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.  4 SWEP and SG SWEP (1975-?)

3. A Typical OBEP Model Credit: Lars E. Engvik

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 form 1 : J is the meson filed self interaction current. I (i=1…n) are self interaction coupling constants. 1. Reference: P.B. Burt, Quantum Mechanics and Nonlinear Waves( Harwood Academic, N.Y. 1981) @ ¹ @ ¹ © + m 2 © + J ( ¸ i ; © ) = 0

5. The  4 and sine-Gordon Field Equation The simplest examples are: J =  4 which leads to the cubic KG equation:      + m 2  +  3 = 0 And J sG = (sin  ) m 2 / -m 2  which leads to the sine-Gordon equation: @ ¹ @ ¹ © + m 2 ¸ s i n ¸ © = 0 When approaches zero they both reduce to the usual linear KG equation:     +m 2  =0

6. L. Jade, H. V. Geramb, M. Sander (Hamburg U) P.B. Burt( Clemson U) and M. Sebhatu Winthrop U), Presented at Cologne, March 13-17,1995Hamburg UClemson UWinthrop U

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 : © ( § ) = 4 ¸ t an ¡ 1 · ¸ 4 Á ( § ) ¸ : This solution can be obtained by direct integration and/or by the Method of Base Equations developed by Burt and Reid 1 Á ( § ) = A ( ¨ ) k e § · K ¢ · X ( D k ! k ) 1 = 2

8. SG Solitary Wave Solution as a Series Once the SG solution is expressed as a tan -1 series (as shown below). © ( § ) = 4 ¸ N X n = 0 ( ¡ 1 ) n 2 n + 1 · ¸Á ( § ) 4 ¸ 2 n + 1 A propagtor is constructed in analogy with the the procedure followed in the linear (free) field theory. See e.g., Bjroken and Drell

9. The SG Propagator  = /16m, M n = (2n+1)m and  F =[K 2 -M n + i  ) -1 is the usual Feynman Propagator in momentum space. The SG poropagator like other solitary wave propagators is essentially a superposition of Feynman propagators it has, however, the advantage of automatically exchanging a series of massive bosons with a minimum of parameters. The sequence of bosons lead to superposition of attractive as well as repulsive Yukawa and exponential potential terms in configuration (position) space 2.series of massive 2 M. Sebhatu, Nuovo Cimento A33, 508(1976);Lett. Nuovo Cim. 16, 463(1976); Lett. Nuovo Cim. 36, 513 (1986); nuc-th9409015 nuc-th9409015 P s G ( K 2 ; M 2 n ) = N X n = 0 [ m ¯ ] 2 n ( 2 n + 1 ) ! ( 2 n + 1 ) 2 n ¡ 4 K 2 + M 2 n ¢ F ( K 2 ; M 2 n )

10. Animation of a 2 nd Order Feynman Diagram Credit: J. Eric SloaneJ. Eric Sloane

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 (1964)] The only change is that the Feynman propagator is replaced by the SG propagator. n p  n p  n’ P’ n’ +

12. SG SWEP IN MOMENTUM SPACE The momentum space SG SWEP obtained from the diagrams shown earlier with leading non static terms is 3 : 3 nuc-th9409015nuc-th9409015

13. NONSTSTIC SG SWEP IN COORDINAE SPACE V ( X n ) = G ( ¿ 1 ¢ ¿ 2 ) r 2 ¼ N X n = 0 C n X n n [( ¾ 1 ¢ ¾ 2 ) V C ( X n ) + S 12 V T ( X n ) + ( 2 ­ ¡ 1 ) L 12 V `` ( X n )]( 1 )

14. Terms and Variables in SG SWEP In general, V NN (x) = V C + V T + V LS + V LL The SG SWEP is missing the V LS. Vector or scalar mesons can provide the V LS. I plan to include the  and the  meson in a future work.

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+1 L 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 1 S 0, 1 D 2, 1 G 4, …are leading even singlet states 1 P1, 1 F 3, 1 H 5, …are leading odd singlet states 3 S 1 - 3 D 1 is 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 See e.g Arfken, Math Methods for Physicists

17. G. 1 S 0 SG SWEP For singlet NN states (S=0, T=1) S 12 =0, V LS =0 and =-3 nuc-th9409015 3. L 12 = ¡ 2 ` ( ` + 1 ) ; f or ` = 0 ; L 12 = 0

18. SWEPs vs REID SC

19. NN Data Bases and References CNS @ George Washington U.CNS CNS maintains the world data base for experimental NN etc. Phase shifts NN On-line from NetherlandsNN On-line They maintain NN Nijmegen Potentials, Phase shifts, Deuteron Properties. U of Hamburg from GermanyU of Hamburg 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). One Solitary Boson Wave Exchange Potentials (OSBEPs). Some basic undergraduate level 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)Computers in Physics

20. SG SWEP PHASE SHIFTS

21. L. Jade, H. V. Geramb, M. Sander (Hamburg), P,B. Burt (Clemson) and M. Sebhatu( Winthrop) Presented at Cologne, March 13-17,1995. Jade, H. V. Geramb, M. SanderHamburgClemsonWinthrop

22. 3 S 1 Phase Shifts

23. 3 S 1 - 3 D 1 Mixing Parameter (  1 )

24. 3 D 1 Phase Shifts

25. Deuteron Wave Functions U(x) & W(x) Reference

26. Concluding Remarks I hope this presentation has demonstrated that realistic potentials ( with a minimum number of parameters —three or less) can be derived using nonlinear pion meson fields. The SG SWEP was used as an example. However, most people prefer the  4 SWEP. It yields almost identical results as the SG SWEP. More general nonlinear extensions to the Klein Gordon equation of the form 1  2p+1 + 2  4p+1 exist. (See Burt’s webpage and references there in).Burt’s Burt has found a simple mass formula for pseudoscalar mesons : M n = (3n+1)m  n= 0, 1,2, 3…, m  is the pion mass. This formula is obtained from a propagator based on the equation      +m 2  + 1  5 + 2  7 =0. It will be interesting to derive and test a corresponding SWEP. It may also be necessary to incorporate  and  mesons. Vector meson can contribute the missing V LS term. The  meson can also weaken the pion tensor contribution which is too strong as it is now. The Hamburg nuclear theory group has done all these and more. However, they also include two fictitious  mesons. This may not be necessary. With an appropriate choice of the self interaction coupling constants, SWEPs can provide sufficient intermediate attraction. Fictitious one or more  mesons are used to simulate multi-meson exchanges in OBEPs. The Hamburg nuclear theory Once expressions for SWEPs are derived, it is possible to involve undergraduate students in the physical science to do most of the work.

27. 1 S 0 SG SWEP vs INVERSE V

28. Deuteron Wave Functions

29. 1 D 2 SG SWEPS vs INVERSE V