Recent Research on the Thomas Fermi Quark Model
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Recent Research on the Thomas
Fermi Quark Model
• Work with Andy Liu and Nate Larson!
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Motivations
• Where are the mesons, baryons or mixed states with more than the usual two or three quarks?
• There is a need for quark models which can help lead expensive lattice QCD calculations in the right direction in the search for high quark states.
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TF models:• Treats particles as a Fermi gas at T=0• Builds in Fermi statistics, but not fully quantum
mechanical. (“semi-classical”)• Gets more accurate for a larger number of
particles• Gives accurate atomic binding energies• Relativistic versions can be formulated• TF model better than “bag models”, which do not
include the Coulomb interactions, for large numbers of particles
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Atomic TF model basics
particle density :n(r)=2
d3p(2πh)3
pF
∫ =1
π 2h3
pF3
3
KE density : 2
d 3p
(2πh)3
pF
∫pF
2
2m=
1π 2h3
110m
(3π 2h3n(r))5 /3
E = d3r
1π 2h3
110m
(3π 2h3n(r))5 /3 −Ze2n(r)
r+
12
d3r '∫e2n(r)n(r ')|rr −
rr '|
⎡
⎣⎢
⎤
⎦⎥∫
Fix : d 3r∫ n(r) =Z (neutral systems)
r =ax, a=a0
Z1/3
12
3π4
⎛⎝⎜
⎞⎠⎟
2 /3
, f(x) =xa0aZ
(3π 2 )2 /3
2(n(x))2 /3
h2
me2=a0
⎛
⎝⎜⎞
⎠⎟
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~144
x3
E =−20.9 Z7 /3eV
d 2 f (x)
dx2=
( f(x))2 /3
x
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Some TF model quark equations
f (r)=ra
2 ×43αs
3π 2n(r)( )2 /3
a =
hmc
r =Rx
d 2 f (x)
dx2=−N f
( f(x))2 /3
x
Fix : d 3r∫ n(r) =A / 3
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Unconfined NRTF Quark Model, f(x)
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Unconfined NRTF Quark Model, density
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Unconfined NRTF Quark Model, potential
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Unconfined NRTF Quark Model, 2 degenerate flavors
N1:N2=2:1 x2=1.67188; x1=3.41003
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Unconfined NRTF Quark Model, 2 degenerate flavors, different number ratios
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Unconfined NRTF Quark Model, 2 different mass flavors
Unequal mass case: N1:N2=2:1 m2:m1=2:1, x2=0.99430, x1=3.18200
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E =−97
43
g2A
rmax
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟; A : Baryonnumber
rmax ~ A2 /3
E ~ −A1/3 No!!
Introduce a bag energy term ~ B V
⇒
Realization
=B4π
3rmax3
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Confined NRTF Quark Model, f(x)
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Confined NRTF Quark Model, f(xmax) vs. xmax.
xmax ~ A−1/3
E ~ A
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Confined ultraRTF Quark Model, w(x)
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Confined ultraRTF Quark Model, density
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Confined ultraRTF Quark Model, potential
rmax ~ A1/3
E ~ A
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Unconfined ultraRTF Massive Gluon Quark Model, w(r)
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Model can be extended:
• To include nonzero mass quarks in relativistic case
• To include anti-quarks
• To include heavy quarks
• To have spin-dependent forces
• To look at exotic forms of matter (eg., “strange” matter, color-flavor locked quarks from Cooper pairs, or massive gluon models, or combinations)
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TF Quark Model Goals:
• To survey the parameter space looking for relative stability and connections to known phenomenology
• To shed light on the question: Are there states of many quarks and/or anti-quarks?
• To prepare the way for detailed lattice calculations
• First step: Confined NR model
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Can show that E ~ A for large A.
E
A
TF Quark Model Energies
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Baylor paper:Thomas-Fermi quark model: Nucl.Phys.A826:49-73,2009.