the quark & bag models simona stoica kvi, september 17, 2008
TRANSCRIPT
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