1	Two Nucleon Solitary Wave Exchange Potentials (NN SWEPs) Mesgun Sebhatu Winthrop University Rock Hill, SC 29733 sebhatum@winthrop.edu
2	INTRODUCTION 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-?)
3	A Typical OBEP Model
4	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:
5	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:
6	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
7	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 :
8	SG Solitary Wave Solution as a Series Once the SG solution is expressed as a tan^-1 series (as shown below).
9	The SG Propagator
10	Animation of a 2^nd Order Feynman Diagram
11	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.
12	SG SWEP IN MOMENTUM SPACE The momentum space SG SWEP obtained from the diagrams shown earlier with leading non static terms is³:
13	NONSTSTIC SG SWEP IN COORDINAE SPACE
14	Terms and Variables in SG SWEP In general, V_NN(x) = V_C + V_T+ V_LS + V_LL
15	N-N STATES L= O, 1, 2, 3, 4, 5,… = S, P, D, F, G, H,… J = L+S; S= O or 1 ^2S+1L_J When S =0, 2S+1 =1, Singlet States When S=1, 2S+1=3, triplet States L= 0, 2, 4, … Even States L= 1, 3, 5, … Odd States ¹S₀, ¹D₂, ¹G₄, …are leading even singlet states ¹P1 , ¹F₃ , ¹H₅ , …are leading odd singlet states ³S₁-³D₁is the most interesting example of a coupled triple state. It has the only bound NN State—the deuteron.
16	Modified Bessel Functions SWEPs yield good results with just the leading four terms n=0,1,2,3, &4
17	G. ¹S₀ SG SWEP For singlet NN states (S=0, T=1) S₁₂=0, V_LS=0 and <(t₁ ¢ t₂)(s₁ ¢ s₂)>=-3
18	SWEPs vs REID SC
19	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: Derivation of OPEP Radial Schrödinger equation and Phase Shifts Deuteron Wave Functions and Properties M. Sebhatu and E. W. Gettys, A Least Squares Method for the Extraction of Phase Shifts, Computers in Physics 3(5), 65 (1989)
20	SG SWEP PHASE SHIFTS
21	L. Jäde, H. V. Geramb, M. Sander (Hamburg), P,B. Burt (Clemson) and M. Sebhatu( Winthrop) Presented at Cologne, March 13-17,1995
22	³S₁ Phase Shifts
23	³S₁-³D₁Mixing Parameter (e₁)
24	³D₁ Phase Shifts
25	Deuteron Wave Functions U(x) & W(x)
26	Deuteron Properties
27	Concluding Remarks