Quark Models:
Quixotic Madness or
Questionable Miracle?
高崇文
中原大學物理系
22/11/2007 交通大學物理所
Brief history of subatomic physics




I
discover
the
neutron
1911 Rutherford discovered atomic nucleus.
1932 Chadwick discovered the neutron.
1932 Heisenberg introduce the concept of isospin.
1933 Stern measured the magnetic moment of the proton
and showed the proton is not point-like.
I am lucky not
to care what
Pauli said
μ: magnetic moment, s: spin
gs=2 for point-like particle
gp=5.59, gn=-3.83
Brief history of subatomic physics (2)



1935 Hideki Yukawa introduced the meson
mediating the force between nucleons.
1947 Powell discovered the first meson: pion.
1949 Fermi and Yang suggest that pion is bound state of
proton and neutron.
But it is ME to discover the pion!
This potential is
named after me !
p
n
n
π0
p
p
p
π+
n
p
n
π-
nn
p
Brief history of subatomic physics (3)
1947 Rochester and Butler
discovered the first strange particles
Kaon and Hyperon in cosmic rays at
Manchester University under the
director Lord Blackett who later
received Noble prize.
Well done, boys!
   p  0  K 0
π++πP+π-
It is very
strange…indeed,,,
Discovery of strangeness


Strangeness was introduced, by Murray Gell-Mann and
Kazuhiko Nishijima, originally to explain the fact that
certain particles, such as the kaons or certain hyperons
were created easily in particle collisions, yet decayed
much more slowly than expected for their large masses
and large production cross sections.
Noting that collisions seemed to always produce pairs
of these particles, it was postulated that a new
conserved quality, dubbed "strangeness", was
preserved during their creation, but not conserved in
their decay.
And more particles are coming…..
Δ(1232) 1st, most prominent and
non-overlapping resonance
Discovered by Fermi in
1952 in πp scatterings
Who order
those
particles?
2
Pauli’s frustration

The development of new particle accelerators and
particle detectors in the 1950s led to the discovery
of a huge variety of hadrons, prompting Wolfgang
Pauli's remark:
"Had I foreseen this, I would have gone into
botany. “
But I obtain my Nobel
prize by finding
them!!
Alvarez
Nuclear resonances
Question: Are they all fundamental particles?
Sakata model
It is such a good
idea … Karl Marx
will be proud of me!
1956 Sakata suggested that all
hadrons are composed of
proton (p), neutron (n) and
hyperon (Λ0).and their antiparticles. For example, K+ is
bound states of proton and
anti-hyperon
坂田昌一
SU(3) symmetry
In Sakata model the symmetry is no longer SU(2) isospin
But SU(3).
exp(iθa‧σa)
exp(i θa‧λa)
0 1 0
 0 i 0 
1 0 0
1   1 0 0  ,  2   i 0 0  ,  3   0 1 0  ,
0 0 0
0 0 0
0 0 0






0 0 1
 0 0 i 
 0 0 0
 4   0 0 0  ,  5   0 0 0  ,  6   0 0 1  ,
1 0 0
i 0 0 
0 1 0






0 0 0 
1 0 0 
1 

 7   0 0 i  ,  8 
0
1
0
.


3
0 i 0 



 0 0 2 
Difficulty of Sakata Model
By this way mesons can be described as
one octets and one singlet.
However, for baryon we have one
15et, two triplets and one sextet.
There are many missing baryons
in the spectrum.
And we shall not forget that proton and neutron are not point-like
But composite particles since they have complicated inner structures.
Eight-fold way (八正道)

1961 Gell-Mann and Ne’eman independently
suggested that one should put nucleon and
hyperon with Ξ and Σ as octet.
Baryon as SU(3) multiples
Debut of Quark

1964 Gell-Mann and Zweig independently
proposed a model in which baryons and
mesons are composites of a fundamental
triplet of U(3), Gell-Mann called them
“quarks” and Zweig called them “aces”.
Fractional charge !!
Gell-Mann explained the origin of “quark”:
In 1963, when I assigned the name "quark" to the fundamental constituents of the
nucleon, I had the sound first, without the spelling, which could have been "kwork".
Then, in one of my occasional perusals of Finnegans Wake, by James Joyce, I
came across the word "quark" in the phrase "Three quarks for Muster Mark". Since
"quark" (meaning, for one thing, the cry of the gull) was clearly intended to rhyme
with "Mark," as well as "bark" and other such words, I had to find an excuse to
pronounce it as "kwork". But the book represents the dream of a publican named
Humphrey Chimpden Earwicker. Words in the text are typically drawn from several
sources at once, like the "portmanteau" words in "Through the Looking Glass".
From time to time, phrases occur in the book that are partially determined by calls
for drinks at the bar. I argued, therefore, that perhaps one of the multiple sources of
the cry "Three quarks for Muster Mark" might be "Three quarts for Mister Mark," in
which case the pronunciation "kwork" would not be totally unjustified. In any case,
the number three fitted perfectly the way quarks occur in nature.
Quark Model
-
Discovery of Ω
Thanks to Ω
I obtained a
Nobel prize!
++
Puzzle of Δ
Δ++ is a spin 3/2 fermion
Δ++=|u u u＞ |↑↑↑＞
Why Δ++ has symmetric wave
function?
Han and Nambu suggested that quarks are triplet of
new hidden quantum number.(1965)
Possible solutions to Δ
puzzles, 1964
By O.W. Greenberg, hep/ph-0212174
++
One interesting remark from
Schwinger…
By O.W. Greenberg, hep/ph-0212174
But some people did follow Schwinger’s insight…
I told you…….
Debut of Color SU(3) symmetry


1950s C.N. Yang and Mills suggested to
localize SU(2) isospin and built a non-Ableian
gauge theory.
1965 Han, Nambu suggested that quark
possess an additional SU(3) gauge degree of
freedom: color and quarks would interact via
an octet of vector gauge bosons: the gluons.
Our
world is
colorful!
Nambu and Han
QCD Langrangian
n=1,2,3; a=1,2…8
Resistance to quark and color



Unobserved fractionally charged quark
seems outrageous!
A new hidden degree of freedom is
doubly outrageous!!
Even Gell-Mann kept ambiguous
attitude toward the reality of quarks!!!
Hmmm…To be or not to be…I am not
sure…….Hmmm….
Deep Inelastic Scattering and Parton

1966 Deep Inelastic Scattering (DIS)
showed the proton consists of many
weakly interacting point-like particles.
Quantum Chromodynamics (QCD)




1973 Gross, Wilczek, and Politzer discovered
asymptotic freedom which explains DIS data.
Namely the coupling constant g becomes small when
the momentum transfer is large.
On the other hand when the momentum transfer is
small the coupling constant is large! It is called
infrared slavery .
At the strongly coupling regime only colourless object
is allowed. It is called confinement. So far there is no
rigor mathematical proof.
So fundamental theory is at hand,
but….
q
q
q
Mystery remains:
Of the many possibilities for
combining quarks with colour into
colorless hadrons, only two
configurations were found, till
now…
Because we cannot apply QCD at
low Q2 since then g is large and
the underlying theory is strongly
coupling Quantum field theory
which means no one can solve it
analytically !
Just do it !
Mission impossible?
QCD is a very successful theory, but can we use QCD to
study the nucleon structure and even the nuclear force?
baryon,meson
quark, gluon
Low Q2, meson-exchange
2
High Q , perturbative QCD
Asymptotic freedom
?
confinement
Non-relativistic Quark Models

Assume baryons are composed of three massive constituent
quarks bound in a confining potential.




The constituent quarks carry the quantum numbers of QCD
quarks but much heavier.
Although the non-relativistic quark model lacks any field
theory basis, its phenomenological value is beyond doubt.
One traditional success of this kind of models is the
anomalous magnetic moments of the proton and neutron.
There are many variants due to the choices of the
potentials.
Isgur-Karl Model
One of most successful non-relativistic quark model is invented by Nath Isgur(’78)
The Hamiltonian consists of kinetic term, mass term, confinement potential and one
Hyperfine interaction whose form is one-gluon-exchange type:
“Wave function” of the proton:
μ0 is the Bohr magneton of the quark:
The anomalous magnetic moment of the proton is:
Similarly one obtains:
Actually this result solely relies on the SU(6)SF symmetry
SU(6)SF Spin-Flavour Symmetry
Meson:
Baryon:
Totally symmetric
Mixed symmetry
Anti-symmetrical
Classification of excited baryons
Quark model predictions for baryons
To describe the known baryon spectrum a lot of quark models
have been developed. General symmetry principles of quark
models as SU(6)*O(3) predict more states than were observed
in the experiment. Different models predict different number
and positioning of these states.
The search for the missing states can provide a
good test for basic principles of quark models and
the effects of quark-quark correlation.
“string” linear confinement
+ Coulomb hyperfine
interaction as SU(6)
configuration mixing
Isgur-Karl, Isgur-Capstick and
collaborators
linear confinement.
SU(6) configuration
mixing by spinflavour-dependent
interaction (GBE)
Glozman-Riska; Graz
group
linear confinement +
Coulomb potential
3-body forces (expected
based on QCD)
Giannini–Santopinto and
collaborators
Large Nc QCD and SU(6)



QCD is a SU(3) gauge theory
If one studies SU(Nc) gauge theory, then
makes 1/Nc expansion, then one finds
when Nc becomes infinity, the baryon
sector owns a symmetry SU(2Nf).
It is amusing to find constituent quark
model owns same symmetry with large
Nc QCD
How to make models more “QCD-like”?



Baryon is a complicated many-body system in QCD but
miraculously one can use constituent quarks and obtains
many good results. One justification is to treat the
constituent quarks as quais-particles which are collective
excitation modes.
Therefore one needs more understanding of QCD
vacuum to construct more realistic models.
However, QCD vacuum is very complicated so one can
only try to grasp some aspects of QCD vacuum from our
limited knowledge, such as spontaneously breaking of
Chiral symmetry (χSB)
Chiral Symmetry of QCD if mq=0
Left-hand and right-hand quark:
QCD Lagrangian is invariant if
Massless QCD Lagrangian has SU(2)LxSU(2)R chiral symmetry.
Quark mass effect
If mq≠0
QCD Lagrangian is invariant if θR=θL.
Therefore SU(2)LXSU(2)R →SU(2)V, ,if mu=md
SU(2)A is broken by the quark mass
Spontaneous symmetry
breaking
Spontaneous symmetry breaking:
a system that is symmetric with respect to some
symmetry group goes into a vacuum state that is not
symmetric. The system no longer appears to behave in a
symmetric manner.
Example:
V(φ)=aφ2+bφ4, a<0, b>0.
Mexican hat potential
U(1) symmetry is lost if one expands around the degenerated vacuum!
Furthermore it costs no energy to rum around the orbit →massless
mode exists!! (Goldstone boson).
2007 APCTP workshop at POSTECH 26~28 Feb. 2007
Instanton
Instanton vacuum configuration
Gluonic potential of QCD
Self-duality condition: minimizing the potential
Topological number realted to the ground state
Guage transformation of the ground state via
Instanton and SχSB
Winding number from homotopic SU(N) gaugetransformation
Tunneling between vacuua
Instanton solution for the self-duality condition
Natural mechanism for SSB
An analogy: Ferromagnetism
Above Tc
Below Tc
＜M＞=0
＜M＞≠0
Dynamical symmetry breaking
and fermion mass generation
Chiral condensation
Gap Equation
Dynamical Quark Mass～350 MeV
Pion as Goldstone boson
π
is the lightest hadron. Therefore it plays a dominant the
long-distance physics.
 More important is the fact that soft π interacts each other
weakly because they must couple derivatively!
 Actually if their momenta go to zero, π must decouple with
any particles, including itself.
Start point of an EFT for
pions.
～t/(4πF)
2
Double faces of pion



Pion in constituent quark model is treat as
quark-antiquark pair.
However it is Goldstone boson associated
with SχSB.
Pion plays dominant role in the low-energy
QCD phenomenology ! There are two
examples…
Nucleon E.M form factors

Hofstadter determined the precise size of the proton
and neutron by measuring their form factor.
Pion cloud surrounding the nucleon


Both the proton and neutron have a central,
positively charged core surrounded by a double cloud
of π-mesons.
Both clouds are positively charged in the proton, but
in the neutron the inner cloud is negatively charged,
thus giving a net zero charge for the entire particle.
∣n＞=∣n＞0+Z∣pπ -＞+…
N→Δ(1232) transition form factor
S wave →S wave

S wave →D wave
In a symmetric SU(6) quark model the
E.M excitation of the  could proceed only
via M1 transition. If the  is deformed, then
the photon can excite a nucleon into a 
through electric E2 and Coulomb C2
quardrupole transitions.
REM = E2/M1 ≈ -2.5 %, (MAMI, LEGS)
( indication of a deformed  )
Both N and ∆ are members of the [56]-plet
and the three quarks are in the (1s)3 states
Deformation in constituent quark model
[ Rij(2)  Sij(2) ](0)
H  T  Vconf  VOGEP ,
2 s
VOGEP (ij ) 
2mi m j

1
 8 (3)

rij S i  S j  3 3S i  r ij S j  r ij  S i  S j

rij

 3
 




,


Vconf  VH .O.
D-state component
PD(%)
N(938)
0.4
1232
1.9
-0.8% < REM < -0.3%
Q(fm2)
0
-0.089
Too
small !!
Pion cloud plays essential role!
How to make Quark model more “chiral”?
Coupling of spins, isospins
etc. of 3 quarks
mean field  non-linear system
 soliton  rotation of soliton
Coherent :1p-1h,2p-2h,....
A crazy idea from Tony Skyrme



1962 British scientist Tony Skyrme created a very
interesting idea, namely can one create a fermion from a
scalar field? The answer is YES and the rsult is skyrmion.
A skyrmion is a homotopically non-trivial classical solution
of a nonlinear sigma model with; i.e., a particular case of
a topological soliton.
If spacetime has the topology S3×R (for space and time
respectively), then classical configurations are classified
by an integral winding number because the third
homotopy group: π3(SU(N)xSU(N)/SU(N))=Z
Skyrme model: SU(2) Skyrmion
Starting from Nonlinear sigma model, Skyrme write down the following
Langrangian:
Skyrme found a family of class solutions of the above Langrangian:
N; Integer valued topological charge
Quantization of SU(2) Skyrmion
Chiral Quark Model
constituent quark
mass ~ 350 MeV
pions
fermions
integrate out quarks
Skyrme Model
Quantizing SU(3) Skyrmion and QM
time-dependent rotation
angular velocities:
Fiasco of Pentaquark
Theoretical predictions for pentaquarks
1. Bag models [R.L. Jaffe ‘76, J. De Swart ‘80]
Jp =1/2- lightest pentaquark
Masses higher than 1700 MeV, width ~ hundreds MeV
Mass of the pentaquark is roughly 5 M +(strangeness) ~ 1800 MeV
An additional q –anti-q pair is added as constituent
2. Soliton models [Diakonov, Petrov ‘84, Chemtob‘85,
Praszalowicz ‘87, Walliser ‘92]
Exotic anti-decuplet of baryons with lightest S=+1
Jp =1/2+ pentaquark with mass in the range
1500-1800 MeV.
Mass of the pentaquark is rougly 3 M +(1/baryon size)+(strangeness) ~ 1500MeV
An additional q –anti-q pair is added in the form of excitation of nearly massless
chiral field
The anti-decuplet
Width < 15 MeV !
uud ( d d  ss )
uus( d d  ss )
uss(uu  d d )
Diakonov, Petrov, Polyakov, 1997
(St.Petersburg, Bochum)
Praszalowicz 1987
Pentaquark publicity 2003
Evidence for Pentaquark states
Pentaquark publicity 2005
Q+: positive and negative results
Experiments: Mass
Width of
+
Q
Pentaquarks: Experiments
Summary
Summary





Should we trust quark models?
Should we continue to use quark models?
Can we tell which model is more suitable than
others for some certain physical quantity?
Can we learn anything from quark models either
when it works or not?
Is it possible for us to solve no-perturbative QCD
in the future?
To dream the impossible dream
要敢夢不可能實現的夢
To fight the unbeatable foe
要敢對抗無法擊敗的敵人
To bear with unbearable sorrow
忍受那無法忍受的苦楚
To run where the brave dare not go
奔向那勇者不敢前去的地方
To right the unrightable wrong
改正那無法改正的錯誤
To love pure and chaste from afar
追求遠方的純潔與高雅
To try when your arms are too weary
當雙臂疲累不堪時
To reach the unreachable star
更要試著去靠近那遙不可及的星星
This is my quest
這是我的追求
To follow that star
去追隨星星
No matter how hopeless
不論希望多麼渺茫
No matter how far
不管目標多麼遙遠
To fight for the right
Without question or pause
我將毫無遲疑的為正義而戰
To be willing to march into Hell
For a heavenly cause
為神聖的使命而奮不顧身
And I know if I'll only be true
To this glorious quest
我知道只要堅持對此榮耀的追求
That my heart will lie peaceful and calm
When I'm laid to my rest
當我躺下之時我心將永享寧靜
And the world will be better for this
世界也因此變得更好
That one man, scorned and covered with scars受到輕視且滿身傷痕的人
們
Still strove with his last ounce of courage 為追求那遙不可及的星星
To reach the unreachable star 將依然全力奮戰直到耗盡所有的勇氣