
2
2
q
(from ref. 6)
2
Q
independent of
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 x =1/3.
But there is no peak in the data !?
If we consider the difference between proton and neutron
structure functions F 2p ( x )  F 2n ( x ) , 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.
F 2 (x ) =x  eq q(x)
2
q
sea quarks
valence quarks
sea
quarks
valence
quarks
x
1/3
(see ref. 4, p. 23)
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
In the difference F 2 ( x )  F 2 ( x ) , 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(x)=q (x)+q (x),
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 ( x ) = q  ( x ) + q  ( x ) , 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 
 dx
q(x)
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
sea quarks.]
s
is only to the
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)
g (?)
unpolarized
Q 2 = 4 GeV 2
dv
us
ds
u v
u s  ds
d v
q(x)=q (x)+q (x)
obtained by analysis of unpolarized
DIS experiments at Q 2 = 4 GeV 2 .
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 x .
uv
g (??)
Parton distributions
polarized
Q 2 = 4 GeV 2
Parton distributions
q(x)  q (x)  q (x)
obtained by analysis of polarized
DIS experiments at Q 2 = 4 GeV 2
The gluon contribution g
is not well known.
(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
d=  0.43  0.02
(82% of the proton spin)
s=  0.10  0.02
( - 10% of the proton spin)
(- 43% 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
proton
proton
“Gluon fusion processes’’
will give information on the
contribution of gluons to the
spin of the proton.
(from ref. 10)
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)
electron
p
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
electron
s
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
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.