Two Nucleon
Solitary Wave Exchange
Potentials (NN SWEPs)
Mesgun Sebhatu
Winthrop University
Rock Hill, SC 29733
sebhatum@winthrop.edu
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-?)
A Typical OBEP Model
Credit: Lars E. Engvik
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:
¹
2
@ @¹© + m © + J(¸i; ©) = 0
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)
The 4 and sine-Gordon
Field Equation
•
•
•
•
The simplest examples are:
J = 4 which leads to the cubic KG equation:
 + m2 +3 = 0
And JsG = (sin ) m2/-m2 
• which leads to the sine-Gordon equation:
2
m
@¹ @ ¹ © +
sin ¸© = 0
¸
When
 approaches
linear KG equation:
zero they both reduce to the usual
+m2=0
L. Jade, H. V. Geramb, M. Sander (Hamburg U)
P.B. Burt( Clemson U) and M. Sebhatu Winthrop U),
Presented at Cologne, March 13-17,1995
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
¡1 ¸ (§)
:
= tan
Á
4
¸
This solution can be obtained by direct integration and/or by
the
Method of Base Equations developed
§ ·by
¢ ·Burt and Reid1
(§)
Á
(¨ )
= Ak
e KX
(Dk !k )1=2
SG Solitary Wave Solution
as a Series
• Once the SG solution is expressed as
· below).
¸
a tan-1 series (as shown
§)
(
©
4
=
¸
X
N
n=0
§)
¡
(
n
( 1)
¸Á
2n + 1
4
2n+1
A propagtor is constructed in analogy with
the the procedure followed in the linear (free)
field theory. See e.g., Bjroken and Drell
The SGXPropagator
N
PsG(K2; Mn2) =
¡4
2n
(2n
+
1)!(2n
+
1)
[m¯]2n
K2 + Mn2
¢F (K2; Mn2)
n=0
 = /16m, Mn = (2n+1)m and F=[K2-Mn+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) space2.
M. Sebhatu, Nuovo Cimento A33, 508(1976);Lett. Nuovo Cim. 16,
463(1976); Lett. Nuovo Cim. 36, 513 (1986); nuc-th9409015
2
Animation of a 2nd Order Feynman Diagram
Credit: J. Eric Sloane
Derivation of the SG SWEP
The lowest order NN interaction is represented by the
2nd 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.
P’
n’
+

n
n’
P’
p

n
p
SG SWEP IN MOMENTUM SPACE
The momentum space SG SWEP obtained from the
diagrams shown earlier with leading non static
terms is3:
V (k; q)
=
h
g2 m
½·
4¼ 2M
i
2
(¿1 ¢ ¿2 )PsG (k 2 ; Mn2 )
¸·
¸
¢ k)(¾ ¢ k)
2
k
(¾
1
2
1¡
4M 2
M2
2 ¡1
¢ k £ q)(¾ ¢ k £ q)
+
[(¾
1
2
2(M m)2
¤ª
¡ (¾ ¢ ¾ )(k £ q)2 (1)
1
2
3 nuc-th9409015
NONSTSTIC SG SWEP
IN COORDINAE SPACE
r
2
¼
V (Xn ) = G(¿1 ¢ ¿2 )
X
N
Cn Xnn
n=0
[(¾1 ¢ ¾2 )VC (Xn )
+S12 VT (Xn ) + (2 ¡ 1)L12 V`` (Xn )]
Where :
h
g2 m
G=
4¼ 2M
i
(2n)!¯ 2n
m; Cn =
;
n!2n
¡
¡
VC (Xn ) = Xn 1=2 Kn¡5=2 (Xn ) ¡ 3Xn 3=2 Kn¡3=2 (Xn );
2
¡
L12
VT (Xn ) = Xn 1=2 Kn¡5=2 (Xn );
h i
1 m 2 VT (Xn )
V`` =
;
2 M
x2
= (¾1 ¢ ¾2 )L2 ¡ [¾1 ¢ L ¾2 ¢ L + ¾2 ¢ L ¾1 ¢ L]=2:
(
Terms and Variables in SG SWEP
In general, VNN(x) = VC + VT+ VLS + VLL
In the SG SWEP shown earlier:
Xn = (2n + 1)x, x = m¼ r;
VC , VT , and V``
are central, tensor, and quadratic spin-orbit
potential components respectively.
The SG SWEP is missing the VLS. Vector or scalar mesons
can provide the VLS. I plan to include the  and the 
meson in a future work.
N-N STATES
• L= O, 1, 2, 3, 4, 5,…
• = S, P, D, F, G, H,…
• J = L+S; S= O or 1
• 2S+1LJ
•
•
•
•
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
1S , 1D , 1G , …are leading even singlet states
0
2
4
1P1 , 1F , 1H , …are leading odd singlet states
3
5
3S -3D is the most interesting example of a
1
1
coupled triple state. It has the only bound
NN State—the deuteron.
Modified Bessel Functions
SWEPs yield good results with just the
leading four terms n=0,1,2,3, &4
The Kº (z) are modi¯ed Bessel function
¤ only ones needed are:
When N · 4£ the
1=2 ¡z
¼
K§1=2 (z) = 2z
e ;
K§3=2 (z) = (1 + z1 )K§1=2 (z);
and K§5=2 (z) = z3 K§3=2 (z) + K§1=2 (z)
See e.g Arfken, Math Methods for Physicists
G.
1S
0
SG SWEP
• For singlet NN states (S=0, T=1) S12=0, VLS=0 and
<(1 ¢ 2)(1 ¢ 2)>=-3
L12 = ¡2`(` + 1); f or` = 0; L12 = 0
e.g,an explicit expression for the 1 S 0 state SG SWEP
with the ¯rst ¯ve (n = 0 to 4)terms is shown below:
V C (x)
=
¡G
·
e
¡x
x
+ ¯ 2 (3x ¡ 2)
e
¡ 3x
3x
¡
+ 3¯ 4 (5x ¡ 3)e 5x
¡ 7x
¡
6
2 ¡
21x 3)e
+ 15¯ (49x
¡
+ 945¯ 8 (81x 3 ¡ 18x2 ¡ 9x ¡ 1)e 9x
nuc-th9409015
¤
SWEPs vs REID SC
NN Data Bases and References
•
CNS @ George Washington U.
•
NN On-line from Netherlands
•
U of Hamburg from Germany
•
CNS maintains the world data base for experimental NN etc. Phase shifts
•
They maintain NN Nijmegen Potentials, Phase shifts, Deuteron
Properties.
•
•
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 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)
SG SWEP PHASE SHIFTS
L. Jade, H. V. Geramb, M. Sander (Hamburg), P,B. Burt (Clemson) and
M. Sebhatu( Winthrop) Presented at Cologne, March 13-17,1995
3S
1
Phase Shifts
160
140
Phase Shift(Deg/)
120
100
3S1Th
80
3S1Ex
60
40
20
0
-20
0
100
200
300
E-Lab (MeV)
400
500
3S
3
1- D1 Mixing Parameter
(1)
3S1-3D1 Mixing Parameter E1
7
6
E1(Deg.)
5
E1Th
4
E1Exp
3
2
1
0
0
100
200
300
E_Lab(Mev)
400
500
3D
1
Phase Shifts
3D1 Phase Shifts
5
Phase Shift(Deg.)
0
-5 0
100
200
300
400
500
-10
-15
3D1Th
-20
3D1Ex
-25
-30
-35
-40
E-Lab(MeV)
Deuteron Wave Functions U(x) & W(x)
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 has found a simple mass formula for pseudoscalar mesons :
Mn = (3n+1)m n= 0, 1,2, 3… , m is the pion mass. This formula is obtained
from a propagator based on the equation +m2+15+2 7=0.
It will be interesting to derive and test a corresponding SWEP.
It may also be necessary to incorporate

can contribute the missing VLS term. The

and

mesons. Vector meson
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.
Once expressions for SWEPs are derived, it is possible to involve
undergraduate students in the physical science to do most of the work.