Transcript The Quark & Bag Models

The Quark & Bag Models
Simona Stoica
KVI, September 17, 2008
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
2
The Quark Model
• By the early 60’s there was a large zoo of
particle found in bubble chamber
experiments
3
Sorting them out
• We could classify them by various
quantum numbers
– Mass
– Spin
– Parity
– C parity
– Isospin
– Strangeness
4
First steps
It was realized that even these new particles fit certain patterns:
pions:
p+(140 MeV)
p-(140 MeV)
po(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 Mp+ > Mpo
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)
p + = pn, p - = np, p o = pp - n n
note this model was proposed
before discovery of anti-proton !
5
First steps
With the discovery of new unstable particles (L, k) a new quantum
number was invented:
strangeness
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/2
hypercharge (Y) = (S+B)
Interesting patterns started to emerge
when I3 was plotted vs. Y
Y
I3
6
Original Quark Model
1964 The model was proposed independently by Gell-Mann and Zweig
Three fundamental building blocks 1960’s (p,n,l)  1970’s (u,d,s)
mesons are bound states of a of quark and anti-quark:
Can make up "wave functions" by combing quarks:
p+ = ud, p- = du, po = 1 (uu - d d), k = ds, k = ds
2
+
o
baryons are bound state of 3 quarks:
proton = (uud), neutron = (udd), L= (uds)
anti-baryons are bound states of 3 anti-quarks:
p  uud
n  u d d L u d s
p  (du )

Λ= (uds)
7
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)
8
Early 1960’s Quarks
Successes 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. W-)
• 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. spp/spp = 2/3)
Failures of the 1960's model:
• No evidence for free quarks (fixed up by QCD)
• Pauli principle violated (D++= (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?)
9
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
10
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
12
Number of particles
The additive quark quantum numbers are given below:
Quantum #
u
d
s
c b
t
electric charge 2/3
-1/3 -1/3 2/3 -1/3 2/3
I3
1/2
-1/2 0
0 0
0
Strangeness
0
0
-1
0 0
0
Charm
0
0
0
1 0
0
bottom
0
0
0
0 -1 0
top
0
0
0
0 0
1
Baryon number 1/3
1/3 1/3 1/3 1/3 1/3
Lepton number
0
0
0
0
0
0
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How to form mesons?
3  3  1 8
14
Baryons?
3  3  3  1  8  8  10
15
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) D++ = u u u
• The D++ wave function is symmetric
16
Color
• Pauli exclusion principle?
– two or more identical fermions may not exist
in the same quantum state
– what about the u quarks in D++ ?
It must be antisymmetric under Pauli
principle!
• More questions on the quark model
17
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
D++ = uR uG uB
18
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
19
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
20
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
21
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
22
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
23
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
24
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
25
M.I.T. Bag Model
• The boundary condition generates discrete
energy eigenvalues.
R - radius of the Bag
xn
n 
x =2.04
R
1
xn
Ekin ( R )  N q
R
4 3
E pot ( R )  pR B
3
Nq = # of quarks inside the bag
B – bag constant that reflects the
bag pressure
26
M.I.T. Bag Model
• Minimizing E(R), one gets the equilibrium radius
of the system
14
 N q xn 

Rn  
 4pB 
4
3 3
En 
4pBNq xn
3


14
Fixing the only parameter of the model B, by
fitting the mass of the nucleon to 938MeV we
have first order predictions
27
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?)
sM q
EX 
R
αs – the strong coupling constant
Mq depends on the quantum no. of
the coupled quarks
28
The Casimir Term
• The zero point energy of the vacuum
ECas
Z

R
• 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
29
Predictions
The masses of N, Δ, Ω, ω
were used to fit the
parameters.
30
Quark confinement
q
q
31
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

  3
dr
2
Wc ~  Ec  Dc d r ~ Qc  2
r  (r )
0
• Integral diverges, unless Qc=0
32
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.
33
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
34
Postrionium Energy Levels
• Can be done with non-relativistic Schrodinger
equation & Coulomb Potential
En  
 2 c2
2n 2
– Principal quantum number n=1,2,3…
  mM /  m  M   m / 2
– Reduced mass
• So result for positronium is
En  
 mc
2
4n
2
2
35
Relativistic Corrections
• Spin-orbit couplings
– Fine structure
V ~ LS
• Spin-spin couplings
V ~ 1  2 ~ S1  S2
• These interactions split levels into
– Hyperfine structure
– Triplet (3S1) (orthopositronium)
– Singlet (1S0) (parapositronium)
DE fine ~
 mc
4
n
3
2
36
Positronium Levels
23P 2
23P 1
S=1
23P 0
L=1
n=2
S=0
L=0
S=1
S=0
n=1
L=0
21P 1
23S1
21S0
S=1
13S1
S=0
11S0
37
Comparison with Charmonium
38
Why should these be similar?
• Coulomb Potential has been shown
before: mediated by massless photons
Vem  

r
• QCD has been found numerically to have
a similar form
VQCD
4 s

 kr
3r
39
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!
40
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
41