1	Two Nucleon Solitary Wave Exchange Potentials (NN SWEPs) Mesgun Sebhatu Winthrop University Rock Hill, SC 29733 sebhatum@winthrop.edu
2	The Major Goals of Physics Identify the ultimate constituents of matter. Discover the fundamental forces through which the basic entities interact. Unify the fundamental forces and come up with a theory of everything.
3	Ingredients: 6 Quarks & 6 Leptons Spices: Four or Three Forces Ref: Particle Adventures The Standard Model Module m305.pdf
4	Electricity
5	Unification of Forces
6	NN Interaction Historical background : The Yukawa Potential and OPEP (1935) Phenomenological potentials: e.g. The Reid Soft-core potential(1968) One Boson Exchange Potentials (OBEPs ): e.g. Bonn Potential (1970 – Present) QCD and/or Effective field theory inspired potentials ( present) Solitary Wave Exchange Potentials (SWEPs): e.g. lF⁴ SWEP and SG SWEP (1975-?)
7	Yukawa Potential
8	A Typical OBEP
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10	Nonlinear Generalizations of the Klein Gordon Equation The field equations for spin-zero meson fields used in the derivation of SWEPs are nonlinear generalization of the well known Klein-Gordon equation. They are of the form¹:
11	The lF⁴ and sine-Gordon Field Equation The simplest examples are: J = lF⁴ which leads to the cubic KG equation: ¶^m¶_mF + m²F +lF³ = 0 And J_sG = (sin lF) m²/l-m²F which leads to the sine-Gordon equation:
12	L. Jäde, H. V. Geramb, M. Sander (Hamburg U) P.B. Burt( Clemson U) and M. Sebhatu Winthrop U), Presented at Cologne, March 13-17,1995
13	SG Solitary Wave Solution with the KG solution as a base A Pair of Quantized Solitary Wave Solution for the SGE from which the SG SWEP is derived are :
14	SG Solitary Wave Solution as a Series Once the SG solution is expressed as a tan^-1 series (as shown below).
15	The SG Propagator
16	Animation of a 2^nd Order Feynman Diagram
17	George Gamow’s Cartoon for Meson Exchange
18	Derivation of the SG SWEP The lowest order NN interaction is represented by the 2^nd order Feynman diagrams shown below. Using Feynman rules an expression for an NN scattering amplitude is written down. [See e.g. Bjorken and Drell, Relativistic Quantum Mechanics (1964) ] The only change is that the Feynman propagator is replaced by the SG propagator.
19	SG SWEP IN MOMENTUM SPACE The momentum space SG SWEP obtained from the diagrams shown earlier with leading non static terms is³:
20	NONSTSTIC SG SWEP IN COORDINAE SPACE
21	Terms and Variables in SG SWEP In general, V_NN(x) = V_C + V_T+ V_LS + V_LL
22	Modified Bessel Functions SWEPs yield good results with just the leading four terms n=0,1,2,3, &4
23	N-N STATES L= O, 1, 2, 3, 4, 5,… L = S, P, D, F, G, H,… J = L+S; S= O or 1 ^2S+1L_j is how NN states are specified When S = 0, 2S+1 =1, Singlet States When S =1, 2S+1= 3, triplet States L= 0, 2, 4, … Even & L= 1, 3, 5, … Odd States 1S0,1D2 ,1G4, …are leading even singlet states 1P1 , 1F3 , 1H5 , …are leading odd singlet states 3S1-3D1 is the most interesting example of a coupled triple state. It has the only bound NN State—the deuteron.
24	G. ¹S₀ SG SWEP For singlet NN states (S=0, T=1) S₁₂=0, V_LS=0 and <(t₁ ¢ t₂)(s₁ ¢ s₂)>=-3
25	SWEPs vs REID SC
26	NN Data Bases and References CNS @ George Washington U. CNS maintains the world data base for experimental NN etc. Phase shifts NN On-line from Netherlands They maintain NN Nijmegen Potentials, Phase shifts, Deuteron Properties. U of Hamburg from Germany They have potentials obtained by inverting experimental phase shifts. The have also greatly extended my work on SWEPs they call them One Solitary Boson Wave Exchange Potentials (OSBEPs). Some general references: M. Sebhatu and E. W. Gettys, A Least Squares Method for the Extraction of Phase Shifts, Computers in Physics 3(5), 65 (1989) Calculation of Phase Shifts
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28	SG SWEP PHASE SHIFTS
29	L. Jäde, H. V. Geramb, M. Sander (Hamburg), P,B. Burt (Clemson) and M. Sebhatu( Winthrop) Presented at Cologne, March 13-17,1995
30	³S₁ Phase Shifts
31	³S₁-³D₁Mixing Parameter (e₁)
32	³D₁ Phase Shifts
33	The Deuteron A Neutron-Proton Bound Sate A deuteron is is the only two-nucleon bound state. It is the simplest nucleus. It plays a role in nuclear physics that that resembles the role the hydrogen atom plays in atomic physics. It is a nucleus of deuterium (²₁H)--an isotope of hydrogen with an abundance of 0.015%. It consists of a proton and neutron with total spin S=1 (parallel), a total angular momentum J=1, and angular momentum ℓ =0 ,2. It is a coupled state of mostly ³S₁ and a small admixture of ³D_1. Using a novel algorithm by a Padua Group a coupled Schrödinger Equation (AKA Rarita-Schwinger Equation[RSE]) can be solved to obtain a pair of wave functions u(³S₁) and w(³D₁) as eigenvectors and the binding energy as an eigenvalue.
34	The Rarita-Schwinger Equation for a Deuteron—An ³S₁-³D₁ NN State
35	The Rarita-Schwinger equation can be rewritten as as a two-point boundary matrix eigenvlaue equation of the form A is matrix that contains the RS equation Y is an eigenvector that consists u(x) and w(x) λ= α²is an eigenvalue that contains the binding energy
36	The ³S₁-³D₁ state Reid Soft Core Potential is chosen for simplicity for testing the Padua Algorithm V(³S₁-³D₁)=V_c+S₁₂V_T+L.SV_LS
37	Central (V_C), Tensor (V_T) and Spin-Orbit (V_LS) Reid Soft Core Potentials
38	The ODE describing the deuteron (RSE) is now a two-point boundary eigenvalue matrix equation
39	The key element in the Padua Algorithm the transformation t=tan^-1(x). Since tan(π/2)= ∞. This transforms r =∞ to t= π/2 i.e., 0<x<∞ is transformed to 0<t<π/2
40	The RS Equation in Difference Equation Form After the 1st and 2nd derivatives in the RSE are replaced by central difference approximations. h=Δt The RSE can be cast into the standard form
41	The A(2Nx2N) Matrix
42	The Matrix Elements a(n,n) of the square matrix A of order 2N are zero except the following:
43	The eigenvectors u(nh) and w(Nh) are:
44	A is an 2N by 2N Matrix For the purposes of this presentation a short FORTRN 90 program to fill up the matrix elements of the A matrix was written.
45	Deuteron wave functions u(x) and w(x) form Reid Soft Core NN potential …
46	Deuteron wave functions u(x) and w(x) The dashed line are from Reid SC and the solid lines form SG-SWEP
47	Deuteron Parameters
48	Deuteron Properties
49	Concluding Remarks
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