1. Cliffor Benjamín Compeán Jasso UASLP
Effective gluon propagator from a Fourier transform of the cot confinement potential
Cliffor Benjamín Compeán Jasso
UASLP
XII Mexican Workshop on Particles and Fields (November 2009)
Mazatlan, Sinaloa

2. Outline
Introduction
SO(4) symmetry in the baryonic spectra
Cornell potential and cot+csc2 potential
cot potential as angular function in S3 curved space
Transform from S3 to momentum space
Gluon instantaneous effective propagator
Summary

3. All radial potentials are SO(3)L
Description of light baryons in ground state wave function (QCD)
Spin (S) 
SU(2)S
Flavor (F)
u, d, s
SU(3)F
Spatial (L)
SO(3)L
Color (c)
R, B, G
SU(3)c
Focus: |L>.
All radial potentials are SO(3)L
Specifying the potential the symmetry grow:
Examples:
Harmonic Oscillator SU(3)NL, L=N,N-2, … 0 ó 1.
Coulomb SO(4)KL, L=K,K-1, … 0
Natanson SO(2,2), etc.
The symmetry of a potential define the degenerations of the spectra. Each potential show an specific spectra.

4. SU(3)F SU(2)S SO(3)L SU(3)F SU(2)S SO(4)KL
Description in light baryons spectra
SU(3)F
SU(2)S
SO(3)L
SU(3)F
SU(2)S
SO(4)KL
Flavor factorization
Lorentz group representations
(Ground state)
u, d, s
Mariana Kirchbach’s talk
SO(4)/SO(2,1)

5. The quantum numbers in each grouping is a representation of SO(4)KL.
Same degeneration and structure in N and D (isospin is factorized) K
S31(2150)
P33(----)
D35(2350)
F37(2390)
G39(2400)
H3,11(2420) 5
P31(----)
D33(----)
F35(2000)
G37(2200)
H39(2300)
K = Parity partners
S31(1900)
P33(1920)
D35(1930)
F37(1950) 3
P31(1910)
D33(1940)
F35(1905)
S31(1620)
D33(1700) D 1
P31(1750)
P33(1232) 1 2 3 4 5 l
Observation:
The quantum numbers in each grouping is a representation of SO(4)KL.
(K/2,K/2)((1/2,0)(0,1/2))

6. The quantum numbers in each grouping is a representation of SO(4)KL.
Same degeneration and structure in N and D (isospin is factorized) K
S11(2090)
P13(1900)
D15(2200)
F17(1990)
G19(2250)
H1,11(----) 5
P11(2100)
D13(2080)
F15(2000)
G17(2190)
H19(2220)
K = Parity partners
S11(1650)
P13(1720)
D15(1675)
F17(----) 3
P11(1710)
D13(1700)
F15(1680)
S11(1535)
D13(1520) 1
P11(1440)
Nucleon
P11(939) 1 2 3 4 5 l
Observation:
The quantum numbers in each grouping is a representation of SO(4)KL.
(K/2,K/2)((1/2,0)(0,1/2))

7. The idea is find a potential that reproduce the Nucleon and D spectra and the level separation that respect the SO(4) symmetry.

8. cot+csc2 potential
Cornell

9. -1/r r + r dp New -cot(r/d) Perturbative regime perturbative regime
Coulomb
(+lineal) +
Infinite well
Perturbative
regime
-1/r
-cot(r/d)
perturbative regime
Non r
Gluon flux tube (strings) +
New r dp
Infinite well (finite range confinement scenario)

10. VS Our model 3 parameters Others >30 parameters
Wave functions: Close form
Others
>30 parameters
Wave functions: Numerical VS

11. = + + Baryon = 3 q + (3q + (qq)n) + (3q + gn)
QCD
Baryon =
3 q +
(3q + (qq)n) +
(3q + gn)
A baryon is a many body system.
Many body system can be described in terms of two effective grades of freedom (two body) in a curved space.
Our conjecture: The interactions between the grades of freedom of QCD (q-g, q-q, g-g, …) in a baryon is changed by a q-qq system on a S3 curved space which has a cot kind interaction. In this way, the curvature codify the QCD configurations.

12. The Nucleon and D spectra was described in term of the SO(4) symmetry.
There was described the degeneration and level separations.
Cot potential reproduce the Nucleon and D spectra degeneration and the level separation and respect the SO(4) symmetry.

13. Mathematical aspects of S3
Our system is in E4 on the hyper-sphere, S3, of constant radius, R.  R
: Second polar angle S3
Parameterization

14. Schrödinger equation on S3
Angular momentum operator on E4
Energy levels
Wave functions
(close form)

15. Wave functions on S3
|| c q

16. Parameterizations of  in terms of r… (change of variable)
… for the purpose of obtaining the spatial part of the wave function (ordinary position space) we need c(r) depending on S3 parameterization
Quantum dots
SUSY-QM
Robertson-Walker
 = arcsin r/R
 = arctan r/R
 = r/R
0  r/R  1
0  r/R < 
0  r/R  p E4 r E3 r E4 E4    R R x4 R r E3 S3 S3 S3
All the spectra are same. But different phenomena.
………..

17. Cot potential under the Robertson-Walker parameterizations
Possibility for position dependent masses
Cot potential under the Robertson-Walker parameterizations
 = arcsin r/R
V() = -2Bcot c
= -2B x4/r
0  r/R  1
 = arcsin r/R
Warning: Position dependent masses appear if one wants to work in r space

18. Deconfinement From “curved” cot potential to “flat” Coulomb potential
k = 1/R2
Curvature parameter lets the description of deconfinement.
Binding states (flat space), came from the limit
Dispersion states (flat space)
(K+1)2 k  k2/h for

19. Robertson-Walker integration volume
 = arcsin r/R
0  r/R  1 E4 
E3 integration volume x4 R r E3 S3

20. QCD - Physics in momentum space
Fourier transform
QCD - Physics in momentum space E4 
Position Flat Space
Momentum Flat Space x4 R r E3 E3 E3 S3

21. Transformation from S3 to momentum space
Transform f(r) to momentum space
Inverse transform
S3 - Robertson-Walker E3

22. Instantaneous effective propagator
Instantaneous photon propagator
Transition amplitude
Virtual photon Propagator
Instantaneous Coulomb Interaction
Born approximation on S3 (to the amount it is acceptable) 22

23. Instantaneous effective gluon propagator
Needed for three body Faddeev calculations
cot potential on Robertson-Walker parameterization
Born Approximation

24. Instantaneous effective gluon propagator
Finite value q2=0
(QCD requirement for confinement)

25. Summary
Cot potential reproduce the Nucleon and D mass spectra and the level separation. Such potential respect the SO(4) symmetry.
The dynamical properties of cot-csc2 potential was analyzed and its interpretation was justify as the exactly solvable extension of Cornell potential.
The quark and gluon dynamics of quarks and gluons of QCD was related with our potential interacting with the cot potential.
In order to relate the curved space on S3 with the position space there was study the Robetson-Walker parameterization.
The transform of a function from S3 to momentum space was given.
The instantaneous effective gluon propagator was build and it was taken as Bohr amplitude. The instantaneous effective gluon propagator grow in the infrared region but is finite on the infrared region in concordance with lasted Lattice QCD predictions.

26. Proton electric form factor (E3)
Dipole
Our case, GE is equal to F1 because F2=0, non-spin interaction (technical problem but non-conceptual problem) 26

27. Proton electric form factor (S3)27

28. Proton electric form factor (Our vs experiment)
Green line: Predicted with the nucleon spectra parameters (mq=1/3 MN)
Blue line: Calculus obtained with the proton charge radii <r2> 28

29. Electric form factors of excited states
n = 1
l = 0
n = 2
l = 0
n = 2
l = 1
n = 3
l = 2
n = 3
l = 0
n = 3
l = 1

30. Curvature dependence of the proton electric charge radius
k1=0.104 fm-2 calculated to <r2>=0.87 fm
k2 =0.187 fm-2 adjust to the spectra
Proton mean square charge radius

31. Summary (applications)
We calculated the proton electric form factor which is in concordance with experimental data.
We has electric form factors of excited states.
Proton charge radii <r^2>1/2 is a function of curvature k from the hyper-sphere S3.

32. Tank you