Recent Research on the Thomas

Fermi Quark Model

• Work with Andy Liu and Nate Larson!

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.

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

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

⎛

⎝⎜⎞

⎠⎟

~144

x3

E =−20.9 Z7 /3eV

d 2 f (x)

dx2=

( f(x))2 /3

x

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

Unconfined NRTF Quark Model, f(x)

Unconfined NRTF Quark Model, density

Unconfined NRTF Quark Model, potential

Unconfined NRTF Quark Model, 2 degenerate flavors

N1:N2=2:1 x2=1.67188; x1=3.41003

Unconfined NRTF Quark Model, 2 degenerate flavors, different number ratios

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

E =−97

43

g2A

rmax

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟; A : Baryonnumber

rmax ~ A2 /3

E ~ −A1/3 No!!

Introduce a bag energy term ~ B V

⇒

Realization

=B4π

3rmax3

Confined NRTF Quark Model, f(x)

Confined NRTF Quark Model, f(xmax) vs. xmax.

xmax ~ A−1/3

E ~ A

Confined ultraRTF Quark Model, w(x)

Confined ultraRTF Quark Model, density

Confined ultraRTF Quark Model, potential

rmax ~ A1/3

E ~ A

Unconfined ultraRTF Massive Gluon Quark Model, w(r)

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)

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

Can show that E ~ A for large A.

E

A

TF Quark Model Energies

Baylor paper:Thomas-Fermi quark model: Nucl.Phys.A826:49-73,2009.