高崇文中原大學物理系

22/11/2007 交通大學物理所

Quark Models: Quixotic Madness or Questionable Miracle?

Brief history of subatomic physics

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.

gs=2 for point-like particle

gp=5.59, gn=-3.83

μ: magnetic moment, s: spin

I discover

the neutron

I am lucky not to care what Pauli said

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.

p

p n

n

p

p n

n

π0 π+

n

p n

p

π-

This potential is named after me !

But it is ME to discover the pion!

00 Kp

Brief history of subatomic physics (3)

π++π-

P+π-

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!

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.

Discovery of strangeness

It is very strange…indeed,,,

And more particles are coming…..

2

Discovered by Fermi in 1952 in πp scatterings

Δ(1232) 1st, most prominent and non-overlapping resonance

Who order those

particles?

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

1956 Sakata suggested that all hadrons are composed of proton (p), neutron (n) and hyperon (Λ0).and their anti-particles. For example, K+ is bound states of proton and anti-hyperon

It is such a good idea … Karl Marx will be proud of

me!

坂田昌一

SU(3) symmetry

In Sakata model the symmetry is no longer SU(2) isospinBut SU(3).

1 2 3

0 1 0 0 0 1 0 0

1 0 0 , 0 0 , 0 1 0 ,

0 0 0 0 0 0 0 0 0

i

i

4 5 6

0 0 1 0 0 0 0 0

0 0 0 , 0 0 0 , 0 0 1 ,

1 0 0 0 0 0 1 0

i

i

7 8

0 0 0 1 0 01

0 0 , 0 1 0 .3

0 0 0 0 2

i

i

exp(i θa‧λa)exp(iθa σ‧ a)

By this way mesons can be described as one octets and one singlet.

Difficulty of Sakata Model

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-likeBut 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 !!

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.

Gell-Mann explained the origin of “quark”:

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.

QCD Langrangian

n=1,2,3; a=1,2…8

Our world is colorful!

Nambu and Han

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.

q

q

q

Mystery remains:Of the many possibilities for combining quarks with colour into colorless hadrons, only twoconfigurations 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 itanalytically !

So fundamental theory is at hand, but….

Mission impossible?

QCD is a very successful theory, but can we use QCD to study the nucleon structure and even the nuclear force?

quark, gluon baryon,meson

High Q2, perturbative QCD Low Q2, meson-exchange

?Asymptotic freedom confinement

Just do it !

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 oneHyperfine 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 symmetryAnti-symmetrical

Classification of excited baryons

Quark model predictions for baryons

To describe the known baryon spectrum a lot of quark modelshave been developed. General symmetry principles of quarkmodels as SU(6)*O(3) predict more states than were observed in the experiment. Different models predict different number and positioning of these states.

“string” linear confinement + Coulomb hyperfine interaction as SU(6) configuration mixing Isgur-Karl, Isgur-Capstick and collaborators

linear confinement + Coulomb potential 3-body forces (expected based on QCD)

Giannini–Santopinto and collaborators

linear confinement. SU(6) configuration mixing by spin-flavour-dependent interaction (GBE) Glozman-Riska; Graz group

The search for the missing states can provide a good test for basic principles of quark models and the effects of quark-quark correlation.

Large Nc QCD and SU(6) QCD is a SU(3) gauge theory If one studies SU(Nc) gauge theory, the

n 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.

Therefore SU(2)LXSU(2)R →SU(2)V, ,if mu=md

Quark mass effect

If mq≠0

SU(2)A is broken by the quark mass

QCD Lagrangian is invariant if θR=θL.

Spontaneous symmetry breaking

Mexican hat potential

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.

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).

20

07

AP

CT

P w

ork

sho

p a

t P

OS

TE

CH

2

6~

28

Fe

b.

20

07

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

Winding number from homotopic SU(N) gaugetransformation

Tunneling between vacuua

Instanton solution for the self-duality condition

Natural mechanism for SSB

Instanton and SχSB

An analogy: Ferromagnetism

Below TcAbove Tc

＜ M ＞≠0

＜ M ＞ =0

Dynamical symmetry breaking and fermion mass generation

Dynamical Quark Mass ～ 350 MeV

Gap EquationChiral condensation

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.

～ t/(4πF)2

Start point of an EFT for pions.

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 exam

ples…

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∣ - ＞ +…

Both N and ∆ are members of the [56]-plet and the three quarks are in the (1s)3 states

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 )

N→Δ(1232) transition form factor

S wave →S wave

S wave →D wave

(3)3

. .

,

2 8 13( ,

2 3)

conf

sij iji j i j i jij

i

OGEP

Oj ij

conf

P

H O

GE

VH T V

r S S S r S r S Sm m r

V V

V ij

QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ

Fermi contact term

(2) (2) (0)[ ]ij ijR S

D-state component

PD(%) Q(fm2)

N(938) 0.4 0

1.9 -0.089

Too small !!

-0.8% < REM < -0.3%

Deformation in constituent quark model

Tensor force

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

fermions

pions

integrate out quarks

Skyrme Model

constituent quarkmass ~ 350 MeV

time-dependent rotation

angular velocities:

Quantizing SU(3) Skyrmion and QM

Fiasco of Pentaquark

1. Bag models [R.L. Jaffe ‘76, J. De Swart ‘80]Jp =1/2- lightest pentaquarkMasses higher than 1700 MeV, width ~ hundreds MeV

2. Soliton models [Diakonov, Petrov ‘84, Chemtob‘85, Praszalowicz ‘87, Walliser ‘92]Exotic anti-decuplet of baryons with lightest S=+1Jp =1/2+ pentaquark with mass in the range

1500-1800 MeV.

Mass of the pentaquark is roughly 5 M +(strangeness) ~ 1800 MeVAn additional q –anti-q pair is added as constituent

Mass of the pentaquark is rougly 3 M +(1/baryon size)+(strangeness) ~ 1500MeVAn additional q –anti-q pair is added in the form of excitation of nearly masslesschiral field

Theoretical predictions for pentaquarks

The anti-decuplet

( )uud d d ss

( )uus d d ss

( )uss uu d d

Width < 15 MeV !

Diakonov, Petrov, Polyakov, 1997 (St.Petersburg, Bochum) Praszalowicz 1987

Pentaquark publicity 2003

Evidence for Pentaquark states

Pentaquark publicity 2005

+: positive and negative results

Experiments: Mass

Width of +

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