The Quark & Bag Models

Simona Stoica

KVI, September 17, 2008

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Outline

• The Quark Model– Original Quark Model– Additions to the Original Quark Model– How to form mesons and baryons– Color

• Quantum Chromodynamics (QCD)– Color Charge– Quark confinement

• M.I.T. Bag Model– Assumptions– Predictions– Failures of the MIT Bag model

• Heavy quark spectra

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The Quark Model• By the early 60’s there was a large zoo of

particle found in bubble chamber experiments

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Sorting them out

• We could classify them by various quantum numbers– Mass– Spin– Parity– C parity– Isospin– Strangeness

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First steps

It was realized that even these new particles fit certain patterns:pions: +(140 MeV) -(140 MeV) o(135 MeV)kaons: k+(496 MeV) k-(496 MeV) ko(498 MeV)

If mass difference between proton neutrons, pions, and kaons is due to electromagnetism then how come:

Mn > Mp and Mko > Mk+ but M+ > Mo

Lots of models concocted to try to explain why these particles exist: Model of Fermi and Yang (late 1940’s-early 50’s):

pion is composed of nucleons and anti-nucleons (used SU(2) symmetry)

+ = pn, - = np, o = pp - n n note this model was proposed before discovery of anti-proton !

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First steps

Gell-Mann, Nakano, Nishijima realized that electric charge (Q) of all particles could be related to isospin (3rd component), Baryon number (B) and Strangeness (S):

Q = I3 +(S + B)/2= I3 +Y/2hypercharge (Y) = (S+B)

Interesting patterns started to emerge when I3 was plotted vs. Y

With the discovery of new unstable particles (, k) a new quantum

number was invented: strangeness

Y

I3

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Original Quark Model

1964 The model was proposed independently by Gell-Mann and Zweig Three fundamental building blocks 1960’s (p,n,) 1970’s (u,d,s)

mesons are bound states of a of quark and anti-quark:Can make up "wave functions" by combing quarks:

+ = ud, - = du, o =12

(uu - d d), k+= ds, ko= ds

baryons are bound state of 3 quarks:proton = (uud), neutron = (udd), = (uds)

anti-baryons are bound states of 3 anti-quarks:

p u u d n u d d u d s

Λ= (uds))( ud

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Quarks

These quark objects are:• point like• spin 1/2 fermions • parity = +1 (-1 for anti-quarks)• two quarks are in isospin doublet (u and d), s is an

iso-singlet (=0)• Obey Q = I3 +1/2(S+B) = I3 +Y/2• Group Structure is SU(3)• For every quark there is an anti-quark• The anti-quark has opposite charge, baryon number and strangeness• Quarks feel all interactions (have mass, electric charge, etc)

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Early 1960’s QuarksSuccesses of 1960’s Quark Model: • Classify all known (in the early 1960’s) particles in terms of 3 building blocks• predict new particles (e.g. -)• explain why certain particles don’t exist (e.g. baryons with spin 1)• explain mass splitting between meson and baryons• explain/predict magnetic moments of mesons and baryons• explain/predict scattering cross sections (e.g. p/pp = 2/3)

Failures of the 1960's model:• No evidence for free quarks (fixed up by QCD)• Pauli principle violated (++= (uuu) wave function is totally symmetric) (fixed up by color)• What holds quarks together in a proton ? (gluons! )• How many different types of quarks exist ? (6?)

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Additions to the Original Quark Model – Charm

• Another quark was needed to account for some discrepancies between predictions of the model and experimental results

• Charm would be conserved in strong and electromagnetic interactions, but not in weak interactions

• In 1974, a new meson, the J/Ψ was discovered that was shown to be a charm quark and charm antiquark pair

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More Additions – Top and Bottom

• Discovery led to the need for a more elaborate quark model

• This need led to the proposal of two new quarks– t – top (or truth)– b – bottom (or beauty)

• Added quantum numbers of topness and bottomness

• Verification– b quark was found in a meson in 1977– t quark was found in 1995 at Fermilab

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Numbers of Particles

• At the present, physicists believe the “building blocks” of matter are complete– Six quarks with their antiparticles

– Six leptons with their antiparticles

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Number of particles

The additive quark quantum numbers are given below:Quantum # u d s c b telectric charge 2/3 -1/3 -1/3 2/3 -1/3 2/3I3 1/2 -1/2 0 0 0 0Strangeness 0 0 -1 0 0 0Charm 0 0 0 1 0 0bottom 0 0 0 0 -1 0top 0 0 0 0 0 1Baryon number 1/3 1/3 1/3 1/3 1/3 1/3Lepton number 0 0 0 0 0 0

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How to form mesons?

8133

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Baryons?

10881333

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Color

• Baryon decuplet (10) states consist of lowest mass J=3/2 states,

assume that the quarks are in the spatially symmetric ground state (=0)

• To make J=3/2, the quark spins must be ‘parallel’(ex) ++ = uu u

• The ++ wave function is symmetric

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Color

• Pauli exclusion principle?– two or more identical fermions may not exist

in the same quantum state– what about the u quarks in ++ ?

It must be antisymmetric under Pauli principle!

• More questions on the quark model

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Color• Another internal degree of freedom was

needed “COLOR”• Postulates

– quarks exist in three colors:– hadrons built from quarks have net zero color

(otherwise, color would be a measurable property)

• We overcome the spin-statistics problem by dropping the concept of identical quarks; now distinguished by color

++ = uR uG uB

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Color & strong interactions• We have assigned a “hidden” color

quantum # to quarks.– “hidden” because detectable particles are all

“colorless”

• It solves the embarrassment of fermion statistics problem for otherwise successful quark model.

• Most importantly, color is the charge of strong interactions

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Quantum Chromodynamics (QCD)

• QCD gave a new theory of how quarks interact with each other by means of color charge

• The strong force between quarks is often called the color force

• The strong force between quarks is carried by gluons– Gluons are massless particles– There are 8 gluons, all with color charge

• When a quark emits or absorbs a gluon, its color changes

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More About Color Charge

• Like colors repel and unlike colors attract– Different colors attract, but not as strongly as a color

and its opposite colors of quark and antiquark

• The color force between color-neutral hadrons (like a proton and a neutron) is negligible at large separations

– The strong color force between the constituent quarks does not exactly cancel at small separations

– This residual strong force is the nuclear force that binds the protons and neutrons to form nuclei

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Quantum Chromodynamics (QCD)

• Asymptotic freedom– Quarks move quasi-free inside the nucleon– Perturbation theoretical tools can be applied

in this regime

• Quark confinement – No single free quark has been observed in

experiments– Color force increases with increasing distance

• Chiral symmetry

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Quark confinement

• Spatial confinement– Quarks cannot leave a certain region in space

• String confinement– The attractive( color singlet) quark-antiquark

• Color confinement

• The quark propagator has no poles

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M.I.T. Bag Model

• Developed in 1974 at Massachusetts Institute of Technology

• It models spatial confinement only

• Quarks are forced by a fixed external pressure to move only inside a given spatial region

• Quarks occupy single particle orbitals

• The shape of the bag is spherical, if all the quarks are in ground state

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M.I.T Bag Model

• Inside the bag, quarks are allowed to move quasi-free.

• An appropriate boundary condition at the bag surface guarantees that no quark can leave the bag

• This implies that there are no quarks outside the bag

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M.I.T. Bag Model

• The boundary condition generates discrete energy eigenvalues.

R

xnn

R - radius of the Bag

x1=2.04

BRRE

R

xNRE

pot

nqkin

3

3

4)(

)(

Nq = # of quarks inside the bag

B – bag constant that reflects the bag pressure

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M.I.T. Bag Model

• Minimizing E(R), one gets the equilibrium radius of the system

4133

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4

4

nqn

nqn

xBNE

B

xNR

Fixing the only parameter of the model B, by fitting the mass of the nucleon to 938MeV we have first order predictions

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One gluon exchange

• Model so far excluded all interactions between the quarks

• There should be some effective interaction that is not contained in B( how do we know that?)

R

ME qs

X

αs – the strong coupling constant

Mq depends on the quantum no. of the coupled quarks

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The Casimir Term

• The zero point energy of the vacuum

• The Casimir term improves the predictions of the MIT bag model.

• However, theory suggests the term to be negative

• Best fits provide a slightly positive value

R

ZECas

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Predictions

The masses of N, Δ, Ω, ω were used to fit the parameters.

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Quark confinement

q

q

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Color confinement

• The non-perturbative vacuum can be described by a color dielectric function k(r) that vanishes for r→∞.

• The total energy Wc of the color electric field Ec of a color charge Qc is

0

223

)(~~

rr

drQrdDEW cccc

• Integral diverges, unless Qc=0

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Failures of the Bag Model

• Chiral symmetry is explicitly broken on the bag surface( static boundary condition)

• Chiral extensions of the MIT-Bag model have been suggested: Cloudy bag model

• Introduces a pion field that couples to the quarks at the surface.

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Heavy quarks. Positronium Results

• Positronium is an e+e- state that forms an “atom”

• Two important decay modes– Two photon (singlet)

• J=0 by Bose Symmetry• C=1 since C(photon)=-1

– Three photon• J=1• C=-1

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Postrionium Energy Levels

• Can be done with non-relativistic Schrodinger equation & Coulomb Potential

– Principal quantum number n=1,2,3…– Reduced mass

• So result for positronium is

/ / 2mM m M m

2 2

22n

cE

n

2 2

24n

mcE

n

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Relativistic Corrections

• Spin-orbit couplings– Fine structure

• Spin-spin couplings– Hyperfine structure

• These interactions split levels into– Triplet (3S1) (orthopositronium)

– Singlet (1S0) (parapositronium)

~V L S

1 2 1 2~ ~V S S

4 2

3~fine

mcE

n

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Positronium Levels

n=1

n=2

L=0

11S0

13S1

L=0

L=1

S=0

S=1

S=0

S=1

S=1

S=0 21P1

21S0

23S1

23P0

23P1

23P2

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Comparison with Charmonium

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Why should these be similar?

• Coulomb Potential has been shown before: mediated by massless photons

• QCD has been found numerically to have a similar form

emVr

4

3s

QCDV krr

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Conclusions

• The quark model – classifies all known particles in terms of 6 building blocks– Explains mass splitting between meson and baryons– Explain/predict magnetic moments of mesons and baryons– Explain/predict scattering cross sections

• The MIT Bag Model – predicts fairly accurate masses of the particles– Explains color confinement– Helps predict heavy quark spectrum

Simple models can give us a very good picture!

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Bibliography

• Y. IWAMURA and Y. NOGAMI, IL NUOVO CIMENTO VOL. 89 A, N. 3(1985)

• Peter HASENFRATZ and Julius KUTI, PHYSICS REPORTS (Section C of Physics Letters) 40, No. 2 (1978) 75-179.

• T. Barnes, arXiv:hep-ph/0406327v1 • Carleton E. DeTar, John 12. Donoghue, Ann. Rev. Nucl. Part. Sci.

(1983)• E. Eichten et al. , Phys. Rev. D, 203 (1980)• E. Eichten et al. , Phys. Rev. Lett, 369 (1975)• Stephan Hartmann, Models and Stories in Hadron Physics