Nucleon-Nucleon Interaction: A Typical/Concise Review

Abstract

Nearly a recent century of work is allocated to Nucleon-Nucleon (NN)
interaction issue. We review some overall perspectives of NN interaction with a
brief discussion about deuteron, general structure and symmetries of the NN
Lagrangian as well as the equations of motion and solutions. The main NN
interaction models, as the frameworks to build the NN potentials, are reviewed
concisely. We try to include and study almost all well-known potentials in a
similar way, discuss more on the various commonly used plain forms of
two-nucleon interaction with an emphasis on the almost phenomenological and
meson-exchange potentials as well as the new ones based on chiral effective
field theory and working in coordinate-space mostly. The potentials are
constructed in a way that fit NN scattering data, phase shifts, and are
compared in this way usually. An aim of this study is to start comparing
various potentials forms in a unified manner. So, we also comment on the
advantages and disadvantages of the models and potentials partly with citing
some relevant works and probable future studies.

Full text

arXiv:nucl-th/0702078v4 28 Mar 2014
Nucleon-Nucleon Interaction:
A Typical/Concise Review
M. Naghdi
∗
Department of Physics, Faculty of Basic Scie nces,
University of Ilam, Ilam, West of Iran.
(Last Revised: March 19, 2014)
Abstract
Nearly a recent century of work is divided to Nucleon-Nucleon (NN) interaction issue. We
review some overall perspectives of NN interaction with a brief discussion about deuteron, g en-
eral structure and symmetries of NN Lagrangian as well as equations of mo t ion and solutions.
Meanwhile, the main NN interaction models, as frameworks to build NN potentials, are re-
viewed concisely. We try to include and study almost all well-known potentials in a similar way,
discuss more on various commonly used plain forms for two-nucleon interaction with an empha-
sis on the phenomenological and meson-exchange potentials as well as the constituent- quark
potentials and new ones based on chiral effective field theory and working in coordinate-space
mostly. The potentials are constructed in a way that fit NN scattering data, phase shifts,
and are also compared in this way usually. An extra goal of this study is to start comparing
various potentials f orms in a unified manner. So, we also comment on t he advantages and
disadvantages of the models and potentials partly with reference to some relevant works and
probable future studies.
∗
E-Mail: m.naghdi@mail.ilam.ac.ir

Cont ents
1 Int roduction 3
2 A Brief of Nucleon-Nucleon Interaction 5
2.1 Three Interaction Parts in Two-Nucleon Systems . . . . . . . . . . . . . . . . 5
2.2 Deuteron: The Sole Bound-State of Two-Nucleon Systems . . . . . . . . . . . 5
2.3 General Symmetries of Two-Nucleon Hamiltonian . . . . . . . . . . . . . . . . 6
2.4 More About NN Intera ctio n . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4.1 Potential Forms, Equations of Motion, and Wave Functions in r-Space . 7
2.4.2 Scattering-Length and Effective-Range . . . . . . . . . . . . . . . . . . 9
2.4.3 P-Space, Relativistic Scattering and so on . . . . . . . . . . . . . . . . 10
3 Nucleon-Nucleon Interaction Models 11
3.1 Almost Full Phenomenological Models . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Boson Exchange Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3 The Models Based on QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.4 Effective Field Theory Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4 Nucleon-Nucleon Interaction Potentials 16
4.1 Basic Potentials and General Remarks . . . . . . . . . . . . . . . . . . . . . . 16
4.2 NN Potential’s Road . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.3 Hamada-Johnston Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.4 Yale-Group Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.5 Reid68 and Reid-Day Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.5.1 Reid68 Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.5.2 Reid-Day Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.6 Partovi-Lomon Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.7 Paris-Group Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.8 Stony-Brook Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.9 dTRS Super-Soft-Core Potentials . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.10 Funabashi Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.11 Urbana-Group Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.12 Argonne-Group Pot entials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.12.1 ArgonneV14 and ArgonneV28 Potentials . . . . . . . . . . . . . . . . . 29
4.12.2 ArgonneV18 Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.13 Bonn-Group Pot entials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.13.1 Full-Bonn Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.13.2 CD-Bonn Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.14 Padua-Group Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.15 Nij megen-Group Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.15.1 The First Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2

4.15.2 Nijm78 Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.15.3 Nijmegen Partia l-Wave-Analysis . . . . . . . . . . . . . . . . . . . . . . 41
4.15.4 Nijm93, NijmI and NijmII Potentials . . . . . . . . . . . . . . . . . . . 43
4.15.5 Reid93 Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.15.6 Extended Soft-Core Potentials . . . . . . . . . . . . . . . . . . . . . . . 47
4.15.7 Nijmegen Optical Potentials . . . . . . . . . . . . . . . . . . . . . . . . 48
4.16 Hamburg-Group Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.17 Moscow-Group Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.18 Buda pest(IS)-Group Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.19 MIK-Group Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.20 Imaginary Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.21 QCD-Inspired Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.22 The Oxford Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.23 The Fir st CHPT NN Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.24 Sao Paulo-Group CHPT Potentials . . . . . . . . . . . . . . . . . . . . . . . . 59
4.25 Munich-Group CHPT Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.26 Idaho-Group CHPT Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.27 Bochum-Julich-Group CHPT Potentials . . . . . . . . . . . . . . . . . . . . . 63
4.27.1 LO, NLO and NNLO Po tentials . . . . . . . . . . . . . . . . . . . . . . 63
4.27.2 NNNLO Po t entials and More . . . . . . . . . . . . . . . . . . . . . . . 64
5 Some Other Models and Potent ials 66
6 Outlook 68
1 Introduc tion
In 1953, Bethe stated [1] that in the quarter of the current century, many experiments, labor
and mental wo r ks is allocated to the Nucleon-Nucleon (NN) problem; probably more than any
other question in the history of humankind. NN interaction is the most fundamental problem
in nuclear physics yet. In fact, since the discovery of neutron by Chadwick in 1932, the subject
has been in t he focus of attention; as, at the first, ”nuclear physics” was often equal to ”nuclear
force”. The reasons for this outstanding ro le are clear. The main reason is that describing the
atomic-nuclei properties in terms o f the interactions between the nucleon pairs is indeed the
main goal of nuclear physics.
In nuclear structure studies, ”nucleons” are always considered as ”fundamental” objects, which
is of course reasonable in the scale of nuclear physics with MeV energies. Although by t he com-
ing of Quantum Chromo Dynamics (QCD), it is established that nucleons are no t fundamental,
but by comparing the results from this traditional approach with the more fundamenta l ones,
one may still understand better the advantages and disadvantag es of t he approaches. NN
interaction is nowadays known more than any other parts of strong int era ctio n both because
of long-term researches (more than 80 years) and many experimental data as well as improved
3

theoretical understanding of its various aspects.
The oldest theory of nuclear forces was presented by Yukawa [2] based on which t he mesons me-
diate the NN (pp, pn, nn) int era ctio ns. Again, although the meson theory is not fundamental
in the view of QCD, t he meson-exchange approach has improved our understanding of nuclear
forces besides giving some good qualitative results. Still, the mesons need in today’s standard
NN models/potentials, with the quarks and gluons, is avoidable to describe well many nuclear
interactions and to build better models/potentials with mo r e satisfactory results. In fa ct, by
the advent of Effective Field Theory (EFT) and applying it to t he low-energy QCD, we are
somehow coming back to the meson-exchange theories with the aid of Chiral Perturbat ion
Theory (CHPT).
Most basic questions were settled in the 1960’s - 1990’s. In recent years, the focuses are on
the subtleties and various extensions of the idea for this special f orce leading to setting up
more sophisticated two- and few-nucleon potentials. As a result, various high-quality models
and forms for NN interaction are present nowadays. According to this, we can absolutely not
address all on this rich and long-lived subject here but some basic f acts and impor t ant issues
of our favorite of course. By the way, we will discuss various potentials in mo r e details in tha t
one may intend to study and compare them in future studies–For some general and up-to-dat e
views to the subject, look, for instance, at [3], [4] and [5].
This note is organized as follows. In Section 2, we briefly discuss some basics of NN
interaction, deuteron as the unique bound-state of two-nucleon systems, the symmetries of
two-nucleon Lag rangian, general forms of NN potentials in configuration/coordinate-space (r-
space from now on), equations of motion and partial-wave analysis. There, we also present
a brief view to the scattering-length, effective-range and moment um space (p-space from now
on) formalisms as well as relativistic NN scattering. In Section 3, we review the f our main
NN interaction models qualitatively. There are the Phenomenological models with many free
parameters to be fitted to experimental NN data, the Boson-Exchange models based on the
field-theoretical and dispersion-relations methods, the QCD-inspired models based on the fun-
damental quarks and gluons degrees of freedom, and the models based on EFT by using the
chiral symmetry of QCD. As t here are many NN interaction models and potentials forms and
detailed studies need more times and places so, in Section 4, we try to review almost all-
important two-nucleon potentials together with addressing the original papers for technical
studies. There, we also mention the ro ad of modeling and improving exact NN potentials. In
addition, we study some high- precession potentials in more details as samples of the various
existing potentials to do further studies and comparisons in an almost common scheme. Next,
in Section 5, we mention few other models and potentials not mentioned in Section 4, which
are the Mean F ield Theory (MFT) methods and the Renormalization Group (RG) approaches
as well as the Lattice QCD techniques. Finally, in Section 6, we make few comments about
the current status and problems as well as the pro bable futures tries to be made on the rich
way of nuclear force studies.
4

2 A Brief o f Nucleon-Nucleon Interaction
One can estimate, with an introductory evaluation (e.g. by uncertainty principle) tha t two-
nucleon interaction has the greatest contribution to nuclear fo r ce and four- and few-body
interactions have almost negligible roles in most nuclear calculations.
In this section, we discuss some basics about NN interaction mainly in r-space a nd nonrelativis-
tic theory. The aim is to introduce the beginners with the subject by referring the interested
readers to the relevant textbooks and lecture notes for various technical and advanced studies.
2.1 Three Interaction Parts in Two-Nucleon Systems
Nucleon-Nucleon int era ctio n is always divided into three parts, first in [6], as follows:
a) The long-range (LR from now on) part (r & 2fm): In most models, it is considered as
One-Pion-Exchange Potential (OPEP) and is added to the other parts of the potential as a
tail. In a simple form in r-space, it reads
V
(1)
OP EP
(r) =
g
2
pi
3
(~τ
1
.~τ
2
)

e
−µr
r
(~σ
1
.~σ
2
) +

1 +
3
µr
+
3
(µr)
2

e
−µr
r
S
12

, (2.1)
where µ =
1
r
0
, r
0
=
~
m
pi
c
and S
12
= 3(~σ
1
.ˆr)(~σ
2
.ˆr) −(~σ
1
.~σ
2
) is the usual tensor operator; and g
pi
is the coupling constant, which is obtained from the exp eriments with mesons (meson-nucleon
scattering). This potential has earned some improvements such as considering the difference
between the neutral and charged pions and that it is different for p p, nn, np interactions be-
sides the clear forms raised from some new models of NN interaction.
b) The intermediate/medium (MR f rom now on)-range part (1fm . r . 2f m): It comes from
the va rious single-meson exchanges and mainly from the scalar-meson excha nges (two pions
and heavier mesons).
c) The short-range (SR from now on) part (r . 1f m): It is always given by exchanges of the
vector bosons (heavier mesons and multi-pion exchanges) as well as the QCD effects.
In some of the potential forms, various Feynman diagrams, depended on the considered ex-
changes, in each of the three mentioned parts, are used. A general scheme for NN potential is
shown in Figure 1.
2.2 Deuteron: The Sole Bound-State of Two-Nucleon Systems
One way to study the nuclear two-body interactions is using a two-nucleon system such as
deuteron (the
2
H nuclei). Detailed studies need a general system of two-nucleon, which is, in
turn, fra med through scattering a nucleon from another nucleon. Nevertheless, deuteron is still
fundamental to understand some basic properties of NN interaction. Deuteron is the exclusive
loosely bound-state of two-nucleon system. From the symmetry considerations,
3
S
1
and
3
D
1
5

Figure 1: A gen eral schem e for n ucleon - nucleon potential.
are its states. Its nonzero electric quadrupole-moment [7],
Q =
√
2
10
Z
∞
0
uwr
2
dr −
1
20
Z
∞
0
w
2
r
2
dr, (2.2)
confirms the presence of D-state and leads to introduce the tensor force. As a partial way to
measure the quality of a potential, one may insert the wave functions for the S-state (u(r)) and
the D-state (w(r)), gained from a special potential, into the last equation and then compar es
the results with the experimental values. For a technical study of deuteron, look at [8].
2.3 General Symmetries of Two-Nucleon Hamiltonian
In general, the invariance of NN interaction under both rotation of coordinate system (the
isotropic property of space) and translation of the origin of coordinat e system (the homogeneous
property of space) a s well as time reversal, charge-independent (CI) and charge-symmetry
(CS) ar e considered commonly. There are already some witnesses for symmetry breaking of
the interaction, such as violating CI a nd CS [9]–Look also at section 4 for more references.
Nowadays, almost all accurate and high-precession NN potentials include these violations–We
remember t hat charg e dependence (CD) of NN interaction means that the interactions o f pp
(T
z
= 1), np (T
z
= 0), and nn (T
z
= −1) a re different, whereas CS of NN int era ctio n means
that just the interactio ns of pp and nn are different.
Fr om symmetry considerations, one can find out the various two-nucleon states under the
condition that
P
r
P
σ
P
τ
ψ(~r, ~σ
1
, ~σ
2
, ~τ
1
, ~τ
2
) = −ψ(~r, ~σ
1
, ~σ
2
, ~τ
1
, ~τ
2
), (2.3)
6

where P
r
, P
σ
, P
τ
are t he space-exchange (Maj orana), spin-exchange (Bartlett) and isospin-
exchange operators, respectively. For instance, in n p system, some states read

S = 0 :
1
P
1
,
1
F
3
,
1
H
5
,
1
K
7
,
1
M
9
, ...
S = 1 : (
3
S
1
−
3
D
1
) ,
3
D
2
, (
3
D
3
−
3
G
3
) ,
3
G
4
, (
3
G
5
−
3
I
5
) ,
3
I
6
, (
3
I
7
−
3
L
7
), ... .
(2.4)
The references [10, 11, 12, 13, 14], and [15] may be useful to earn more basic and general
information about NN interaction.
2.4 More About NN Interaction
2.4.1 Potential Forms, Equations of Motion, and Wave Functions in r -Space
Generally, o ne can construct the following combinations





~
A .
~
B (scalar)
~
A ×
~
B,
~
A ±
~
B (vector)
S
ij
=
1
2
(A
i
B
j
+ A
j
B
i
) −
1
3
δ
ij
~
A .
~
B (rank − 2 spherical tensor)
(2.5)
from two vectors of
~
A,
~
B. For spin, isospin, space, and momentum vectors and also their com-
binations, o ne can consider many cases that obey the symmetry conditions as well. General
form for the central potential is a linear combination of I, ~σ
1
.~σ
2
, ~τ
1
.~τ
2
by multiplying each o p-
erator in a suitable radial function such as V (r/a), where the range parameter a is different for
various operators. In general, these spin-isospin operato r s make t he potential state-dependent.
The generic forms for the centr al and noncentr al terms always read
V
central
= V
c
(r) + V
σ
(r)(~σ
1
.~σ
2
) + V
τ
(r)(~τ
1
.~τ
2
) + V
στ
(r)(~σ
1
.~σ
2
)(~τ
1
.~τ
2
), (2.6)
V
non−central
= V
ls
(r)
~
L.
~
S + V
t
(r)S
12
+ V
lsτ
(r)(
~
L.
~
S)(~τ
1
.~τ
2
), +V
lsσ
(r) (
~
L.
~
S)(~σ
1
.~σ
2
)
+ V
lsστ
(r)(
~
L.
~
S)(~σ
1
.~σ
2
)(~τ
1
.~τ
2
) + ..... .
(2.7)
On the other hand, the matrix elements for some of the operators
I
~τ
1
.~τ
2

× V

r
a

×









I
~σ
1
.~σ
2
(~r × ~p) . (~σ
1
.~σ
2
) (spin − orbit)
S
12
= 3(~σ
1
.ˆr)(~σ
2
.ˆr) −~σ
1
.~σ
2
(2.8)
are as follows:
< ~σ
1
.~σ
2
>=
(
1 ; S = 1 (spin − tripletstate)
−3 ; S = 0 (spin −singletstate)
, (2.9)
hℓ
′
Sjm, T M
T



~
L.
~
S



ℓSjm, T M
T
i =
1
2
δ
ℓℓ
′
[j(j + 1) −ℓ(ℓ + 1) − S(S + 1) ] , (2.10)
7

hℓ, S = 1, jm |S
12
| ℓ
′
, S = 1, jmi =
ℓ \ ℓ
′
j
j 2
,
ℓ \ ℓ
′
j − 1 j + 1
j − 1
−2(j−1)
2j+1
6
√
j(j+1)
2j+1
j + 1
6
√
j(j+1)
2j+1
−2(j+2)
2j+1
.
(2.11)
The uncoupled radial Schrodinger equation (without the Coulomb force) reads
d
2
u
dr
2
−
j(j + 1)
r
2
u − hjSjm, T M
T
| υ| jSjm, T M
T
iu + k
2
u = 0, (2.12)
in which υ = −
M
~
2
V and k
2
=
M
~
2
E, where M and E are the nucleon mass and cent er of
mass (c.m. from now on) energy, respectively. The potential of V is indeed f r om the already
mentioned forms of (2.6) and (2.7). Actually, that is composed of a form-function a s V (r/a), a
linear combination of the various exchange operators and noncentral operators such a s (
~
L.
~
S),
(
~
L.
~
S)
2
, S
12
and so on.
Then, the asymptotic solutions to the equations read
r → 0 ⇒ u(r) =

r
−ℓ
r
ℓ+1
; r → ∞ ⇒ u =
1
k
A
ℓ
sin

kr −
1
2
ℓπ −
1
2
+ δ
ℓ

. (2.13)
As one could find an asymptotic solution, the solution for all r’s is obtained by numerical
integrat ion. Next, with phase shifts, one can earn a potential fro m Schrodinger equation.
For the coupled states (without the Coulomb force), in turn, we also have
d
2
u
dr
2
−
j(j − 1)
r
2
u + k
2
u + F (r)u + H(r)w = 0,
d
2
w
dr
2
−
(j + 1)(j + 2)
r
2
w + k
2
w + G(r)w + H(r)u = 0.
(2.14)
It is notable that the ground-state of deuteron is a special case of the last equation with
k
2
= −γ
2
and j = 1, where γ
2
=
M
~
2
E
B
with E
B
for deuteron binding-energy, and u and w
stand now for the radial functions of
3
S
1
- and
3
D
1
- states, respectively. Because two partial-
wave channels are coupled, an incoming wave in either ℓ = j − 1 or ℓ = j + 1 channel is
scattered into either ℓ = j − 1 or ℓ = j + 1 channel. Therefore, we have two phase shifts
(proper phases) δ
α
j
, δ
β
j
, and a mixing parameter of ε. In presence of the Coulomb potential, a
Coulomb phase-shift is also added and the problem becomes a little more complicated [13].
On the other hand, as the c.m. kinetic energy of two-nucleon system is larger than the necessary
amount to produce a meson, inelastic reactions becomes possible (see Figure 2). Since the
lightest meson (π) mass is about 140 MeV, we expect, when the bombar ding energy is upper
than the threshold, some kinetic energy in the system transfers into the pion. By increasing the
energy, the excitations because of the nucleon internal degrees of freedom, and the probable
8

production of other particles, become more and more importa nt. The inelastic scattering
shows losing t he flow from the incident channel and so, the pr obability amplitude is no longer
conserved. Such a condition may be described by a complex scattering po t ential while some
other relativistic effects come into account; therefore the Schrodinger equation for two-nucleon
system is no longer enough. When discussing various direct potential forms, we return to the
issue partly.
Figure 2: T he energy dependence of the cross-section fo r pion production in the np scattering
through the reactions of p + p → d + π
+
, p + p → p + p + π
+
and p + p → p + p + π
0
; [16].
2.4.2 Scattering-Length and Effective-Range
One can simply show, by semiclassical reasoning’s, that for the low-energy scatterings, only
the S-stat e is important. By increasing the energy, the high-momentum states come into
play because o f the short-range properties of NN interaction. If we show the range by a and
the momentum by ~p, the maximum angular-momentum, which is affected by the scattering
potential, is pa trivially. By squaring the last quantity and equating it with ℓ(ℓ + 1)~
2
, one
can easily get the energy in which a given ℓ comes into play. A rough estimate says that, for
ℓ = 1 state, that energy is nearly 10 MeV.
For np scattering below this energy, we have the following expression (see, for instance,
Sec. II.C of [7], or Sec. 9.a of [13]):
k cot δ = −
1
a
+
1
2
r
e
k
2
− P r
3
e
k
4
. (2.15)
As the term in k
4
becomes important (for E & 10 MeV), the P- and D-waves come into the play
and so, it is not easy to break the k
4
dependence of k cot δ for t he S-wave phase-shift. Therefore,
the useful energy region for the legality of (2.15 ) is where the first two terms answer. In the
9

case where the Coulomb potential is present ( pp scattering), a near effective-range expansion
reads (see, for instance, Sec. IV.C of [7], or Sec. 9 .b of [13], and or see [17]):
C
2
k cot δ + 2kηh(η) = −
1
a
+
1
2
r
e
k
2
− P r
3
e
k
4
, (2.16)
where
C
2
=
2πη
(e
2πη
− 1)
, η =
Me
2
(2~
2
k)
,
h(η) = −γ − log η + η
2
∞
X
M=1

M

M
2
+ η
2

−1
, γ = 0.577.
(2.17)
By using these relations (or similar o nes) and three low-energy
1
S
0
phase shifts, one can hold
the parameters of a (scattering-length), r
e
(effective-range) and P (e.g. for a special potential).
2.4.3 P-Space, Relativistic Scatter ing and so on
For many calculations associated with NN int era ctio n, it is suitable to express Schrodinger
equation as an integral equation in momentum-space; loo k, for instance, at [18, 19] for some
typical studies. The nonr elat ivistic scattering theory that leads to Lippmann-Schwinger (LS)
equation and T-ma trix are useful in this approach. Bethe-Salpeter (BS) equation in relativistic
scattering theory, as relativistic counterpart of LS equation, studying separable potentials,
separable expansions for arbitrary potentials, inverse scattering problem (see, for instance, [20 ],
[21] for some particular studies with r eferences therein), are among the topics covered in this
theory. The interested readers may also refer to [7, 13, 14] for more fundamental discussions–
As a side, we should note that many modern NN potentials (mainly meson-exchange and
chiral EFT potentials) are always written in p-space originally and then Fourier transform
into r-space. So, the p-space formalism is important and commonly used in the standard a nd
relativistic approaches to NN interaction.
By the way, let us discuss a little more on the need for r elat ivistic approaches to the prob-
lem. In fact, one may adjust the LS equation by including relativistic considerations. At the
first look, one may suppose tha t these corrections are not so important below the first inelastic
threshold. That is because the c.m. energy of the system is an almost small fraction of the
nucleon rest mass. But, for the high momenta, it does not seem that describing the interaction
through just nonrelativistic equations is satisfactory. In other words, the short-range repul-
sion of NN interaction, known from the various models based on at least phenomenological
investigations, rapidly reduces the S-wave functions at the distances less than almost 0.5 f m.
Therefore, it brings the high-momentum components into the wave functions for all energies.
Meanwhile, one should note t hat to regularize the potentials at the origin, various parameter-
izations or form-factors with cutoffs are used; altho ugh in the potentials based on chir al EFT,
more standard approaches are employed.
Nevertheless, as one uses the phenomenological approach to describe NN interaction, the short-
10

ages in nonrelativistic approaches are not so impo r t ant. That is b ecause the parameterizations
of the phenomenological models/potentials have enough flexibility to describe NN scattering
in terms of the mesons with various coupling constants, masses and ot her free parameters.
These phenomenological approaches are valid until they provide at least good quantitatively
descriptions of experimental scattering data, and then they could be good alternatives for the
complete relativistic descriptions.
On the ot her hand, while t here is not a comprehensive theory for strong interactions, looking
for a relativistic equation is somehow not ional. Indeed, one may start fr om adapting the LS
equation to satisfy the least needs of every relativistic equation. The basic want is that the
scattering amplitude must satisfy the relativistic unitary along the elastic cutoff. The r esultant
equation is not unique ever; though, it is a relativistic version of the LS equation. For some
studies on the relativistic NN scattering, look, for instance, at [22], [23] and references therein.
3 Nucleon-Nucleon Interaction Models
There are some substantial models to build NN interaction potentials. In t his section, we
specify some qualitative features a nd, for more technical and quant itative studies, refer the
interested readers to other relevant studies.
3.1 Almost Full Phenomenological Models
These models always use the general form o f a potential allowed by the symmetries like rota-
tion, translation, isospin, and so on. In general, the phenomenological potentials often have
the following features:
a) They are somewhat in a similar spirit as EFT, as we describe below, but much older and
restricted to the space-time, spin, and isospin symmetries. b) Four important terms in the
potentials are the central (I), spin-spin (~σ
1
.~σ
2
), spin-orbit (
~
L.
~
S), and tensor (S
12
) int era ctio ns.
c) Each term occurs twice; o ne time without isospin-dependence and one time with the de-
pendence (~τ
1
.~τ
2
), which in turn measures total isospin of NN system. d) The potential terms
are responsible to describe various phenomena remarked in NN interactions. For example, the
tensor term is important for the LR part of potential and ar ises nat ur ally f rom pion-exchange.
In these potentials, the MR and SR parts are usually determined in a fully phenomenological
way while for the LR part, an OPEP is often used. Examples for thepotentials are Hamada-
Johnston potential [24], Yale-group potent ial [25], Reid potentials (Reid68 [26], Reid68-Day
[27], Reid93 [28]), Urbana-group potentials (e.g., UrbanaV14 [29]), Argonne-group potentials
(e.g., ArgonneV14 [30], ArgonneV18 [31]), etc. Look at [32] for a new study of phenomeno-
logical NN potentials. In the next section, we concentrate more on some samples of these
phenomenological po t entials.
The phenomenological potentials have almost many free parameters to be fitted to experimen-
tal scattering data and phase shifts. Less physics one may earn fro m t hem rather than the
physics one may earn from the other potentials with tight theoretical grounds. Nevertheless,
11

their ability to describe the practical facts of NN (pn, n n, pp) interactions, their flexibility and
convenience for using in nuclear structure calculations, are notable. These properties have still
kept them in work mo r e than the other potentials nowadays.
3.2 Boson Exchange Models
The potential acting between a pair of particles, because of a meson exchange, has the range o f
the meson’s Compton-wavelength; which is in turn proportional to the meson-mass inversely.
Because the pion π is the lightest meson exchanged between nucleon pairs, it contributes the
LR part of NN interaction, beyond its Compton’s wavelength. So, the resultant potential is
called OPEP. Similarly, to describe the MR and SR part s of the interaction, one should con-
sider the exchanges of the heavier mesons than the pion as well a s two, three a nd more pion
exchanges. However, because considering these exchanges exactly is rather difficult, most peo-
ple consider them phenomenologically while, in meson-exchange potentia ls, they are included
clearly. Look at [33, 34] for the first meson and multi-pion exchange NN potentials.
On the other hand, it is already known that multi-pion systems have some strongly corre-
lated resonances that behave often as a single meson. So, it is supposed that multi-pion
resonances, when exchange between two nucleons, may contribute to the MR and SR parts
of interactions. The potentials built in t his way, by including single meson exchanges, are
called One-Boson-Exchange Potentials (O BEP). Besides the tra ditional one π(138) exchange,
various meson exchanges are considered in OBEP’s. There are t he exchanges of ρ(769) meson
(as a 2π resonance), ω(783) meson (as a 3π resonance), η(549) meson (the same quantum
numbers with π but its isospin that is T = 0), ´η(958) meson (the same quantum numbers with
η but heavier and a resonance of ηππ), δ(983) (as a 4π resonance), φ(1 020) meson (the same
quantum numbers as ω but a resonance of K
+
K
−
system), a nd S
∗
(975) meson (as 2 π, K
¯
K res-
onances). In addition, one always considers the experimentally undetermined scalar-isoscalar
boson of σ(500 − 700), which is usually considered as a good parameterization of 2π system
in S-state. Still, there are two other mesons with the mass above 1 GeV that may act as
2π resonances. They are ǫ(1300) (or f
0
) meson ( with the same quantum numbers as S
∗
(975)
but just as a 2π resonance) and f(1274) meson. Some other two-boson resonances may come
from the mesons of A
1
(1275), A
2
(1318) (as ρπ resonances), B(1 234) (as a ωπ resonance) and
D(1283) (as ηππ, 4π resonances). But, because of the importance of the hadrons’ structure
in the energy region of 1 GeV and with respect to the energies involved in the common NN
interactions, the roles of the heavier mesons may not be so important.
In general, π-meson (and also φ-meson) exchange provides the most LR (tensor) force, whereas
ω-meson exchange provides t he most SR repulsive f orce and SR spin-orbit force. The inter-
mediate attractive force is often explained by 2π (as ρ- and/or fictitious σ-meson) exchang es,
whereas the potential contr ibution by η-meson is weak and always ignorable. Therefore, these
few mesons describe the main features of NN interaction; but to describe well experimental
data and other subtle properties, depended on the case, the exchanges of the other mentioned
mesons are also included.
Fr om the differences among var ious NN potentials, based on meson theory, are their meth-
12

ods to deal with the 2π exchange. In one approach, its effect is simulated through one or two
scalar and isoscalar mesons. Considering the 2π environmental effects as well as employing
scattering-length and effective-range formalism for the S-state of the system are the efforts in
the line. By the way, each group uses its own methods and details to eva lua te the potential.
In general, the settled ”field t heoretical techniques” and the methods based on ”dispersion re-
lations” are two main ways for handling the problem. Look at [35] for a senior field-theoretical
NN model, [36] for an old review on OBEP, [37] f or a comprehensive review, and [38] for
another useful typical study. See also [39, 40, 41, 42] for some other relevant studies.
In other words, by discovering the vector mesons of ρ and ω, with the massed in the
ranges of 770-780 MeV, more progresses in understanding NN interaction were archived and
led to expand OPEP’s. In OBEP’s, mostly, the unrelated contributions of the single-meson
exchanges of the pseudoscalar mesons π, η and the vector mesons ρ, ω as well as the scalar me-
son δ(9 83) are considered and iterated into scattering equation. There are also 2π exchanges,
which are always parameterized by the artificial σ meson with the masses in the range of
400-800 MeV. The core (SR) region o f the potentials is always parameterized through phe-
nomenological parameters and the form factors r elat ed to the meson-nucleon vertices. The
form factors in turn hold on fundament al relations to QCD. Then, such OBEP’s provide good
(at least quantitative) description of scattering data. Many types of these potentials, each
with its own characterizes and features already exist. Nowadays, it is almost clear that the
meson-exchange potentials (MEP’s) a r e almost the standard NN potentials. Some examples
are the Partovi- Lomon model [35], and Stony Brook- group [43], Pa ris-group [44], Bonn-group
[45], Padua- group [46], Nijmegen-group [28 ] and Hamburg-group [47] potentials.
Boson-exchange methods are nowadays extended, besides NN systems, to many Bar yon-Baryon
(BB) interactions such as pion-nucleon, pion-pion, Hyperon-Nucleon (YN) and Hyperon-Hyperon
(YY) interactions as well. Although these models do not refer to QCD deeply, but the baryon
and meson fields ar e already considered as the asymptotic states that absorb all effects from
the quark and gluon dynamics. It is also notable that not only phenomenological models but
also the advanced models of NN interactions, such as QCD-inspired and chiral EFT models,
which we describe below, use boson exchanges in some parts of studies.
To summary, we note that in t he quark-antiquark pair (= meson) exchange model, there
are the following f eat ures: a) It is similar to the quark exchange but the reverse direction of
one quark. b) It gives a good description of many aspects of NN interaction. c) It is preferred
because the meson states are colorless and have a lmo st lower masses or larger ranges. d) It
studies O PEP a nd generalizes it to other mesons that results in OBEP’s and more. e) Next
to full phenomenological potentials and chiral EFT pot entials, BEP’s are the best physical
potentials that give perfect agreement with the data for the LR and MR parts especially.
3.3 The Models Based on QCD
In these models, the aim is to connect hadronic pr ocesses to the underlying t heory of strong
interactions that is QCD. In other words, hadron-hadro n interactions ar e described in terms
of quark and gluo n degrees of freedom. Look at [48], [49], [50], [51], [52] and [5 3], [54] for some
13

reviews and typical studies of NN interaction in QCD and quark models.
In low energies, releva nt to NN interaction, QCD is nonperturbative and could not solved
exactly. Chiral Perturbation Theory ( CHPT) (see, e.g., [55]), Skyrme Model (see, e.g., [56],
[57]), and Nambu-Jona-Lasinio (NJL) models (see, e.g., [58]) are examples of this approach.
The models describe the characteristic phenomena observed in nucleon-nucleon, pion-nucleon,
and pion-pion scattering well qualitatively but they fail quantitatively. Common feat ures of
the ”QCD-inspired” models, that reduce the demand f or them, are cumbersome mathematics,
large numbers of parameters and limits in applying especially to very low energies. Therefore, if
one wants a good quantitative description of experimental data, phenomenological approaches
such as boson-exchange and phenomenological models are preferred. Nevertheless, in some
models just for the short distances, the QCD approach is used whereas for the remaining parts
of interaction, the two fo r mer approaches are used with satisfactory results. We deal with the
issue more when discussing plain potentials.
In summary, there are two subsets of QCD-inspired models, with basics features and main
characteristics, as follows:
1) The gluon a nd quark exchange among nucleons plus the Pauli-repulsion between similar
quarks in overlapping nucleons, with the following features: a) The gluon exchanges based on
”constituent quark model”(CQM) besides one-gluon-exchange-potential (OGEP). It does not
give a good description for reasonable distances because of confining the colorless singlets. b)
The Pauli-repulsion is related to a minimum energy to excite a nucleon (that is t o move a
quark into a different state) o f 300 MeV. c) The quark exchanges between two nucleons and
may change nucleon charges (i.e., n → p and a t the same time p → n). d) It gives a reasonable
and semi-quantita t ive description of the SR repulsive part and maybe the MR part of NN
interaction. Look a t, for instance, [50, 51, 52, 53, 54] f or some general studies. Among the
exact potent ials of this type are the Moscow-group [59] and Oxford-group [60] potentials.
2) Chiral symmetry and CHPT can also be considered as a subset of QCD methods, with the
following features: a) That is based on chiral symmetry of QCD Lagrangian. That symmetry
means that the quarks with opposite helicity are indistinguishable and do not couple to each
other except for their masses. b) Chiral symmetry is spontaneously broken because QCD
prefers the quark-antiquark pairs with negative par ity to the quark-quark pairs with positive
parity. Thus, t he low-mass modes (zero-mass theoretically) of the ”quark condensation” are
called ”Goldstone bosons” (pions, kaons, etc.). This, in turn, limits the Lagrangian to the
processes involving nucleons and pseudoscalar mesons. In other words, for the energies around
Λ
QCD
≈ 1GeV, there is a ”phase transition” from ’fundamental” theory to a n ”effective”
theory through spontaneous breaking of the chiral symmetry of QCD Lagrangian. During
this procedure, pseudoscalar ”Go ldstone” bosons are produced. As a result, in low energies
(E < Λ
QCD
), the proper degrees of freedom are the pseudoscalar mesons and other similar
hadrons, and not the quarks and gluons o f the original theory. The standard effective theory
to describe this process is called CHPT. c) Chiral symmetry is also violated by the (small)
quark masses; so, the Goldstone bosons are not massless totally. Nevertheless, one can expand
the interaction in small parameters to make definite predictions ( as in CHPT). Look, for some
14

related studies, at [61, 6 2] and [55 ] and also references therein. In the following subsection, we
discuss this issue further.
Still, we should note to some other studies, on NN interaction, in language of ”lattice QCD”,
for instance in [63, 64, 65, 66] and [67 ].
3.4 Effective Field T heory Ap proach
Effective-field-theories (EFT’s) are the low-energy descendants to the high-energy parent the-
ories. Some of the features are as follows: a) In general, one notes that there are differ-
ent/separate energy scales in the nature each with its own degrees of freedom. In each energy
level, just some degrees of freedom are relevant and as the energy decreases, some others a r e
frozen and become irrelevant. An example of this is the chiral symmetry. b) About NN in-
teraction, as first hinted by Weinberg [68], EFT means applying a ll symmetries including the
chiral symmetry of QCD Lagrangian but not directly considering the underlying degrees of
freedom like pions or quarks. This gives the most general Lagrangian that cont ains ma ny
parameters to be constrained with data. In other words, the Lagrangian must include all
possible terms to guarantee that the ”effective” theory is indeed the low-energy limit of the
”fundamental” high-energy theory. So, no presumptions about, for example, renormalizability
or simplifying the Lagrangian are permissible. This, in turn, means that we probably have an
infinite set of interactions. Therefore, to have a reasonable theory with well-defined results,
one must organize the perturba t ive expansion up to some defined orders. L ook, for instance,
at [6 9, 70, 71, 72] and references therein, for some reviews of EFT approach to NN interaction.
In general, a systematic improvement in the ability of the model to reproduce NN data is
observed when the orders of chiral expansion increase. One of the first extended models (in
Next-to-Next-to-Leading Order: NNLO) of CHPT described np pha se shifts well up to the
energies about 100 MeV; but, for the higher energies, some inconsistencies occurred in some
partial waves–See [73] for a recent study of this approach and developments. Although NNLO
and the most recent higher- order chiral NN potentials show significant pr ogress towards the
earlier ones and are almost perfect (in fact, the new NNNLO potentials describe data well up
to 350 MeV with similar quality as the high-precession phenomenological and boson-exchange
potentials), still to apply well the resultant po t entials to all nuclear structure calculations,
more quantitative and even qualita tive improvements are necessary. We should, of course,
note that chiral EFT models have more standards and great theoretical bases to be known as
the most reasonable models to describe the strong nuclear interactions.
In summary, we can say that CHPT from EFT in low energies is as fundamental as QCD in
high energies. In addition, because of the perturbative nature of CHPT, it can be evaluated
order by order in chiral expansions. As long as we are looking for a substantial theory of nuclear
forces, a pplicable to nuclear structure calculations as well, CHPT is likely able to overcome the
discrepancies between experiment and theory. For some other typical studies on the subject
of EFT and CHPT, look, for instance, at [69, 74, 75], and [3, 55, 76] for some recent views.
Meanwhile, among the high-quality potentials of this type are those by Texas-group [77 ], Sao
Paulo-group [78], Munich-group [79], Idaho-Group [80], and Bochum-Julich-Group [81].
15

4 Nucleon-Nucleon Interaction Potentials
4.1 Basic Potentials and General Remarks
In this subsection, we discuss on the main preliminary potentials and a br ief on the methods
of making them. As already mentioned, the range of nucleon-nucleon interaction is divided
into three parts, which a re the short-range (SR), the intermediate or medium-range (MR)
and the long-range (LR). For the MR and LR parts, many workers have always taken the
phenomenological and boson-exchange pictures. However, in most models, for the LR part,
one-pion-exchange (OPE) is usually included. For the SR part, phenomenological parame-
terizations are often employed. In some models, form factors are included to regula rize the
potentials at the origin; whereas, in some other models, severe ha r d cores are included. The
first major approach to describe the MR part was to include two-pion-exchange (TPE) contri-
butions. The first samples of TPE potentials were given by Taketani-Machida-Ohnuma [33] and
Brueckner-Watson [34]. However, those TPE potentials did not provide good descriptions of
NN scattering data as one reason was the lack of a spin-orbit p otential therein. Next, Gammel,
Christian a nd Thaler [82] discovered the need t o include a spin-orbit potential when they tried
to fit the NN scattering data at that time with a velocity-dependent local phenomenological
NN potential as
V = V
c
(r) + V
t
(r)S
12
, (4.1)
for each of the four combinations of the spin and isospin. Nevertheless, they failed!
In 1957, the efforts to build f ur t her phenomenological potentials, by including the phenomeno-
logical spin-orbit pot entials as well, got started. The purely phenomenological potential of
Gammel-Thaler [83] provided a good description of the scattering dat a at that time below
T
lab
=310 MeV (note that we use throughout this note t he laboratory energy unless other-
wise be told). At the same time, the semiclassical potential of Singell-Marshak [84], which
was consist of the TPE potential of Gartenhaus [85], next to the phenomenological spin-orbit
potential, provided a satisfactory description of the data below 150 MeV.
Then, Okubo-Marshak [86] showed that t he most general two-nucleon potential, by considering
symmetry conditions, reads
V (~r, ~p, ~σ
1
, ~σ
2
, ~τ
1
, ~τ
2
) = V
c
(r) + V
σ
(r) (~σ
1
.~σ
2
) + V
τ
(r) (~τ
1
.~τ
2
) + V
στ
(r) (~σ
1
.~σ
2
) (~τ
1
.~τ
2
)
+V
ls
(r)(~τ
1
.~τ
2
) + V
lsτ
(r)

~
L.
~
S

(~τ
1
.~τ
2
)
+V
t
(r)S
12
+ V
tτ
(r)S
12
(~τ
1
.~τ
2
)
+V
q
(r)Q
12
+ V
qτ
(r)Q
12
(~τ
1
.~τ
2
)
+V
pp
(r) (~σ
1
.~p) (~σ
2
.~p) + V
ppτ
(r) (~σ
1
.~p) (~σ
2
.~p) (~τ
1
.~τ
2
),
(4.2)
where
~
L.
~
S is the usual spin-orbit operato r and
Q
12
=
1
2
n
(~σ
1
.
~
L)(~σ
2
.
~
L) + (~σ
2
.
~
L)(~σ
1
.
~
L)
o
(4.3)
16

is the quadratic spin-orbit o perator. The twelve terms in the potential ar e g iven by the twelve
radial functions V
c
(r), ... . These functions can be o btained from our knowledge about the
nature of nuclear forces. Information to find out V (r)’s could be fr om, for example, the
exchanges of various mesons or phenomenological mechanisms in which some given radial
functions, with maybe some arbitrary free parameters to be fixed to experimental data, exist.
Once our understanding of underlying theories (such as QCD) improves further, we may be
able to get these functions from the basics. The first four terms in (4.2) stand for the complete
central potential and, in the case, L a nd S are the good quant um numbers. By adding other
terms, the good quantum number is J as the two-nucleon system is now invariant under the
combined space of L and S. The main reason for two terms in the spin-orbit potential of
V
Spin−Orbit
(r) = V
ls
(r)
~
L.
~
S + V
lsτ
(r)(
~
L.
~
S)(~τ
1
.~τ
2
) ( 4.4)
is that the radial dependence of the potentials may be different for the isospin-independent and
isospin-dep endent parts, for examples, because of different meson exchanges. The seventh and
eighth terms stand for the tensor forces while the ninth and tenth terms ar e for the quadratic
spin-orbit forces. The latter two terms enter just when momentum dependence exists in the
potential. The last, 1 1th and 12th, terms are always omitted because, at least for elastic
scattering, they can be written as linear combinations of the other terms. So, their role cannot
be determined from elastic scattering from which most of our information about NN interaction
comes–For a useful study about NN interaction, including the potentials and ideas, before 1960,
look also at [15].
Soon after, better potentials were constructed. Among the 1960’s meson-exchange and field-
theoretical potentials, t he NN potential by Sugawara and others [87] , [88] are also ment ionable.
Other impo rtant phenomenological potentials then were Hamada-Johnston [24] and Yale [25]
potentials and also various hard- and soft-core potentials by Reid [26].
Before going more into discussing some other potentials, it is useful to mention that almost
all experimental elastic phase shifts are derived from the differential cross sections of pp and
np scattering’s. For most potentials the data are oft en fitted in the energy range of 0 -350
MeV. That is because, in the higher energies, inelastic processes (with the threshold of about
280 MeV), such as pion production and other relativistic effects, come into play and so, the
two-body Schrodinger equation is no long er enough. It should be mentioned that the modern
analysis with more improved relativistic equations (for example with BS equation) have tried
to account for all effects at once.
To know the methods of ma king the ear ly pot entials, we note that, f or instance, the Hamada-
Johnston ( HJ) [24] and Yale-group [25] determined all two-nucleon scattering data a nd po-
larization parameters as a function of energy for the energies of a few hundred MeV. The
Yale-group potential was initially framed to reproduce the phase shifts in various states as a
smooth function of energy. As a first step, the phase parameters (that is the phase shifts and
mixing parameters of the coupled states) were determined as functions of energies by fitting
to all experimental scattering and polarization data. The procedure was performed by several
groups mainly Yale-gr oup [89] and Livermore-group [90] then–For a updated analysis of NN
17

scattering data by the latter group look a t [91]. As a second step, the potentials, with their
adjusted parameters, reproduced the phase parameters. The more standard procedure is to
present scattering amplitudes as a sum of all partial waves up to a maximum orbital angular-
momentum, which is more or less ℓ
max
= 5. The contributions of the higher partial waves are
always indicted by O PE contr ibution to the scattering amplitude. In the Yale-group potential,
OPEP was used as a fixed part while the remaining parts of the potential were fixed by fitt ing
the energy-dependent phase parameters up to ℓ
max
. It is mentiona ble that, for the current
up-to-date potentials, the basic analyses are drastically improved although the procedures ar e
more or less similar.
By t he way, in most NN potentials, for the LR part, OPEP is usually used while for the MR
part the multi-pions and single mesons such as ρ, ω, σ, ... are often used. Still for the SR
repulsive part, various methods including neutral vector-meson exchanges, velocity-dependent
potentials, phenomenological parameterizations and QCD substructure techniques are used.
4.2 NN Potential’s Road
The original try to find the fundamental theory of nuclear forces was started around 1935
by Yukawa. The Yukawa [2] meson-exchange model f or nuclear for ce and the other old pion-
exchange potentials, such as those by Taketani-Machida-Ohnuma [33], Brueckner-Watson [34],
Singell-Marshak [84], Gartenhaus [85], etc. were not so successful. That wa s bo t h because
of the failing of their structures and the pion dynamics, which we now know is restricted by
chiral symmetry. By discovering heavy mesons in the early 1960’s, modeling b etter one-bo son-
exchange-potentials (OBEP’s) was start ed in [92], [93] and [94] as well, and was then devel-
oped more by framing some better potentia ls. Therefore, the field-theoretical and quantum-
dispersion methods were involved with making the potentials such as Partovi-Lomon model
[35], Stony Brook-group [43], Paris-group [44], Nijmegen-group [95] and Bonn-gro up [45] po-
tentials
1
. But there were still some problems with the boson-exchange potentials. Among
them was the σ-boson exchange for which experimental evidence was polemic. Nevertheless,
because that equals a 2π resonance, there were many efforts to find two-pion contributions to
the interactions. Anyhow, then, more high-precession potent ials such as parameterized Paris
potential [98], the high-quality potentials of Nijm93, NijmI, NijmII [28], CD-Bonn [99] and
many other interesting potentials based on meson-exchange pictures were constructed. So, it
seemed that the nuclear force problem was solved! But, no!
With the coming of QCD and its underside quarks and gluons degrees of f r eedom, the stud-
ies came into new phases. Still, the problem with QCD was its nonperturbative structure
when applying to the MeV low-energy limit, where nuclear physics is valid. The QCD-inspired
quark models were the first tries in the pha se [48]. Lattice QCD was/is also a way to deal with
the problem; see for instance [63]. Still, the QCD-inspired potentials were/are qualitat ively
successful but no quantitatively well a s are the phenomenological meson-exchange potentials.
Among these potentials, the potential set up by some members of the Paris-group in [100],
1
Among the other boson-exchange potentials are those in [96] and [97], where the former is a relativistic
OBE model and the latter is constructed from the meson-exchange and nucleon structure properties.
18

the Moscow-group potentials [59] and the Oxford potential [60] are mentionable. Nevertheless,
some potentials, such as the high-quality Nijmegen-group ones [28] (and even two former ones)
use a mixture of the mesons and quarks in some parts of the interaction.
Meanwhile, many phenomenological potentials composed of meson-exchanges, operators and
functions with adjusted parameters to fit experimental data, with wide applications in nuclear
computations, were constructed. Among them a r e the Reid [26] a nd UrbanaV14 [29] potentials,
and the high-precession Nijmegen-group potentials [28] and ArgonneV18 [31] potential.
By coming EFT and applying it to the low-energy QCD, first by Weinberg [68], the new
phase to set up NN potentials got started. In such models, one usually starts by writing the
most general Lagrangian including the assumed symmetries and especially chiral symmetry of
QCD. In low-energies, chiral symmetry breaks down and then the suitable degrees of freedom
are not quarks a nd gluons but there are pions and nucleons, while heavy mesons and nucleon
resonances are integrated out. So, it seems that we are going back to the meson theory! of
course with much more experiences.
The chiral effective Lagrangian is composed of a set of the sentences increasing in derivative
terms or nucleon fields. Indeed, one use a perturbative expansion in (Q/Λ
QCD
)
ν
, where Q
refers to the soft scale a ssociat ed with external momenta or pion mass, Λ
QCD
≈ 1 GeV is the
chiral symmetry breaking scale and ν ≥ 0. By applying the Lagrangia n to NN scattering,
there are the suiting Feynman diagrams whose importance becomes less as the order of the
chiral perturbation theory (χPT) expansion increases. Besides describing the nuclear two-body
problem, the model makes some good predictions for nuclear few-body forces as well. The first
potential of this type was constructed by Texas-group (Ordonez, Ray and van Kolck) [77] and
among the further developed ones are those by Idaho-group [80] and Bochum-Julich-group [81]
up to NNNLO. These new CHPT potentials are quantitatively and qualitatively best so far
candidates to describe two-nucleon as well as few-nucleon interactions.
It is also notable that there are some tr ies t o construct NN potentials ba sed on renormalization-
group (RG) appro ach to NN interaction by another Stony-Brook group [101]. As a result, t hey
have earned many creditable and satisfactory r esults that we comment more in Section 4.
Anyway, in what follows we continue studying some of the potentials which are of course more
important with established results in nuclear structure calculations, briefly.
4.3 Hamada-Johnston Potential
The Hamada-Johnston (HJ) potential [24] is a leading phenomenological NN (pp+np here)
energy-indep endent po t ential. It described well the scattering data below 350 MeV and
deuteron properties as well as the effective-range parameters. The general form of HJ po-
tential [24] reads
V = V
c
(r) + V
t
(r)S
12
+ V
ls
(r)
~
L.
~
S + V
ll
(r)L
12
, (4.5)
where
S
12
= 3(~σ
1
.ˆr)(~σ
2
.ˆr) − (~σ
1
.~σ
2
), L
12
= (δ
ℓj
+ ~σ
1
.~σ
2
) L
2
− (
~
L.
~
S)
2
, (4.6)
19

and









V
c
(r) = 0.08

1
3
m
pi

(~τ
1
.~τ
2
) (~σ
1
.~σ
2
) Y (x) [1 + a
c
Y (x) + b
c
Y
2
(x)],
V
t
(r) = 0.08

1
3
m
pi

(~τ
1
.~τ
2
) Z(x) [1 + a
t
Y (x) + b
t
Y
2
(x)],
V
ls
(r) = m
pi
G
ls
Y
2
(x) [1 + b
ls
Y (x)],
V
ll
(r) = m
pi
G
ll
x
−2
Z(x) [1 + a
ll
Y (x) + b
ll
Y
2
(x)],
(4.7)
in which m
pi
, x and M are the pion mass (139.4 MeV), the internucleon distance measured in
the units of the pion Compton’s wavelength (r
0
= 1.415fm), and the nucleon mass (taken to
be 6.73µ), respectively. Note also that x = µr, µ = m
pi
c/~ = r
−1
0
with respect to Eq. (2 .1),
and that
Y (x) =
e
−x
x
, Z(x) =

1 +
3
x
+
3
x
2

Y (x). (4.8)
We should note that the quadratic spin-orbit potential was mainly introduced to describe np
data satisfactorily. For the r large enough, V
c
(r) and V
t
(r) r educe to the well-known OPEP
with the pseudovector coupling constant of 0.08. The coefficients a
c
, b
c
, a
t
and b
t
represent
the potential diversion from OPEP at small r’s. G
ls
is the strength of the short ranged spin-
orbit potential V
ls
(r) and is depended on the pa r ity of state. G
ll
, as the strength of V
ll
(r),
originated from special eva luations, is determined phenomenologically. All the coefficients are
determined from the detailed fit t o scattering data and are g iven in the original paper [24].
The hard cores are considered for all states with their radius at x
c
= 0.343. The HJ potential,
as originally proposed, included a strong long-range quadratic spin-orbit potential in triplet-
even states, and also a strong short-range spin-or bit potential in triplet (ℓ = j)-odd states,
where it is known that t he latter does not exist. So, the potential for triplet-odd states was
modified as follows [1 02]: It was defined to be - 0.26 744 m
pi
around x
c
< x ≤ 0.487 and by
above standard relations for x > 0.487. The values of the binding energy, electric quadratic-
moment, effective-range, D-state pr obability and the asymptotic D-wave to S-wave ratio of
deuteron were determined by the potential to be 2.226 MeV, 0.285 fm
2
, 1.77 f m, 6.97 % and
A
D
/A
S
= 0.0 2656, respectively.
An improvement of HJ potential was made in [103] (we call it Massachusetts-group
potential) to replace mainly the HJ hard cores (for x ≤ x
c
) by finite square-well cores. Outside
the square-well radius (for x > x
c
), the potential is the same as HJ except for a few changes in
parameters such as considering the pion mass differences, and that the a
c
values of the singlet-
even and triplet-odd states as well as the triplet-odd b
c
are changed slightly. The pion mass
splitting leads to charge-independent breaking (CIB) while CS is still preserved. Now, m
pi
is
replaced by the effective pion mass and x
c
= 0.4852, which in turn implies the larg er core-
radius of 0.7 fm. Describing NN scattering data and deuteron pro perties with the potential
were good. Indeed, the main aim to form the latter potential was to show tha t the hard cores
were not necessary since all data could be described by the finite soft-core potentials.
4.4 Yale-Group Potential
The Yale-group potent ial [25] is a pp+np phenomenological potential similar to HJ potential
[24] tha t is fitted to its time phase parameters as well. There, an OPEP is included directly
20

and the quadratic spin-orbit pot ential is considered in a somewhat different form than that of
HJ. The whole NN potential reads
V = V
(2)
OP EP
+ V
c
(r) + V
t
(r)S
12
+ V
ls
(r)
~
L.
~
S + V
ql
(r)

Q
12
− (
~
L.
~
S)
2

, (4.9)
where
h
Q
12
− (
~
L.
~
S)
2
i
= (
~
L.
~
S)
2
+
~
L.
~
S −
~
L
2
, (4.10)
and
V
(2)
OP EP
=

g
2
pi
12

m
pi
c
2

m
pi
M

2
(~τ
1
.~τ
2
)

(~σ
1
.~σ
2
) + S
12

1 +
3
x
+
3
x
2

e
−x
x
. (4.11)
This OPEP is used for the distances larger t han nearly 3fm, with the same parameter defini-
tions as in HJ potential. For the coupling constant, g
2
pi
/14 = 0.94 is used in singlet-even states
and 1 elsewhere. For singlet-even and triplet-odd states, the neutral-pion mass (m
pi
= m
π
0
)
is used while for singlet-odd and triplet-even states, a mean of the charged- and neutral-pion
masses [m
pi
= (m
π
0
+ 2m
π±
) /3 ] is used. The hard-core radius is considered at x
c
= 0.3 5, and
except in the OPEP part, all the radial functions V
c
, V
t
, V
ls
and V
ql
are taken as
V =
7
X
n=1
a
n
e
−2x
x
n
. (4.12)
The potential’s par ameters are determined by fitting to data for various states and involved
potentials. It is also notable tha t HJ and Yale potentials are OPEP for L > 5, and that the
Yale potentia l sets V
ls
= 0 for J > 2.
4.5 Reid68 and Reid-Day Potentials
4.5.1 Reid68 Potential
Among the failures o f HJ [24] and Yale [25] hard-core po tentials were that they could not repro-
duce reasonable results when applying to many-body calculations. It appeared that the Reid
soft-core potent ials [26] were better. The Reid potentials are static and local phenomenological
potentials similar to those of HJ and Yale. Reid determined the potential for each two-nucleon
state independent of the other states. So, one may suppose that this approach is problematic
in that, with many two-nucleon states each with its own potent ial, fitting the experimental
data could be probably meaningless. But, because the highest energy in the analyses was
about 350 MeV, just the two-nucleon states with J ≤ 2, which are more important in nuclear
calculations, were considered in practice.
Reid used only a central potential in the singlet- and uncoupled triplet-states while, for the
coupled triplet-states, he used
V = V
c
(r) + V
t
(r)S
12
+ V
ls
(r)
~
L.
~
S, (4.13)
21

which has the centra l, tensor and usual spin-orbit components. For the LR part, he used the
OPEP of (4.11) as a tail attached to the potential, with g
2
pi
= 14, m
pi
= 138.13 MeV, M =
938.903 MeV and µ = 0.7 f m
−1
. On the other hand, to remove the x
−2
and x
−3
behaviors a t
small distances, an SR potential was subtracted from the tensor part o f the potential. For t he
MR’s, the potentials were expressed as the sums of the Yukawa’s functions of e
−nx
/x, where n
was an integer. The SR repulsions were also some combinations o f the severe ha r d-core and the
Yukawa soft-core pot entials–It is mentionable that the criterion for a pot ential to be soft-core
is t hat the wave functions do not vanish in nonzero radiuses. For the hard- cor e radius, when
needed, the radiuses of x
c
≤ 0.1 could be used there. One should, of course, note that because
of fitting the potentials to the energies often below 350 MeV, finding a unique formalism for
the SR part was almost difficult. Finally, it is notable that the Reid potentials did no t describe
well some of the scattering data and deuteron properties a t that time. It was also hinted the
need for velocity-dependence a nd nonlocality in NN p otentials, imposed by experimental data .
4.5.2 Reid-Day Po tential
In 1980, B. D. Day [27] expanded the Reid68 soft-core potentials up to the higher partial waves
to solve three-body equation in nuclear matter calculations. In fact, he used three two-nucleon
potentials in calculations. The first one (called V
2
) was just the central part of the R eid68
potential in
3
S
1
−
3
D
1
channel for all states. The second one (called V
6
(Reid)) had four forms
for the four (S, T ) states. Indeed, in the latter case, for all S = 0 states, just two centr al
V
c
(r) potentials (Reid68
1
S
0
and
1
P
1
for T = 1 and T = 0 respectively) were used; meanwhile
for all S = 1 states, just two central V
c
(r) and two tensor V
t
(r) potentials (Reid68
3
P
2
−
3
F
2
and
3
S
1
−
3
D
1
for T = 1 and T = 0 resp ectively) were used. The third one (called Full-R eid
potential that we call Reid-Day potential) used the original Reid68 potentials for all J ≤ 2
states; meanwhile for the states with J ≥ 3, he set up the potent ials based on the Reid68 ones
almost roughly. Clearly, for the states up to J = 5, the potential structures were similar to
the original Reid68 ones. For example, in the coupled sate of
3
D
3
−
3
G
3
, he used
V
c
(r) = −10.463 Y (x) − 103.4 Y
2
(x) − 419.6 Y
4
(x) + 9924.3 Y
6
(x), (4.14)
V
t
(r) = −10.463

Z(x) −

12
x
+
3
x
2

Y
4
(x)

+ 351.77 Y
4
(x) − 1673.5 Y
6
(x), (4.15)
V
ls
(r) = 650 Y
4
(x) − 5506 Y
6
(x), (4.16)
where x = 0.7r, and r is the internucleon distance measured in fm as usual. For all other not
clearly mentio ned states, he used the V
6
(Reid) potentials. Therefore, that new expansion was
not based on any fundamental underlying argument on NN interaction, and was just to sake
of applying the wanted potentials in some nuclear calculations.
22

4.6 Partovi-Lomon Potential
Partovi-Lomon potential [35] is among t he early NN potentials based on quantum field theory
methods. The advantages of boson exchanges and multi-pion resonances especially in short
distances were considered. They also considered some TPEP’s and OPEP’s to improve the
quality of previous similar potentials. The resultant (Schrodinger-equation) potentia l was
originated from reducing BS equation to an LS equation. In fact, by starting from a relativistic
tow- body equation, they arrived in a nonrelativistic LS equation and presented a potential as a
solution of the integral equation. Then, they tried to build r-space potentials with momentum
operators, resulting in a potentia l composed of the central, spin-orbit, t ensor and spin-spin
parts. Contributions of ρ, ω, η bosons were included and then, by using experimental masses
and coupling parameters, the complete potential was calculated. The potential has some
likeness to Hamada-Johnston potential [24], and appears to dissolve some of the problems
hinted in Reid68 potential [26 ].
4.7 Paris-Group Potentials
Paris-gr oup potentials are ba sed on dispersion relations and field-theoretical techniques. In
their first major potential, Paris72 [44], they included some TPE contributions for the poten-
tials by considering pion-nucleon phase shifts and pion-pion interactions. They also included π-
and ρ-meson exchanges. Indeed, for the LR and MR parts, the accurate potentials of π+2π +ρ
exchanges were used; while for the SR pa r t of r ≤ 0.8fm, a constant soft-core potential wa s
used. The Paris72 potential includes the central, spin-spin, spin-orbit, tensor and quadratic
spin-orbit compo nents for each isospin state. Fitting t he po t ential to the pp, np scattering
data of the Livermore-group [104] of 1969, needed 12 adjustable parameters. The potential
described the data with similar qualities as the phenomenological potentials of HJ and Yale-
group with more adjustable para meters. Describing the LR and MR parts by the potentials
was more sensible. Nevertheless, describing short distances was no t satisfactory besides the
problems in its applications to many-body nuclear calculations.
The next improved version of the potential came in 1 979, named as ”parameterized Paris
potential” or Paris79 potential. In that version, they employed a unique expression for the
whole potential, which was a sum of the Yukawa’s functions that had simple forms in both
configuration and moment um spaces. Indeed, those 12 local Yukawa functions could provide
a semi-phenomenological description of the Paris72 potentia l. Meanwhile, the older contri-
butions for the LR and MR parts were used yet. The potentials for both va lues o f isospin
(T = 1, 0) have the following nonrelativistic for ms in r-space:
V (~r, p
2
) = V
0
(r, p
2
)SS
1
+ V
1
(r, p
2
)SS
2
+ V
ls
(r)
~
L.
~
S + V
t
(r)S
12
+ V
q
(r)Q
12
, (4.17)
where
SS
1
=

1 − ~σ
1
.~σ
2
4

, SS
2
=

3 + ~σ
1
.~σ
2
4

. (4.18)
23

Clear for ms for the velocity-dependent functions of V
0
and V
1
, and especial forms fo r the
Yukawa functions of V
ls
, V
t
, V
q
, as well as coupling constants and other parameters, under
special conditions, are given in the original paper [98]. The potent ial in p-space, by Four ier
transform of (4.17), reads
˜
V (~p
i
, ~p
f
) =
˜
V
0
(~p
i
, ~p
f
)SS
1
+
˜
V
1
(~p
i
, ~p
f
)SS
2
+
˜
V
ls
(k
2
)
˜
LS
1
+
˜
V
t
(k
2
)
˜
S
12
+
˜
V
q
(k
2
)
˜
Q
12
, (4.19)
where
˜
LS
1
= i
~
S.~n,
˜
S
12
=
h
k
2
(~σ
1
.~σ
2
) − 3(~σ
1
.
~
k)(~σ
2
.
~
k)
i
,
˜
Q
12
= (~σ
1
.~n)(~σ
2
.~n), (4.20)
with the definitions
~
k = ~p
f
− ~p
i
, ~q =
1
2
(~p
f
+ ~p
i
), ~n = ~p
i
× ~p
f
= ~q ×
~
k,
~
S =
1
2
(~σ
1
+ ~σ
2
), (4.21)
and especial Fourier transfor mations for the velocity- dependent central and noncentral compo-
nents. Note that ~p
f
and ~p
i
are in- and outgoing two-nucleon momentum transfers, respectively.
The results for fitting its time pp and np scattering data were good up to the energies about
350 MeV except for the low energies below about 13 MeV.
In the next related work [105], in 1984, a separable representation of the Paris79 potential,
through using a special method, was presented. That representation offered a good approxi-
mation of the on-shell and off-shell properties of the potential. In 1985, a nother adjustment of
the separable representation for the states of
1
S
0
and
3
P
0
was performed [106] to improve the
previous problem in representations.
4.8 Stony-Brook Potential
Stony-Brook potential is also among the original NN potentials based on dispersion relations
and field theory. The group included the contributions from π, ω and ππ exchanges. They
tried to set up a local and energy-dependent regularized potential in p-space by using the field
theoretical elastic NN scattering amplitudes. Indeed, by solving Blankenbecler-Sugar (BbS)
equation, by using a proposed interaction potential, they estimated NN phase parameters.
Some phenomenological par ameters were a dj usted to get satisfactory results compared with
exp erimental data. The short-range repulsion was weaker t han the phenomenological poten-
tials such a s HJ and Reid68 as well as its time OBE potent ials, mainly because of ω- boson
exchange. It is also mentionable that describing experimental dat a and deuteron properties by
the potential was not as good as the phenomenological potentials at that time. For detailed
studies on the potential and the t echniques used there, look at [7] and [43].
4.9 dTRS Super-Soft-Core Potentials
The earlier super-soft-core (SCC) potential, called dTS pot ential in [107], described physical
observables better than the harder-core potentials. But in dTS potential, only the OPEP was
24

purely theoretical while in the next SCC potential [108 ], called dTRS B potent ial, by the same
group, more theoretical components were added. In addition, dTRS B improved fitting NN
scattering da t a besides giving better results for nuclear-matter and many-body calculations
rather than dTS potential.
In dTRS B, the OBEP’s, because of π, ρ, ω exchanges, were considered dir ectly; and the
remaining contributions for the MR part due to the other probable OBE’s and TPE’s were
parameterized phenomenologically by special OBE functions that we mention below. In other
words, the O BEP functions with 32 free ranges and amplitudes were used instead. In the
SR part, below about 1 fm, the potentia l components were regularized, and the core region
phenomenological po t entials were chosen so that the previous results for the LR and MR parts
could not be disrupted. The general form of dTRS B potential in (S, T ) space, reads
V = V
c
(r) + V
t
(r)S
12
+ V
ls
(r)
~
L.
~
S + V
q
(r)Q
12
+ V
ll
(r)L
2
, (4.22)
in which the L
2
potential term is to account the difference between
1
S
0
and
1
D
2
potentials. In
the LR pa r t, the radial dependence of every component reduces t o the OPE contribution of
V
c
, V
t
. For the phenomenological OBEP’s, they used the radial functions of V
c
(r), ... a s linear
combinations of the following functions:
Y
c
(x) =
e
−x
x
= Y (x), Y
ls
(x) = (
1
x
+
1
x
2
)Y (x),
Y
q
(x) =

1
x
2
+
2
x
3

Y (x), Y
t
(x) =

1 +
3
x
+
3
x
2

Y (x) = Z(x),
(4.23)
and
F (r) =
(1.2r)
20
[1 + (1.2r)
20
]
20
, (4.24)
for various states in subspaces of (S, T ) independently. x is the same as that we a lready used
in HJ, Yale and Reid68 potentials except that we have to use m
λ
with λ = pi, ρ, ω here instead
of m
pi
there. The constant coefficients in the linear combinations are, in turn, some functions
of the involved masses and other parameters determined by fitting to experimental data and
from other sources. In the last relation, F (r) is a step-like function t hat is used as a cutoff
to define the core region. Meanwhile, for pp scattering, M = M
p
, m
pi
= m
π
0
; and for np
scattering, M = M
r
= 2M
p
M
n
/(M
p
+ M
n
) and m
pi
=

m
π
0
+ 2m
π
±

/3 are used.
Describing the experimental data below 3 50 MeV and NN bound states were good as the other
phenomenological potentials at that time. Meanwhile, although describing nuclear matter and
some many-body results by using TRS B potent ial, as seen further in [109], were reasonable,
more improvement were yet required. It is also mentionable that there are some likenesses
between this and Paris72 potential [44] framed first.
It is good here to mention another potential built in 1981 , with a similar meson content
and opera t ors as dTRS potent ials, which we call it Melbourne potential [110]. In fact, it is
esp ecially a np potential that includes the OBE’s of π, ρ, ω and TPE of 2π next to some phe-
nomenological features to reproduce experimental elastic scattering data and neutron proper-
25

ties (mainly its binding energy) and low-energy pa r ameters (mainly scattering-length). There,
a special form function was used for each meson contributing to a special energy range. Re-
producing the data and deuteron properties as well as the basic results from nuclear-structure
and -matter calculations were satisfactory.
4.10 Funabashi Potentials
Funabashi pot entials are among meson-exchange po tentials based on field-t heoretical methods.
They included the OBEP o f π, ρ, ω, η and the scalar mesons of δ, σ for LR and MR parts. For
the core region, they included the hard cores, Ga ussian soft cores and velocity-dependent cores.
Indeed, the potentials are nonstatic OBEP’s with retardation in r-space. The no nstat ionary,
mainly because of recoiling, is considered by including the spin-orbit, quadratic spin-orbit
and velocity-dependent terms; whereas the retardation of the meson propag ations causes the
off-energy shell effects that in turn contribute to two-nucleon processes, and ar e even more
important in many-body systems.
The general form of the Funabashi OBEP’s in r-space reads [1 11]
V = V
core
+ V
c
(r) + V
t
(r)S
12
+ V
ls
(r)
~
L.
~
S + V
qll
(r)
`
Q
12
+ V
ll
(r)L
2
−
1
M

∇
2
V
p
+ V
p
∇
2

, (4.25)
in which
`
Q
12
= Q
12
−
2
3
L
2
~
S, V
i
(r) = U
i
(r) + R
i
(r), i = c, t, ls, qll, ll, p, (4.2 6)
where U
i
(r) and R
i
(r) stand for the usual Yukawa and Retarded potential functions, respec-
tively. These functions are in turn expressed as combinations of the functions in (4.2 3) where
the involved masses, coupling constants and other par ameters are used in combinations as co-
efficients in various two-nucleon states and for the various included mesons. The cor e potent ial
also reads
V
core
= V
c
core
(r) + V
ls
core
(r)
~
L.
~
S, (4.27)
where, depended on the case, three different cores are included. a) The hard core (step-like)
potential (OBEH) plus a spin-orbit core as
V
c
core
(r) =

8, r ≤ r
c
,
0, r > r
c
,
, V
ls
core
(r) = −V
ls
(0)
exp

−(rr
ls
)
2

. (4.28)
b) The Gaussian soft core (OBEG) plus a special spin-o rbit core as
V
c
core
(r) = V
G
(0)
exp

−(rr
G
)
2

, V
ls
core
(r) =
1
M
2
1
r
∂V
c
core
(r)
∂r
. (4.29)
26

c) The velocity-dependent core (OBEV) plus a spin-or bit core as
V
c
core
(r) =
p
2
M
φ(r) + φ(r)
p
2
M
, φ(r) = φ
p
0
exp

−(rr
p
)
2

,
V
ls
core
(r) = −V
ls
(0)
exp

−(rr
ls
)
2

.
(4.30)
In addition, to remove singularities and to make OBEP’s in the core region, the V
i
(r)’s in
(4.26) are multiplied by the following cutoff factor
F
i
(r) = 1 − exp

−(rr
cc
)
2

n
, n =

1, when i = c, t, ls,
6, when i = qll, ll,
. (4.31)
The parameters of r
c
, r
cc
, r
ls
, r
G
, r
p
, φ
p
0
, V
ls
(0)
, V
G
(0)
, .... are the constants properly chosen for the
potentials. It is also ment ionable that, to have nonrelativistic potentials, the higher-order
terms than p
2
/M
2
are avoided, where ~p = ~p
i
here is the nucleon momentum.
Next, in [112], the velocity-dependent tensor potentials were included to discuss better non-
static effects. So, the improved potentia l reads
V
(2)
= V −
1
M

∇
2
V
pt
S
12
+ V
pt
S
12
∇
2

. (4.32)
In addition, the core potentials were modified, rather tha n those in the first version of [111],
to improve the phase shifts of
3
P
3
state by including t he a t tractive spin-orbit cores in (4.2 7),
which were in turn set to zero in the first potentials. With these improvements, the properties
of neutron- and nuclear-matt er were evalua ted. Describing experimental scattering data and
low-energy parameters as well as deuteron properties with the latter potentia l was better than
the first one. More improvements to give even better results in nuclear structure calculations
were then done in [113].
Later, a development of the potentials was given in [114]. In fact, it was shown t hat the
radial dep endence of the OBEP’s, which was smooth and finite at origin, could be represented
by a superposition of special Gaussian f unctions. In other words, the Yukawa functions in
Funabashi’s potentials were expanded as
Y (µr) =
e
−µr
µr
=
N
X
n=1
a
n
exp

−(rr
n
)
2

, (4.33)
where the coefficients of a
n
were determined by fitting the data; and for N and r
n
, some special
finite values were chosen. As the authors have claimed, the new potentials give better fit of
NN scattering data.
4.11 Urbana-Group Potentials
UrbanaV14 ( Urb81) potential [29] is a charge-independent fully phenomenological potential
including the operators of central, spin-spin, tensor, spin-orbit, centrifugal, centrifugal spin-
27

spin with general dependence on isospin. Besides an LR OPE part and a representation of MR
part as TPE’s with 14 parameters, the SR part is describ ed by two Woods-Saxon pot entials
with free parameters fitted to experimental data. The whole potentia l reads
V =
n
X
i=1
V
i
O
i
, ( 4.34)
in which the fourteen operato r s (n = 14 here) read
O
i=1,...,14
=1, ~σ
1
.~σ
2
, ~τ
1
.~τ
2
,

~σ
1
.~σ
2

~τ
1
.~τ
2

, S
12
, S
12

~τ
1
.~τ
2

,

~
L.
~
S

,

~
L.
~
S

~τ
1
.~τ
2

,
L
2
, L
2

~σ
1
.~σ
2

, L
2

~τ
1
.~τ
2

, L
2

~σ
1
.~σ
2

~τ
1
.~τ
2

,

~
L.
~
S

2
,

~
L.
~
S

2

~τ
1
.~τ
2

,
(4.35)
and the radial potentials are
V
i
= V
i
π
(r) + V
i
M
(r) + V
i
S
(r), (4.36)
where V
π
, V
I
, V
S
stand for the pion- exchange po tential, MR and SR potentials, respectively.
Furt her, we should note that the first eight operators of (4.35) are obtained from fitting the
phase shifts of ℓ < 4 up to the labora t ory energies of 425 MeV, and deuteron properties.
The next six ”quadrat ic-L” operators are introduced to do many-body calculations with the
potentials and have almost weak effects. We shorten the operat ors as c, σ, τ, στ, t, tτ, ls, lsτ,
ll, llσ, llτ, llστ, ls2, ls2τ , to simplify their using, from now on.
The LR OPEP of V
i
π
(r) is nonzero just for i = στ, tτ with
V
στ
π
(r) = 3.488
e
−0.7r
0.7r

1 − e
−cr
2

, (4.37)
V
tτ
π
(r) = 3.488

1 +
3
0.7r
+
3
(0.7r)
2

e
−0.7r
0.7r

1 − e
−cr
2

2
= 3.488T
π
(r), (4.38)
with a note t hat x = µr with µ = 0.7fm
−1
is considered. The cutoff parameter of c is obtained
by fitting the experimenta l phase shifts, and that the 1/r and 1/r
3
singularities of OPEP’s
are removed. A remarkable point is that (1 − e
−cr
2
)
2
, as argued in [115], simulates ρ-meson
exchange effect. Another po int is that because nucleon is not a point source, the two-nucleon
interaction should not have the singular behavior of 1/r at small distances.
The MR potential of V
i
M
(r) is considered as
V
i
M
(r) = S
i
T
2
π
(r). (4.39)
With respect to the T
2
π
(r) included, this potential is usually owned to the second-order OPEP’s.
This form of V
i
M
(r) is suitable to include three-nucleon (3N) interactions as argued in [11 6].
Besides, the strengths of S
i
are determined by fitting experimental phase shifts.
For the SR potential of V
i
S
(r), in contrast to the custom method where Yukawa functions are
28

used, here a sum of two Woods-Saxon potentials is considered as
V
i
S
(r) = S
i
1
W
1
(r) + S
i
2
W
2
(r), (4.40)
where
W
1
(r) =

1 + exp

r − R
1
a
1

−1
, W
2
(r) =

1 + exp

r − R
2
a
2

−1
. (4.41)
For all i’s, except for ls and lsτ, a good fit of data is achieved with S
i
2
= 0.
There are some likenesses between pa r ameterizing the Urb81 potential with those used in
Hamada-Johnston [24], Yale- group [25], Reid68 [26] and also Bressel et al. [103] potentials.
The values of free pa r ameters are obtained mainly by fitting the np phase shifts by Arndt et
al. [117] and its time analysis by Bugg et al. [11 8], with some differences and adjustments.
Describing scattering data and deuteron properties with Urb81 potential are satisfactory with
similar results as the Reid68 a nd Paris79 [98 ] potentials. For more details see [2 9].
4.12 Argonne-Group Potentials
4.12.1 ArgonneV14 and Ar gonneV28 Potentials
The basic potential of Argonne-group, ArgonneV14 ( Arg84) pot ential [30], has a similar struc-
ture with UrbanaV14 (Urb81 ) [29] potential with a few differences. The first difference is that
the used πNN coupling is larger than that used in Urb81 potential. Second, in Arg84, con-
trary to Urb81 potential, S
i
2
is nonzero for i = t, tτ. Third, as a pro bable result of the former
constraint, that is no need to insert a second SR Woods-Saxon function for i = ls, lsτ as is in
Urb81 potential. As a result, in high-energies (low-distances), the phase shifts are fitted well
and especially the D-state of deuteron takes mor e contribution than that in Urb8 1 potential.
Furt her, the effects of t he six quadratic-L operators are confirmed in some nuclear structure
calculations. The pot ential was fitted to the phase-shift analyses of Arndt and Ro per in 1981
(an update of the analyses in [117]). Still, in the energy range of 25 -350 MeV, the Arg84
potential provides an improvement over Urb81 potential.
It is good to mention another potential of Argonne-group, called ArgonneV28 [30], framed
simultaneously with ArgonneV14 potential. It includes the ∆(1232) degrees of freedom, which
play important roles in both TPE processes in the MR part of NN systems as well as TPE and
repulsive parts of 3N systems. The effects because of including these degrees of freedom are
shown by 14 extra o perators next to the 14 operators used in Arg84 potential. From these 14
extra operato r s, 12 transition operators are for all πN∆ and π∆∆ couplings while 2 central
operators ar e for N∆ and ∆∆ channels. The extra operators are so chosen that no other
free parameters than those used in Arg84, but the coupling constants of πN∆ and π∆∆, are
required to fit the experimental scattering data. In general, ArgonneV28 potential has a more
complicated structure and gives better results especially in many-body calculations.
29

4.12.2 ArgonneV18 Potential
ArgonneV18 (Arg94) NN potential [31] is an improved and upda t ed version of Arg84 NN
potential [30]. It addition to 14 operators of Urb81 and Arg84 pot entials, it includes three
charge-dependent and one charge-asymmetry operators next to a complete electromagnetic
interaction. Arg94 potential ha s the following general form:
V = V
EM
+ V
N
= V
EM
+ V
π
+ V
R
, (4.42)
where V
π
is for an LR OPEP, V
R
is for MR and SR parts (called the Remaining part), and
V
EM
is for electromagnetic (EM) part.
The EM part, in turn, reads
V
EM
= V
C1
(r) + V
C2
(r) + V
DF
(r) + V
V P
(r) + V
MM
(r), (4.43)
where the terms with the indices C1, C2 , DF, V P and MM stand for one-pho ton, two-photon,
Darwin-Foldy, vacuum-polarization and ma gnetic-moment interactions. In these interactions,
some short-r ange terms and the effects due to finite size of nucleon are also included. The
terms of V
C2
, V
DF
, V
V P
are used just for pp scattering while the other terms have its own forms
for each t hr ee scattering cases; and that for nn scattering, just V
MM
is used. The various
EM potentials are given through some combinations of the following functions, with masses,
coupling constants, coefficients and other constants determined from other sources or from
exp erimental data,
F
c
(r) = 1 −

1 +
11
6
x +
3
16
x
2
+
1
48
x
3

Y
(0)
, (4.44)
F
δ
(r) = b
3

1
16
+
1
16
x +
1
48
x
2

Y
(0)
, (4.45)
F
t
(r) = 1 −

1 + x +
1
2
x
2
+
1
6
x
3
+
1
24
x
4
+
1
44
x
5

Y
(0)
, (4.46)
F
ℓs
(r) = 1 −

1 + x +
1
2
x
2
+
7
48
x
3
+
1
48
x
4

Y
(0)
, (4.47)
where x = br, b = 4.27fm
−1
. They are related as
F
δ
= −∇
2

F
c
r

, F
t
=

F
c
r

′′
−

F
c
r

′
/r, F
δ
= −∇
2

F
c
r

, (4.48)
and for a point-like nucleon go to 1. In other words, these SR f unctions show the finite-size
of the nucleon charge distribution with a dipole for m-factor. It is also mentionable that for
pp case, V
C1
, V
C2
, V
V P
are in terms of F
c
, with the mentioned adjusted parameters; V
DF
is in
terms of F
δ
, while V
C1
is in terms of
F
np
(r) =
b
2
384

15x + 15x
2
+ 6x
3
+ x
4

Y
(0)
. (4.49)
30

V
MM
(pp) includes F
δ
, F
t
, F
ls
together with spin-spin (~σ
1
.~σ
2
), spin-orbit (
~
L.
~
S) and tensor (S
12
)
operators, while V
MM
(np) includes the same functions a nd operators as in pp case besides
the CS operator of
~
L.
~
A with
~
A =
1
2
(~σ
1
−~σ
2
) [119]; and V
MM
(nn) includes F
δ
, F
t
with just
spin-spin (~σ
1
.~σ
2
) and tensor (S
12
) operators. One should note that the r adial dependencies,
various coefficients and combinations are different for all three cases. It is also notable that
the vacuum-polarization and two-photon interactions are useful to fit the low-energy scattering
data, and that F
c
, F
2
c
used in V
V P
, V
C2
, are the estimated ways to remove the singularities of
1/r, 1/r
2
, respectively.
The LR OPE part of Arg94 potential (V
π
) is charge-dependent, because of the differences
between the neutral- and charged-pion masses. It reads
V
π
(N
1
N
2
) = f
N
1
N
1
f
N
2
N
2
V
(3)
OP EP
(m
π
0
) + (−1)
T +1
2f
2
c
V
(3)
OP EP
(m
π
±
), (4.50)
where the second term on the RHS is used just for np system, i.e. with N
1
= n, N
2
= p and
that f
pp
= −f
nn
= f
c
≡ f [12 0] with f
2
= 0.0 75, and
V
(3)
OP EP
(m
pi
) =

m
pi
m
π
±

2
1
3
m
pi
c
2
[Y
µ
(r)~σ
1
.~σ
2
+ T
µ
(r)S
12
] , (4.51)
in which
Y
µ
(r) =
e
−µr
µr

1 − e
−cr
2

, (4.52)
T
µ
(r) =

1 +
3
µr
+
3
(µr)
2

e
−µr
µr

1 − e
−cr
2

2
, (4.53)
where Y
µ
(r) and T
µ
(r) are the common Yukawa and tensor functions with exponential cut-
offs similar to those in Urb81 and Arg84 potent ials, µ = m
pi
c/~ as before (with m
pi
=
m
π
0
, m
π
±
in the formulas); and the scaling mass of m
π
±
in (4.51) makes the coupling-constant
dimensionless–Look at the diff erences between (4.51) and (4.1 1), a s well a s among (4.52) and
(4.53) with (4.37) and (4 .38), respectively.
Similar to Urb81 and Arg8 4 potentials, the remaining (MR and SR) phenomenological
parts could be written as a sum of all eighteen terms as
V
R
=
n=18
X
i=1
V
i
O
i
, (4.54)
where 14 out of the 18 operators are those in (4.35) of the Urb81 potential and the 4 remaining
ones are
O
i=15,16,17,18
= T
12
, T
12
(~σ
1
.~σ
2
), T
12
S
12
, (τ
z1
+ τ
z2
), T
12
= 3τ
z1
.τ
z2
−~τ
1
.~τ
2
, (4.55)
that in turn we mark them with the indices i = T, σT, tT, τ z, respectively.
Now, we note t hat in the operator form of (4.54), the whole CD po t ential could be separated
into a CI part and a CIB part, where the latter in turn could be separated into a CD part with
31

three CD operato r s (i = T, σT, tT ) and a charge-asymmetry (CA) part with one operator
(i = τz). In this procedure, the radial potentials of V
i
could be expressed in terms of the
following potentials, with suitable weighting coefficients,
V
i
ST,N N
= S
i
ST,N N
T
2
µ
(r) +

P
i
ST,N N
+ µrQ
i
ST.N N
+ (µr)
2
R
i
ST,N N

W
3
(r), (4.56)
where now µ =
1
3
(m
π
0
+2m
π
±
)c/~ – It is also mentionable that the pot ential is basically written
in (S, T, T
z
) space for various two-nucleon states [31]. The same as in Urb81 potential, T
2
µ
(r)
is to simulate TPE force, and the only Woods-Saxon function is
W
3
(r) =

1 + exp

r − R
3
a
3

−1
. (4.57)
All constant coefficients of S
i
ST,N N
, P
i
ST,N N
, Q
i
ST,N N
, R
i
ST,N N
are obtained by fitting experimen-
tal dat a for each two-nucleon state. Further, they imposed the regularization conditions at the
origin as
V
i
ST,N N
(r = 0 ) = 0,
∂V
i6=t
ST,N N
∂r





r=0
= 0, (4.58)
which reduces the number of free parameters for each V
i
ST,N N
by one.
It is also no t able that the EM int era ctio n (and also CD ) of Arg94 potential is the same as
that used in Nijmegen partial-wave-analysis (PWA93) [121] besides including the short-r ange
terms and effects for the finite-size of nucleon. This potential is fitted to the Nijmegen pp
and pp scattering database [122], [121], low-energy nn scattering data, and deuteron binding-
energy. It has 40 adjustable pa r ameters and gives a best description of the da ta in the energy
range of 0-350 MeV as a high-quality NN potentia l. The effects of CD and CA ar e explicitly
seen in finite nuclei systems and the results in many-body and nuclear-structure calculations
are more satisfactory tha n the mentioned potentials so far. Another extension of the Arg94
potential, called ArgonneV 18pq potential, is presented in [123], where various choices for the
quadratic momentum-dependence in NN potentials, to fit the phase shifts of the high part ial
waves, are included. There is also a p-space proposal for Arg94 potential presented in [124].
4.13 Bonn-G r ou p Potentials
4.13.1 Full-Bonn Potential
The Bonn-group has used the field-theoretical methods to deal with NN interaction problem. In
the first version [45], in 1987, they presented a comprehensive NN pot ential by including various
meson-exchanges t hat they thought were important below the pion-production threshold. To
do so, the mesons of π, ω, δ as O BE’s and ρ, 2π (as the direct exchange and ∆(12 32)-isobar
excitation) as TPE’s as well as a special combination of πρ were considered. There were also
3π, 4π exchanges that did not have significant contributions. Indeed, the OBE contributions
provided good descriptions of high ℓ phase shifts while the TPE’s with πρ combination provided
good descriptions of low ℓ phase shifts. So, in general, the exchanges of π and ω to gether with ρ
32

and 2π provided good descriptions of t he L R and MR (high ℓ’s) parts while for good describing
the SR part (low ℓ’s), including the πρ combination next to 2π exchange was required. We
should also mention that t he δ meson was needed to provide a consistent description of S-wave
phase shifts, and that including the crossed-box diagrams in the two-boson-exchanges (TBE’s)
made another fitting quality of the potential.
This Full-Bonn (Bonn87 or Bonn-A) potential is originally written in p-space and is energy-
dependent that makes its applications in nuclear calculations problematic. To resolve some
of the problems, a parameterization of the potential in terms of OBE’s in both p-space and
r-space is given, which is always called Bonn-B (or Bonn89) potential. As a first step, the
retardation terms are neglected to suppress the energy dependence by applying the O BE’s in
the framework of reducing BS equation into BbS equation, where the latter equation is similar
to nonrelativistic Schrodinger equation while it is a relativistic equation. The resultant energy-
independent p-space OBE potentials are useful t o apply in nuclear structure calculations. The
details can be found in [45], [37]. The latter expansion changes somewhat the original results
because of some new adjustments. Anyhow, to do so, first the effects of 2π + πρ exchanges
are replaced by the scalar-isoscalar σ-meson exchange; and without the πρ cont ribution in the
new OBE expansion, the η meson is introduced to improve the
3
P
1
phase shifts.
The general form of the expanded potential in r-space, coming from the Fourier transform of
the agreeing p-space contributions, can be written as a sum on the six boson contributions as
V =
X
α=π,ρ,η,ω,δ,σ
V
OBE
α
, (4.59)
which in turn divides into a local and a nonlocal part; or may be written as
V = V
c
(r) + V
t
(r)S
12
+ V
ls
(r)
~
L.
~
S −
1
M

∇
2
V
p
+ V
p
∇
2

, (4.60)
in (S, T ) space, where V
c
(r) includes the contributions from all six mesons of π and η (pseu-
doscalar mesons), δ and σ (scalar mesons), ρ and ω ( vector mesons), a nd is written in terms of
c, σ, τ, στ o perators together with Y
c
(x) (x = m
α
r) and the nucleon and included meson masses
and couplings as well as some constants; and similarly for the other two functions V
t
(r), V
ls
(r).
But V
t
(r) includes the contributions from the four pseudoscalar and vector mesons together
with Y
t
(x); and V
ls
(r) includes the contributions from the four scalar and vector mesons to-
gether with Y
ls
(x). Some of the scalar functions are defined in (4.23) and that here
V
p
=
X
β
C
0
g
2
β
4π
m
β
4M
Y (m
β
r), (4.61)
where β = δ, σ, ρ, ω; g
β
is the suiting meson coupling, and C
0
= 1 for the scalar mesons and
C
0
= 3 for the vector mesons. It is also mentio nable that general structure of the potential
for the pseudoscalar mesons is similar to the OPEP used, for example, in the Yale-group
and Reid68 potentia ls (4.11), which is in turn related to the fact that the pion provides the
33

main long-range part of the interaction here as well. It is also not able that the potentials
are regularized at the origin by the dipole f orm factors, which are coming from the Fourier
transformations of
F
α
(k
2
) =

Λ
2
α
− m
2
α
Λ
2
α
+ m
2
α

n
α
, (4.62)
where for each vertex n
α
= 1 and Λ
α
is the so-called cutoff mass. In other wo rds, the SR part
of the interaction is parameterized through the phenomenological form factors attached to the
p-space Feynman diagrams, while the high-momentum part of the scattering amplitudes are
then regularized with the cutoffs. The cutoff masses (Λ
α
) are adjusted to fit the data and are
given in [45] next to other potential pa r ameters, coupling and constants.
By the way, the Bonn87 potential described very good its time experimental NN data up to
T
lab
=300 MeV, low-energy parameters and deuteron properties. Meanwhile, we note that, t he
weak tenor force there, because of the ρ-meson exchange and including a real πNN form-factor
as well a s introducing the meson retardation, caused a smaller contribution of the deuteron
D-state; and at the same time, the larger quadruple moment and the asymptotic D/S state of
deuteron were in full agreement with experimental results—However, in a work done in 1993
[125] to compare some of the potential forms with pp scattering data, it was shown that the
adjusted r-space versions [37], i.e. Bonn-A and Bonn-B potentials, give a very poor description
of the scattering data (χ
2
/N
data
> 8 in the energy range of 2-3 50 MeV). That was not stra ng e
of course, in that Full-Bonn pot ential was originally fitted to np scattering data and not to pp
scattering data.
In addition, these potentials have many other special adva ntages to describe well NN inter-
actions. The nucleons, isobars (nucleon resonances) and mesons are discussed on an equal
footing. Because of relativistic approach, the meson retardation (r ecoil effects) and the off-
shell behavior of the nuclear force were included besides that a consistent expansion to the
regions above the pion-production threshold was possible. Further, the potential could discuss
about three-body nuclear forces (at least because of an almo st complete set of the diagra ms
contributing to t he NN interaction and expandable to the 3N case), the meson-exchange cur-
rents contributing to the electromagnetic properties of nuclei, the medium effects of t he NN
interaction in many-body calculations and also CSB and CIB issues. It is also notable that
the cutoff masses, used in the meson-nucleon vertex f orm functions, to explain the extended
structure of hadrons, are obtained in a consistent way t o be Λ
α
=1.2-1.5 GeV, where applying
the meson-exchange picture is suppressed. For detailed studies of various aspects of Bonn-A
and Bonn-B potentials and the already mentioned issues, look at the original study of [37],
where the final version of the Full Bonn potential was presented in 1989.
To remind the main differences, we note that Bonn-A potential includes the correlated
2π and πρ contributions with an intermediate ∆-isobar, while Bonn-B potential is a so-called
OBE potential that uses a fictitious σ-meson (and also a η-meson) to simulate these two meson
exchanges. In contrast to Bonn-A potential, Bonn-B potential is energy-indep endent that in
turn simplifies its applications in nuclear structure and nucleon-nucleus scattering calculations.
Despite its greater simplicity, Bonn-B potential gives a good description of its time data, and
34

the other results almost identical with tho se fo und in Bonn-A potential.
Still, we note that the p-space Full-Bonn potent ial was fitted just to np scattering data. In
1989, another development of the potential to apply it to pp scattering dat a was presented in
[126]; see also [127]. To do so, a Coulomb interaction (similar to V
C
1
in Arg94 potential [31])
was introduced in t he p-space calculations. Then, after a few minor adjustments (for example,
the coupling constants of the scalar mesons changed) to face the pot ential with data, a good
description of pp dat a was found as well.
4.13.2 CD-Bonn Potential
The Bonn Charge-Dependent (CD-Bonn) NN potential [99] is an improved and updated version
of the previous Bonn-A and Bonn-B potentials [45, 37]. It is based on the OBE contr ibutions
of π, ρ, ω mesons next to two scalar-isoscalar mesons of σ
1
, σ
2
, which the latter simula t es the
roles of 2π + πω exchanges. The resultant potential is energy-independent in the framework of
nonrelativistic LS equation and produces the results of Full-Bonn potential. In addition, the
predictions of the latter potential such as CSB and CIB (for all partial waves below J ≤ 4)
are involved directly. Fur t her, the predicted off-shell effects because of relativistic Feynman
amplitudes for the meson exchanges, which are important in microscopic nuclear structure cal-
culations, are included. It is notable that the first version of the CD-Bonn potential presented
in [128] involved mor e with the off-shell analyses than the CD issues.
Although CSB in the potentials is mainly due to the difference between the pr oton and neutron
masses (the nucleon mass splitting), in CD-Bonn potential an equivalent contribution is due
to TBE (mainly 2π and πρ excha nge) diagrams. On the other hand, CIB is mainly due to the
difference between the neutral- and charged-pion masses (the pion mass splitting) from OPE
diagram, while in CD-Bonn potential an almost equivalent contribution (about 50%) is due to
TBE and πγ diagra ms for ℓ > 0 (or with the predictions of Full-Bonn model due to 2π as well
as 3π and 4π exchange diagrams). To see CIB in the potential, we first note that although the
OPE amplitudes in the potential are nonlocal, but in the local/static approximation and after
a Fourier transformation, the local OPEP in r -space reads
V
(4)
OP EP
(m
pi
) =
g
2
pi
12

m
pi
2M

2

e
−µr
r
−
4π
µ
2
δ
3
(~r)

(~σ
1
.~σ
2
)
+

1 +
3
µr
+
3
(µr)
3

e
−µr
r
S
12

,
(4.63)
where µ = m
pi
c/~. Now, because of the pion mass splitting (a s the main CIB factor), we have
V
pp
OP EP
= V
(4)
OP EP
(m
π
0
),
V
np
OP EP
= −V
(4)
OP EP
(m
π
0
) ± 2V
(4)
OP EP
(m
π
±
),
(4.64)
where in the second relation, + (−) is for T = 1 (T = 0). We see that because of the pion
mass differences, the np OPEP with T = 1 is weaker than that of pp, leading to CIB. It is also
notable that the ∆-isobar states and multi-meson exchanges in Full-Bonn (Bonn-A) potential
35

caused the energy dependence which was in turn problematic in applying the potentials to
direct nuclear calculations. So, in CD-Bonn potentia l (also in Bonn-B pot ential) this problem
is avoided by using just OBE cont r ibutions.
The three potentials of p p, np, nn are not independent but they ar e related by CSB and
CIB. Each of them is first fitted to the related Nijmegen phase shifts; then by minimizing the
earned χ
2
from the Nijmegen error matrix and finally minimizing the exact χ
2
, which is in
turn obtained from comparing with all related scattering data, the potential parameters are
adjusted. For the Coulomb force in pp case, a similar V
C1
as in [126] is used, and the r elat ivistic
Coulomb interaction besides nuclear phase shifts are considered as well. The base phase shifts
are a sum of the Nijmegen-group ones in [122], [121] up to 1992, used also in Arg94 potential,
besides the published data aft er-1992-date and before-2000-date. So, CD-Bonn po t ential fitted
the world 2932 p p data below T
lab
=350 MeV available in 2000 with χ
2
/N
data
= 1.01 and the
corresponding 3058 np data with χ
2
/N
data
= 1.02. This reproduction of NN data is more
accurate than by any other previous NN potentia ls, according to it s authors of course! For
more details, such as its first applications to few-and many- body nuclear calculations, CIB,
CSB and off-shell effects, see the original papers [128, 99]; and also look at [9].
4.14 Padua-Group Potential
The Padua model for NN interaction, as a mixture of meson-exchange theory and phenomeno-
logical methods, is a special and important effort. The group has tried to set up an NN
potential based on their special model for ”Nucleon”. They have used a nonlocal pot ential
coming from the Padua nucleon-model with similar operato r s as in Hamada-Johnston [24],
Yale-group [25] and dTRS Super-Soft-Core [108] Potentials.
Indeed, the various terms with the operato r s shortened as c, σ, τ, στ, t, tτ, ls, lsτ, ll, llτ, ls2, ls2τ
are included. The general f orm of the potential, in (S, T ) space, can be written as
V = V
c
(r) + V
t
(r)S
12
+ V
ls
(r)
~
L.
~
S + V
ls2
(r)(
~
L.
~
S)
2
+ V
ll
(r)L
2
, (4.65)
where the radial functions have special forms almo st different from the other potentials men-
tioned so far . In fact, various contributions of t he pion and other single mesons as well as
two-pion combinations are introduced through these functions. The functions are in turn in
terms of special combinations of some radial functions and operators with included meson
masses, their coupling constants, amplitudes, and o t her free parameters and constant coeffi-
cients. Plainly, both V
c
(r) and V
t
(r) include the contributions f rom the mesons of π, ρ, ω, η, ´η,
and are written in terms of c, σ, τ, στ and c, τ operators respectively, t ogether with some func-
tions such as Y (x), Z( x), ... (x = m
π
0
r) and the nucleon and included masses as well as some
other coupling and coefficient constants. Similarly, in V
ls
(r), V
ls2
(r), V
ll
(r), the contributions
from the mesons of ρ, ω, s and the operators of c, τ are included. It is also notable that one
may use the operators of
~
L. (~σ
1
−~σ
2
) and/or
~
L. (~σ
1
×~σ
2
) instead of
~
L.
~
S in the Padua model
as they are also consistent.
In g eneral, the involved radial functions in the potential are more based on theoretical knowl-
36

edge by aiding of the nucleon-model rat her than merely fitting experimental data. Nevertheless,
reproducing deuteron pa r ameters and fitting phase shifts are good compared with the coun-
terpart results of its time potentials such as Arg94 [30], Bonn [45] and Paris [44] potentials.
Although it is rarely used in nuclear calculations, the Padua NN potential is a serious try
to find an even more sensible NN potential. For o t her interesting theoretical and numerical
analyses in their method, see the original paper [46].
4.15 Nijmegen-Group Potentials
The Nijmegen-group potentials ar e mainly the mixtures of meson exchanges with phenomeno-
logical characterizes and are often referred to QCD degrees of freedom for the SR part. The
group built various Baryon-Baryon (BB) and Baryon-AntiBa ryon (B
¯
B) potentials among
which are some high-quality NN and Hyperon-Nucleon (YN) potent ials. First, they presented
a few potential before 1990’s and then perfo rmed the partial-wave-analysis (PWA) [121 ], [129]
of the exp erimental scattering data. The insights gained from the analyses were then employed
to set up some improved and better potentials. In their NN potentials, besides the fa mous OBE
parts, many new features and other meson contributions are included. The nucleon- a nd pion-
mass splitting are o ften considered and, for the potentials after PWA93, charge dependence is
used. Because of the short-range par ameterization, because of the vertex form f unctions, the
potentials are in contact with QCD. The potentials may be divided into a t least four classes;
the Har d-Core (HC), Soft-Core (SC), Extended Soft -Core ( ESC), and High-Qua lity (HQ) po-
tentials as well as PWA’s. We address in the following subsections some of their NN potent ials
briefly; look also at [130].
4.15.1 The First Potentials
The main aim was to form BB po t entials below the pion-production threshold. As we know,
the OBEP’s describe almost well the LR and MR parts next to including uncorrelated 2π
or scalar meson exchanges. Further, to describe the dat a better, the fictitious meson of σ
(as a correlated 2 π excha ng e) is always required. In these models, the heavier meson of ǫ is
sometimes used as well. The Schrodinger equation in r-space is solved with local potentials
and Coulomb force (depended on the case) and, in addition, the SR repulsion is considered
through HC potentials. The first potentials of the group, named as NijmA, NijmB, NijmC,
NijmD, NijmE and NijmF, were represented from 1972 to 1978.
NijmA potential [131] is composed of some OBEP’s and a TPEP. Indeed, it includes the
members of the pseudoscalar- and vector- meson nonets as well as the Brueckner-Watson TPEP.
The potential was to describe low-energy YN data though it was not so good to describe the
high NN partial waves. NijmB a nd Nij mC potentials [132] are OBEP’s fully and reproduced
well their time NN scattering data; the group also showed that one can describe the YN
channels with this OBEP approach.
It is notable that in the pure OBEP’s, the mesons were considered in an SU(3) consistent way.
That was mainly because one then could extend the calculations from NN to YN systems as
37

well. For example, in the vector-meson ( pseudoscalar-meson) nonet, one should use ρ, ω, φ
(π, η, ´η) and all knowledge about φ − ω (η − ´η) mixing and coupling constants fro m SU(3).
The OBEP’s were constructed in two classes I and II, where both used the nonets of the
pseudoscalar- a nd vector- mesons but they were different in discussing the scalar mesons. In
the class I, just the singlet scalar meson of ǫ was included while in the class II, an octet of
the scalar mesons was included. The first model of the class I was NijmB potential with
m
ǫ
= 720MeV and Γ
ǫ
= 400MeV that gave almost χ
2
/N
data
= 5.9 for its time NN scattering
data below T
lab
=330 MeV of the Livermore-gr oup [104] of 1969.
NijmD potential [1 33] is belong to the class I OBEP’s and is similar to NijmB potential
except for including the η − ´η mixture, m
ǫ
= 760MeV and Γ
ǫ
= 640MeV, a different ratio of
F/(F + D) for the pseudoscalar octet, the slightly different potential for ms for vector- and
scalar-mesons, as well as some other coupling and parameter changes. Clearly, the NijmD NN
potential includes the nonets of the pseudoscalar mesons of π, η, ´η and the vector mesons of
ρ, ω, φ, each with a singlet-octet mixing angle as well as the unitary singlet scalar-meson of ǫ.
For short distances, it uses some strong r epulsive phenomenological HC potentials, which in
turn should simulate the effects of the absent heavier-meson exchanges, inelastic effects and
so on. This HC para meterization is suitable, rather than the vertex form factors, in that it
is independent of the meson dynamics and is simple to use with Schrodinger equation. The
13 parameters of the potential, which are 8 meson-nucleon couplings and 3 core radiuses, ar e
determined from data fitting.
The general form of NijmD potential can be written in the operator format as (4.34), where
now n = 10 and c, σ, τ, στ, t, tτ, ls, lsτ, q, qτ are the indices for the 10 involved operato rs.
In other words, one may say that the potential includes the central, tensor, spin-orbit and
quadratic spin-orbit terms in (S, T ) space. The po t entials of V
i
are gained from field theory
with some approximations such as ignoring their total energy dependence, and writing the
energy factors as E ≃ M +k
2
/8M, where the no t ations ar e those in (4.21) . This approximation
means that j ust the terms up to the order of k
2
/M
2
are kept in the p-space potentia ls. In
addition, there are the recoil effects to the quadratic spin-or bit potentials that cause the total
energy dependence. Furt her, in Fourier transform to r-space, all the terms that include
~
∇
r
are
neglected except that in L
2
operator. It is also notable t hat the meson bandwidth was settled
with a special propagat or instead of the static meson propagator of 1/(k
2
+ m
2
); and after the
Fourier tr ansform to r-space, a superposition of the Yukawa functions resulted.
Resultant potentials in r-space are in terms of the functions of Y (x), Z(x) in (4.8) and the
proper o perators and coupling constants as well as the nucleon and t he pion averaged masses.
Indeed, we note that the potential fo r the pseudoscalar mesons is similar to the Full-Bonn
potential [45], where both have a similar structure as the OPEP of (2.1) or tha t in the Yale-
group [25] and Reid68 [26] phenomenological potentials. Anyway, it is determined that all
mesons contribute to the central potentials (with the function of Y
c
(x)), the pseudoscalar and
vector mesons contribute to the tensor pot entials (with the function of Y
t
(x)), the scalar and
vector mesons contribute to the spin-orbit potentials (with the function of Y
c
(x)) and to the
quadratic spin-orbit potentials (with the function o f Y
t
(x)). Still, for the short distances of
38

r . 0.5, the HC radius x
c
has four different values for the four channels of
1
S
0
,
3
S
1
−
3
D
1
,
ℓ = 1, and ℓ ≥ 2.
The pp+np scattering data of the energy-independent phase-shift analyses of the Livermore-
group [104] were fitted good with χ
2
/N
data
= 2.4 for NijmD potential, next to good describing
low-energy scattering parameters and deuteron properties. Then, the YN version of the NijmD
potential was shown in [134]. In fact, there, some ΛN and ΣN potentials were presented with
considering charge symmetry between the Λp and Λn channels. The contributions for a scalar
octet in t his YN potential were neglected (just ǫ with an importa nt role in YN scattering was
included) to prevent introducing more free parameters in the potential. It was argued that the
YN interaction there next to NijmD NN potential describe all studied BB systems well.
NijmE potential [135] is almost the same as NijmD po t ential except for the contributions
of the scalars in the nonet; meanwhile the results are almost same. NijmF potential [135]
completed the HC po tentials to describe all experimental known BB systems. Indeed, the
need to settle the scalar-octet coupling constants fo r YN systems, without increasing the
number of pa r ameters, led t o a different HC potential. Further, that need led to strong er
SU(3) constraints between NN and YN analyses t han before. With t he changes, such as those
of the coupling constants and relations among them, they earned better results than Nijm B
potential with NijmF potential [135].
4.15.2 Nijm78 Potential
Nijm78 potential [9 5], published in 1978, is a mixture of OBPE’s and one-Reggon- exchange-
potentials (OREP’s). In fact, it includes the vector mesons of ρ, ω, φ; the pseudoscalar mesons
of π, η, ´η, with the couplings and mixings from their suiting SU(3) relations; the scalar mesons
of δ, S
∗
, ǫ(760 ) ; the dominant J = 0 contributio ns of Pomeron (P) (o r multi-gluon exchanges),
and f,
´
f, A
2
tensor Regge-trajectories. So , this nonlocal and SC potential is indeed based on
Regge-pole theory for low-energy NN interaction and fits high-energy data by using exponential
form factors.
In p-space, the general form of Nijm78 potential, the OBEP’s with p-dependent central
terms and Pomeron- type potentials, reads
˜
V (~p
i
, ~p
f
) =
˜
V
0
(k
2
, q
2
) +
˜
V
σ
(k
2
)~σ
1
.~σ
2
+
˜
V
t
(k
2
)
˜
S
(0)
12
+
˜
V
ls
(k
2
)
˜
LS
1
+
˜
V
q
(k
2
)
˜
Q
12
, (4.66)
where the symbols a r e those in (4.20) except
˜
S
(0)
12
= (~σ
1
.
~
k)(~σ
2
.
~
k). With the last relation,
one should not e that we have just nonlocality in the central potential that means all the
momentum dependence in r-space is in the central part of the potential. Meanwhile, in the
Fourier tra nsform into r-space, the energy-factor is approximated by E ≃ M +k
2
/8M +q
2
/2M
and that just the first order terms in k
2
/M
2
, q
2
/M
2
are kept. Now, by the approximations,
one can write the potentials of
˜
V
i
(i = c, σ, t, ls, q) for all four sets of the involved mesons. The
potentials so are some combinations of k
2
, q
2
, meson and nucleon ma sses, coupling constants
39

and the exponential form-factor o f ∆ as
∆ =
1
~
k
2
+ m
2
mes
e
−
~
k
2
/Λ
2
, ∆
P
=
1
M
2
p
e
−
~
k
2
/4m
2
p
, (4.67)
where m
mes
, m
P
, M
p
, Λ are the meson, Pomeron, proton ( a scale mass) masses and the cutoff
mass (964.52 MeV here), respectively.
The Fo urier transforms of the potentials into r-space, for the central, tensor, spin-orbit and
quadratic spin-orbit potentials are given in the Nijm87 original paper [95]. The pot entials
so are in terms of some f unctions of φ
0
c
(r), φ
1
c
(r), φ
2
c
(r), φ
0
t
(r), φ
1
t
(r), φ
0
ls
(r), φ
1
ls
(r), which are in
turn in terms of m
mes
, m
P
(just for the Pomeron-type potentials) and Λ. Further, the Fourier
transform of the form-f actor of ∆ in (4.67) becomes
˜
∆ =
m
mes
4π

1
4
m
2
mes
φ
1
c
(r) −
1
2

∇
2
φ
0
c
(r) + φ
0
c
(r)∇
2


, (4.68)
and similar for ∆
P
by setting
1
2
Λ = m
P
, m
mes
= 0, φ
P
n
j
(r) = φ
n+1
j
(r) with j = c, t, ls here–For
a study of the Fourier transformation in such cases look, f or instance, at [136].
Now, we can write the r-space potential, in (S, T ) space, as
V = V
c
(r) + V
t
(r)S
12
+ V
ls
(r)
~
L.
~
S + V
q
(r)Q
12
−
1
M
h
∇
2
`
V
p
+
`
V
p
∇
2
i
, (4.69)
where V
c
(r) includes the contributions fro m all mesons and is written in terms of c, σ, τ, στ
operators together with φ
0
c
(r), φ
1
c
(r), φ
2
c
(r) and the nucleon and included meson masses and
couplings as well as some other constants; and similarly for the o ther three functions of
V
t
(r), V
ls
(r), V
q
(r). But V
t
(r) includes the contributions from all involved pseudoscalar and
vector mesons together with φ
0
t
(r), φ
1
t
(r) and t, tτ operators; V
ls
(r) includes the contributions
from all involved scalar, vector and Pomeron-type mesons together with φ
0
ls
(r), φ
1
ls
(r) and ls, lsτ
operators; V
q
(r) includes the contributio ns fr om all involved scalar, vector and Pomeron-type
mesons t ogether with φ
0
t
(r) and q, qτ operators. In is mentionable that the local part of the
Pomeron-type potentials is multiplied by the exponential factor of e
−m
2
p
r
2
≡ φ
0
P
(r). We also
note that in the nonlocal potential, to which all except the pseudoscalar mesons contribute, as
the last part in (4.69), we have
`
V
p
=
X
γ
C
0
g
2
γ
4π
m
γ
4
´
M
φ
0
c
(r) −
g
2
P
4π
√
π
m
3
γ
M
´
M
2
φ
0
P
(r), (4.70)
where γ and g
γ
are the suiting meson indices and couplings, M,
´
M are fo r the proton and/or
neutron mass (M
p
is chosen often), and C
0
= 1 for scalar mesons and C
0
= 3 for vector mesons–
It is also not able that the methods to solve Schrodinger equation with nonlocal potentials ( such
as [∇
2
`
V
p
+
`
V
p
∇
2
] here) is presented in [116].
Anyway, The 1 3 free parameters of the potential were fitted to the Livermore-group 1969
data up to 330 MeV [104] good with χ
2
/N
data
= 2.09 besides good describing low-energy
40

parameters such as
1
S
0
(pp),
3
S
1
(np) scattering lengths as well as deuteron properties. The
results were very good among the best po tentials of its time–The updated and improved
version of Nijm78 potential is Nijm93 potential [28] framed in 1992, which we describe below.
The YN version of Nijm78 potential was presented in 1989 [137] and applied to B
¯
B systems
as well. The form factors, from the Regge- poles, are Gaussian that guarantee the soft behavior
of the potentials near the origin. It gave a good description of YN interactions by using SU(3)
and meson-nucleon coupling constants from the NN analyses.
4.15.3 Nijmegen Partial-Wave-Analysis
The first Nijmegen-group multi-energy phase-shift analysis was published in 1990 for just pp
interaction [122]. Next in 1993, they published a combined analysis of np+pp scattering data
[121]–For a newer PWA look at [138]. Indeed, the basic aim was to provide a more complete
database and then to improve the NN phase-shift analyses. To do so, they surveyed the NN
data published from Jan 1655 to Dec 1992 in the energy range o f T
lab
=0-350 MeV. As a result,
from 2078 pp data and 3446 np data, those survived with an optimized (not a very-high or
a very-low) χ
2
were 1787 pp and 2514 np data. Next, they parameterized a special energy-
dependent NN potential for each partial-wave up to almost J = 4. After that, the radial
Schrodinger equation was solved by the adjusted potential to get the phase shifts as functions
of the adj usted parameters and energy. Then, from the phase shifts, some predictions for
observables, and χ
2
to fit the experimental scattering data, were made. So, one may call the
Nijmegen analysis as an ”optimized p otential” analysis from which the phase shifts are bought
for various partial waves.
By the way, in the Nijmegen PWA’s, the pot entials for each partial-wave are actually
divided into two main parts: A nuclear (N) part and an electromagnetic (EM) part; or a
long-range (LR) part, a medium-range (MR) and a short-range (SR) part . That is
V = V
EM
+ V
N
= V
LR
+ V
MR
+ V
SR
, (4.71)
where the electromagnetic interaction has almost the same structure as that in Arg94 poten-
tial [31] while the nuclear part includes an LR OPEP, an MR heavy-boson-exchange (HBE)
potential and a phenomenological SR potential. The LR potent ial V
LR
is indeed a sum of
the EM and OPE potentials; the MR potential V
MR
is mainly from the HBE contr ibutions of
Nijm78 potent ial [95]; and the SR potential V
SR
is described by an energy-dependent b oundary
condition at r = b = 1.4f m, where the energy-dependent square wells are used.
The involved EM potential here, in general, reads
V
EM
= V
C1
(r) + V
C2
(r) + V
V P
(r) + V
MM
(r), (4.72)
where, as before, the indices of C1, C2, V P, MM stand for the one-phot on, two-photon, vacuum-
polarization and magnetic-moment interactions. More details were given in the subsection of
(4.12.2) with two main differences here with the more improved considerations in Arg 94 case,
where t he effects due to the finite-size of the nucleon and a Darwin-Foldy term (V
DF
(r)) were
41

also included and improved.
On the other hand, the LR nuclear interaction because of OPE’s and the MR nuclear interac-
tion because of HBE’s always read
V
N
=
M
E
V
OP E
+ f
s
med
V
HBE
. (4.73)
Indeed, the energy-dependent factor of M/E (where M is as usual the nucleon-mass, E =
p
M
2
+ q
2
is the c.m. energy and q
2
= MT
lab
/2) is required to get a better fit of the data.
Also, adding the HBE’s (such as ρ, ω, η) from Nijm87 potential for r > b to the OPEP tail,
give a better fit o f the data but the nuclear part is still incomplete. The f
s
med
factor in the last
relation, for the singlet(s) partia l waves, makes further improvement with f
s
med
(S = 0) = 1.8
and f
s
med
(S = 1) = 1.0, where S stands for the tota l spin of NN systems here.
For V
OP E
, we first note that one may face with the four isovector coupling constants of f
ppπ
0
,
f
nnπ
0
, f
npπ
−
, f
pnπ
+
in NNπ vertexes. So, for three po ssible NN scattering’s, one can write
f
2
pp
≡ f
ppπ
0
f
ppπ
0
, f
2
0
≡ −f
nnπ
0
f
ppπ
0
, 2f
2
c
≡ f
npπ
−
f
pnπ
+
, (4.74)
where one may then take f
2
pp
= f
2
0
when CS and f
2
pp
= f
2
0
= f
2
c
when CI is assumed. Now, we
can use the same expression in (4.50) for V
OP E
in (4.73) with a not e that again the second term
on its RHS is used just for np case and f
pp
= −f
nn
≡ f [120] with the CI value of f
2
= 0.0 75;
and also we should replace V
(3)
OP EP
there with V
(4)
OP EP
here as
V
(4)
OP EP
(m
pi
) =
1
3

m
pi
m
π
±

2
e
−µr
r

(~σ
1
.~σ
2
) +

1 +
3
(µr)
+
3
(µr)
2

S
12

, (4.75)
where µ = m
pi
c/~ as before.
For the SR potential V
SR
, used for r < b or lower partial waves, in t he pp PWA’s (Nijm90pp)
[122], the coordina te-independent energy-dependent square wells were used up to J = 4 (see,
Fig. 2 and 3 of [121]). Further, for the isoscalar (T = 0) np partial waves up to J = 4, and
1
S
0
partial-wave, the same parameterization as the pp case was used; whereas fo r the isovector
(T = 1 ) np phases shifts (except for
1
S
0
phase-shift), the suiting pp results by including
the pion Coulomb corrections were used. For the middle partial waves of 5 ≤ J ≤ 8, the
evalua ted phase shifts of t he OPE+HBE of Nijm78 potential [95] were used. And finally, the
higher part ial waves were obtained from the OPE phase shifts by including the electromagnetic
effects depended on the need.
It is also nota ble that the energy dependence of t he square-well depth is parameterized through
three parameters for each partial-wave. Fro m the tot al 49 such parameters f or the states of
J ≤ 4, 21 parameters are for the pp case and 18 parameters are for the np case besides the
pion-nucleon coupling constants (f
π
±
, f
0
) and f
s
med
determined by fitting the data. By the
way, in the combined pp+np Nijmegen PWA’s [121], with 1787 pp data (with 1613 degrees of
freedom) and 2514 np data (with 2332 degrees of freedom) below 360 MeV, published from
nearly 1955 to 1992, the ”perf ect” result of χ
2
/N
data
≈ 1 from the data fitting was achieved.
42

Later, in 2004, the Nijmegen-group made a new PWA of pp and np data up to 500 MeV [138].
There, the NN databa se was enlarged to almost 5000 pp and the same np data below that
energy. Inelastic effects could be included, and o ne could gain both the T = 0, 1 phase shifts
from the np data in contrast to PWA93 [121], where T = 1 phases shifts were gained from the
suiting pp ones with some corrections. In the analysis, a chiral TPE potential was added to
the LR OPEP used in PWA93, with an improvement of the data fitting–For a new PWA of
NN scattering data, by another group, look at [139].
4.15.4 Nijm93, NijmI and NijmII Potentials
The Nijmegen high-quality (HQ) potentia ls, which are Nijm93, NijmI, Nij mII and Reid93 NN
potentials [28], all give almost the perfect value of χ
2
/N
data
≈ 1. Nijm93 potential is indeed an
updated version of Nijm78 potential in that it is fitted to its time Nijmegen np+pp database
[121] (with χ
2
/N
data
= 1.87 ) and includes new OPEP’s with the pion mass splitting. Both
NijmI and NijmII potentials are also built on Nijm78 potential [95] with some differences
and improvements of course. In NijmI potential, in each partial-wave, a few parameters of
the potent ial are adjusted. It includes, like Nijm78 and Nijm93 potentials, the momentum-
dependent terms that result in the nonlocal structure of the potential in r-space. However,
NijmII potential is completely local that means all momentum-dependent terms in p-space are
deliberately removed. These three potentials are regularized at the origin by exponential form
factors, are fitted to the same database and have the same number of fitting parameters (15
free parameters) as in PWA93 [121]. The results of data fitting signal that NijmI and NijmII
potentials have almost the same quality, and that all three potentials reproduce a χ
2
close to
the suiting value for PWA93.
The general form of the NijmI and NijmII potentials in p-space are as in (4.66) except for
some differences. The first difference is adding the new operator of
˜
LA =
i
2
(~σ
1
−~σ
2
).~n = i
~
A.~n
(which is the Fourier t ransform of the charge-symmetry opera tor
~
L.
~
A used in Arg94 potential
[31] as well) and so, the new term of
˜
V
la
(k
2
)
˜
LA is added to the po t ential. It is notable that
for identical-particle scattering, this operator does not contribute; and when CI is supposed,
˜
V
la
(k
2
) vanishes. The second difference is that instead of
˜
S
(0)
12
in Nijm78, one now uses the
complete
˜
S
12
in (4.20), which is in turn the Fourier transform of the r-space tensor operator of
S
12
. The third difference is that because
˜
Q
12
in (4.20) is not an exact Fourier transform of the
quadratic spin-or bit operator Q
12
in (4.3), to have equivalent r- and p-space potentials with
the same phase shifts and bound-stat es,
˜
Q
12
˜
V
q
(k
2
) in (4.66) must be replaced by
˜
Q
12
˜
V
q
(k
2
) −
˜
Q
′
12
Z
k
2
∞
dk
′2
˜
V
q
(k
2
), (4.76)
where
˜
Q
′
12
=

(~σ
1
.~q)(~σ
2
.~q) − q
2
(~σ
1
.~σ
2
)

−
1
4
h
(~σ
1
.
~
k)(~σ
2
.
~
k) − k
2
(~σ
1
.~σ
2
)
i
. (4.77)
One should not e that including Q
12
was indeed necessary there to describe the pha se shifts of
1
S
0
,
1
D
2
simultaneously, and its effect could be simulated by including special nonlocal poten-
43

tials. By the way, the resultant potential forms V
i
of V =
P
i
V
i
O
i
, where i = c, σ, t, ls, q, la,
are supposed to be same for all partial waves, as the differences among the potentials arise from
the vacuum expectation values of the operators in different partial waves. It is also nota ble
that V
i
may be a function of r
2
, q
2
and L
2
in r-space (or
˜
V
i
(
~
k, ~q, ~n, E) in p-space); meanwhile
a r
2
-dependence is always preserved and the q
2
-dependence is included in V
c
, which in t urn
signals the nonlocal structure of the potential in r-space.
The included mesons and Reggo ns, OBEP’s and OREP’s as well as the propagators are
the same as t hose in Nijm78 potential except for a f ew differences. Indeed, the pion- and
nucleon-mass splitting a r e also considered. Taking the mass difference between the neutral-
and charged-pions ( and also for the ρ meson here) leads to CIB. The coupling constants for
the pseudoscalar and vector mesons are related through SU(3) with their special singlet-octet
mixing, whereas for the scalar mesons and the Regge poles, the coupling constants are con-
sidered as fr ee parameters. Also, for each exchange, a n independent cutoff mass is used and
so, with the three cutoffs of Λ
P S
, Λ
V
, Λ
S
, there are a total number of 14 free parameters. It
is also notable that the broa d mesons of ρ and ǫ could be described by a dispersion integral
instead of the static f ormula of ∆(k
2
) = 1/(k
2
+ m
2
mes
). In the OPE part, as in PWA93,
the pion mass splitting is considered and so, the isovector np phase parameters are smaller
than the isovector pp phase parameters, which in turn means CIB. The plain OPEP’s for pp
and np systems are the same a s those in PWA93 ( and also Arg94 potential in (4.12.2)) with
f
2
pp
= f
2
c
= f
2
0
= f
2
π
= 0.0 75 (pointing out CI for the pion-nucleon coupling constants), and
V
(5)
OP EP
(m
pi
) =

m
pi
m
π
±

2
1
3
m
pi
c
2

φ
1
c
(m
pi
, r)~σ
1
.~σ
2
+ 3φ
0
t
(m
pi
, r)S
12

, ( 4.78)
instead of V
(4)
OP EP
(m
pi
) in (4.78).
Describing the data, in the energy range of 0-350 MeV, with the potentials are satisfac-
tory. In fact, Nijm9 3 potential fits 1787 pp data with χ
2
/N
data
= 1.8 and 2514 np data with
χ
2
/N
data
= 1.9 and so, the whole data with χ
2
/N
data
= 1.87. This description is better than
that of para meterized Paris potential [98] and Full-Bonn potentials [45], [37]. This result sug-
gests that just with the conventional OBEP’s one could not describe the data well. On the
other hand, NijmI and NijmII Reid-like po t entials describe the whole pp and n p data with
χ
2
/N
data
= 1.03 with 41 and 47 fitting parameters, respectively. The potentials are called
Reid-like in that, in each partial-wave, just a few para meters a re adjusted that is in turn
similar to the Reid method in parameterizing the potentials in each partial-wave separately.
It is good to remind t hat, in making these HQ potentials, the Schrodinger equation of
(∇
2
+ k
2
)Ψ = 2M
r
V Ψ, (4.79)
is used, which is a r-space approximation of the full four-dimensional scattering equation. In
this equation, M
r
is the nucleon r educed mass, and the relations between the c.m. energy (E)
and the squared c.m. momentum (k
2
) are as E = k
2
/2M
r
and E =
p
k
2
+ M
2
p
+
p
k
2
+ M
2
n
−
(M
p
+ M
n
) for the nonrelativistic and relativistic kinematics, respectively.
44

On the other hand, following the discussion in the previous subsections, we know that
to regularize the potential at t he orig in, the form- factor of F (k
2
) is always used. For the
Nijmegen potentials, and to complete t he discussion, we quote the following useful Fourier
transform (with the λ index for the corresponding meson)
Z
d
3
k
(2π)
3
e
i
~
k.~r
k
2
+ m
2
λ
(k
2
)
n
F (k
2
) ≡
m
λ
4π
(−m
2
λ
)
n
φ
n
c
(r)
=
m
λ
4π
(−∇
2
)
n
φ
0
c
(r),
(4.80)
according to which, for the well-known form functions, we can write
F (
~
k
2
) = 1 ⇒ φ
0
c
(r) =
e
−m
λ
r
m
λ
r
, (4.81)
which is the usual Yukawa potential without the form function (the point-like nucleon);
F (k
2
) =

Λ
2
− m
2
λ

/

Λ
2
+ k
2

⇒ φ
0
c
(r) =

e
−m
λ
r
− e
−Λr

/m
λ
r, (4.82)
as the Monopole for m-factor normalized such that at the pole, F (−m
2
λ
) = 1; and
F (k
2
) =

Λ
2
− m
2
λ

2
/

Λ
2
+ k
2

2
,
⇒φ
0
c
(r) =

e
−m
λ
r
− e
−Λr

1 +
Λ
2
− m
2
λ
2Λ
2
Λr

/m
λ
r,
(4.83)
as the Dipole form-f actor; and
F (k
2
) = e
−k
2
/Λ
2
,
⇒φ
0
c
(r) = e
m
2
/Λ
2

e
−m
λ
r
erfc

m
λ
Λ
−
Λr
2

− e
m
λ
r
erfc

m
λ
Λ
−
Λr
2

/2m
λ
r,
(4.84)
as the Exponential form-factor with
erfc(y) =
2
√
π
Z
∞
y
dte
−t
2
, (4.85)
as the complementary-error-function.
It should be also mentioned t hat, without the form factors, one should use
φ
1
c
(r) = φ
0
c
(r) − 4πδ
3
(m
λ
~r) (4.86)
instead o f φ
0
c
(r) in the presence of the form factors. Besides, with the help of ( 4.80), one can
get the tensor and spin-orbit pot entials in terms of the central function of φ
0
c
(r) as
φ
0
t
(r) =
1
3m
2
λ
r
d
dr

1
r
d
dr

φ
0
c
(r), (4.87)
45

φ
0
ls
(r) = −
1
m
2
λ
1
r
d
dr
φ
0
c
(r). (4.88)
Therefore, one can see that with the dipole form-factor (in Reid93) and the exponential form-
factor (in Nijm93, NijmI, NijmII) to regular ize the potentials, the t ensor function is vanished
at the origin as well.
It is also good to mention the Fourier transform of the momentum-dependent terms (linear in
q
2
in Nijm78, Nijm93, NijmI) in the p-space potentials, which lead to the nonlocal structure
in r-pace as
Z
d
3
k
(2π)
3
e
i
~
k.~r
k
2
+ m
2
λ

q
2
+
1
4
k
2

F (k
2
) = −
m
λ
8π

∇
2
φ
0
c
(r) + φ
0
c
(r)∇
2

, (4.89)
(to see how to handle such nonlocal terms, look at [116]) whereas the absence of the q
2
terms
in NijmII (and also Reid93) po t ential in p-space leads to a radial local potential in r-space.
4.15.5 Reid93 Potential
The so-called r egula r ized-R eid (Reid93) potent ial [28] is fitt ed to the updat ed Nijmegen database,
while the quality o f the original Reid68 [26] np dat a were poor. Besides, there was a 1/r sin-
gularity for all partial waves, which are now removed by including the dipole for m factors
(with the cutoff o f Λ = 8m
pi
); and so the tensor potentials vanish at the origin. For the OPE
part, as in the other Nijmegen high-quality potentials, the neutral-and charged-pion mass dif-
ferences a re considered (with f
2
π
= 0.075 a gain) and so Reid93 potential is charge-dependent.
Meanwhile, in (4.78), φ
1
c
(r) is used just for S-wave while, for other partial waves, φ
0
c
(r) is used
instead of φ
1
c
(r).
Besides the OPEP tail, the potent ial in each par t ial-wave is parameterized separately by choos-
ing suitable combinations o f the central, tensor and spin-orbit terms with arbitrary masses and
cutoff parameters. In Reid93 potential, with the coefficients of ¯m =

m
π
0
+ 2m
π
±

/3, Λ = 8 ¯m,
all potentials are written as linear combinations of the following functions
¯
Y (p) = p ¯mφ
0
c
(p ¯m, r),
¯
Z(p) = p ¯mφ
0
t
(p ¯m, r),
¯
W (p) = p ¯mφ
0
ls
(p ¯m, r), (4.90)
with 50 coefficients of A
jp
, and B
jp
, which are used for isovector potentials, and isoscalar and
np
1
S
0
potentials, respectively. These coefficients are fixed by fitting to the relevant pp+np
scattering data. Here, p is an integer and j labels various partial waves, and that φ
0
t
and φ
0
ls
are some special radial functions [28].
One should note that , as in Reid68 potent ial, in the non-OPE part, for the singlet- and
uncoupled triplet-states, the central pot entials are used; and for the coupled triplet-states, the
potentials having the central, tensor and spin-orbit terms as (4.13) are used. For instance, for
46

the uncoupled states of (T = 1, S = 0, L = J), they used
V
pp
(
1
S
0
) = A
12
Y (2) + A
13
Y (3) + A
14
Y (4) + A
15
Y (5) + A
16
Y (6) ,
V
np
(
1
S
0
) = B
13
Y (3) + B
14
Y (4) + B
15
Y (5) + B
16
Y (6) ,
V (
1
D
2
) = A
24
Y (4) + A
25
Y (5) + A
26
Y (6),
V (
1
G
4
) = A
33
Y (3) , V (
1
J
1
) = V
pp
(
1
S
0
), J ≥ 6,
(4.91)
where the different pp and np
1
S
0
potentials are because of the CIB considered in the potentials.
We should a lso mention that, for the coupled states, the potentials have clear forms up to
J = 4; and for the higher partial waves (J ≥ 5), a similar expansion as done by Day [27]
(look at subsection (4.5.2) is performed. Clearly, for the triplet isovector (isoscalar) partial
waves of J ≥ 5, the central and tensor potentials are those of the corresponding S = 1, T = 1
(S = 1, T = 0) J < 5 par t ial waves while the spin-orbit potential is set to zero.
By the way, with 50 fitting pa r ameters, Reid93 potential r eproduces the result of χ
2
/N
data
=
1.03 such as the other Nijmegen HQ Potentials. It is also remarkable that the predicted values
of the quantities, such as deuteron parameters and low-energy scattering parameters, by Reid93
potential (and also the other HQ potentials of Nijm93, NijmI, NijmII) has a good agreement
with the experimental values [28]. Nowadays, these HQ potentials are extensively used in
nuclear structure calculations with many satisfactor y results.
4.15.6 Extended Soft-Core Potentials
The already mentioned Nijmegen po t entials based on O BE and ORE approaches described
well the data but with many phenomenological argument s included. The extended-soft-core
(ESC) potentia ls include extra exchanges and are more on the theoretical g r ounds. That is
because by adding a few more free parameters, while preserving the previous advant ages, t he
new potentials reproduce the data well. In the first ESC model [140], next to the whole previ-
ous exchanges of Nijm78 potential [95], they included some two-meson-exchange (TME) (and
also 2π-exchange) and meson-pair-exchange ( MPE) contributions. That model described the
Nijmegen PWA93 (pp+np database) in the energy range of 25-3 20 MeV with χ
2
/N
data
= 1.16,
which was the first promising result in the case.
The next completed version was presented in 1995 [141]. In fact, besides the previous OBE’s
and OR E’s of Nijm78 potential [95], TPE’s, TME’s and MPE’s were included a s well. In
general, TME contributions ar e from π ⊗ ρ, π ⊗ ω, π ⊗ η, π ⊗ ǫ, π ⊗ P, ..., where the parallel
and crossed-box Brueckner-Watson diagrams with Gaussian form factors a r e computed. The
MPE contributions are due to one-pair and two-pair (ππ, πρ, πω, πη, πǫ, ...), where a Gaussian
form-factor (as e
−k
2
/2Λ
2
) is attached t o each vertex. The interaction Lagrang ian’s are for the ef-
fective relat ivistic theories, with LS equation. The ESC potentials were compared with Nijm9 3
potential [28] in that a t least both include 14 free parameters. Next, the ESC po t entials gave
better results with more theoretical grounds besides using the chiral symmetry of the invo lved
Lagrangian’s.
The chiral-invariant ESC model for NN interaction with 12 free parameters reproduced the
47

data in t he energy r ange of 0 -350 MeV with χ
2
/N
data
= 1.7 5 [142]. It is notable that the TME
contributions improved the ESC potential quality with respect to the previous OBEP’s. In
addition, one notes that the meson-pair vertexes in the t riangle and double TME diagrams
(supposed to simulate the heavy mesons and resonance degrees of freedom) are analyzed in
principle by chiral-symmetry and so, these cont r ibutions do not introduce any new parameter.
The YN and YY versions of the ESC potentials were then report ed in [143] first, where next
to discussing the usual boson-exchanges, the interactio ns were discussed in the framework of
QCD, flavor SU(3), and chiral SU(3)⊗SU(3) for the low-energy region as well. Then, in 2000,
besides reviewing the Nijmegen SC pot entials, a new ESC model (called ESC00) was presented
to describe NN, YN and YY systems in an unified manner by using SU(3)
f
symmetry [144].
In the energy range of 0-350 MeV, it described YN and NN systems with χ
2
/N
data
= 1.15.
After that, t hey modeled the comprehensive p-space versions of the ESC NN potentials in
[145]. With 20 free parameters (of the masses and coupling constants) there, they reproduced
the NN scattering data in the energy range of 0-350 MeV again with χ
2
/N
data
= 1.15. Some
new improvements were because of including the axial-vector mesons and a zero in the scalar
meson form factor s–It is also mentionable that the SC meson-baryon interactions were dis-
cussed in [146] as well. Fr om 200 5 onwards, some new generations of the ESC BB potentials
(called ESC04) have been presented, where the contributions fr om OPE’s, ORE’s, MPE’s and
two-pseudoscalar-meson-exchange (PS-PS exchange) are also included. There are the NN in-
teraction in [147], the YN interaction in [148], and the BB stat es with the total strangeness of
S = −2 in [149]. For some recent reviews of the Nijmegen ESC potentials, see [150], [151].
4.15.7 Nijmegen Optical Potentials
NN potentials are often considered to be real below the pion production threshold in about
T
lab
=290 MeV. One way to include inelasticity’s at the high energies, above the thresholds, is
to consider optical potentials. On the other hand, we saw that in the Nijmegen PWA93 for the
short distance potential (V
SR
), below r < b (with b = 1.4fm; look at subsection of (4.15.3)),
the energy-dependent square wells were used. Now, one may write [152 ]
V
S
= V
Rel
− iV
Img
, (4.92)
where the real SR potential V
Rel
, which is different for each partial-wave, always reads
V
Rel
=
N
X
n=0
a
n
(k
2
)
n
, (4.93)
and the imaginary SR potential V
I
is taken as
V
Img
= (k
2
− k
2
th
)V.θ(E − E
th
). (4.94)
It was established from t he Nijmegen PWA’s [121, 129] that t he fully real potentials work
up t o about 500 MeV quite well. Nevertheless, the o ptical potentia ls of the Nijmegen-group
48

could be constructed by adding, to the real HQ potentials, the same imaginary part used in
the Nijmegen PWA’s of the np data below 500 MeV, according to the above prescription of
course. But, the resultant optical potentials did not give good results for all partial waves in
that energy range. Clearly, if one considers all np date below 1 GeV, some differences among
the results from t he preliminary PWA’s and the above constructed optical potentials arise in
some partial waves. For instance, as it is clear f r om Figure 3, for all np data below 1 GeV,
the phase shift of
1
S
0
is well described by both PWA and the optical NijmI potential; whereas
for the phase shift of
1
D
2
, the large differences are recognizable clearly. Still, by refitting, the
modified NijmI optical potential, NijmI (mod), is obtained that gives a good fitting of the
1
D
2
phase shift up to 1 GeV. Therefore, it seems that it is not so difficult to model the optical
potentials to fit the np da t a up to T
lab
=1 GeV.
Figure 3: The phase shifts
1
S
0
(left), and
1
D
2
(right) for the NijmI optical potential and a
modified v ersion o f that (quoted from [130]).
4.16 Hamburg-Gr ou p Potentials
Hamburg potentials are also among meson-exchange models related to QCD substructures.
Since, in Bonn-B potential [37], the scattering amplitudes were obtained from the meson-
baryon Lagrangian in a clear and comprehensive way, that model wa s used as a ba se to
build one-solitar y- boson-exchange potential (OSBEP) by this Hamburg-group. In fact, the
Hamburg-group potential somehow refines the common boson exchange picture by seeking for
a procedure that reduces markedly the number of free par ameters of the conventional boson
exchange potentials; whereas t he parameters a r e in turn needed for the quality fittings of the
potentials to scattering data. Because of its special mechanism, there is not any cutoff pa r am-
eter as the only adjustable par ameters are the pion self-coupling constant and meson-nucleon
coupling constants. The first version [47] was presented in 1996, to fit just elastic np scattering
data up to J ≤ 3, and was further developed in [153] to include both np+pp data with more
49

improvements.
In fact, the f eat ures of both QCD-inspired models and common phenomenological boson-
exchange potentials are included in OSBEP. We know that below the pion production thresh-
old, chiral symmetry is supposed to be broken. Now, a meson Lagrangia n, by including all
OBE contributions, with a similar structure as the linear sigma-model, is considered. Be-
cause the symmetry conditions are not imposed on the ma sses and coupling constants, but
the latter are used as free parameters in the Lagrangian, the chiral symmetry is so broken.
Therefore, the spontaneous chiral symmetry breaking leads to nonlinear terms in the meson
part of the Lagrangian. The resultant decoupled nonlinear meson equations are then analyzed,
and semiclassical solutions are quantized leading to defining ”solitary mesons” from which the
propagators come out. Indeed, the nonlinear features of the QCD-inspired models are consid-
ered as the nonlinear boson equations result. Because of the nonlinear property of the boson,
the form factors are no t as those in the Bonn-B potentials; and they are replaced by properly
normalized solitary meson fields. As another result, an experimental scaling law arises that
relates all the meson parameters and so, reduces the number of fitting parameters. The model
also g ives a good quantitative description of experimenta l data and deuteron properties though
the quality is not as high as the other ment ioned HQ potentials [153].
Meanwhile, in [39], NN interaction is discussed from quant um-inversion approach versus meson-
exchange picture and especially from OSBEP. In general, we know that by inserting a potential
with its special operators into LS equation, one can earn phase shifts and other observables.
In full generality, this method involves quark and gluon substructures, and then both on- shell
and off- shell data are describ ed well. However, the inversion potentials are in general local
and energy-dependent in r-space, whereas BEP’s lead to nonlocal potentials in p-space mainly.
It is established there that the results from quantum-inversion and boson-exchange potentials
are almost same. A main difference is t he larger D-state probability for the local potentials,
which is in turn related to the different tensor part of the potentials.
The next improvement in studying OSBEP’s was to extend them for pion-nucleon interactions
as well, as in [154], where the OSBEP model was recast into a unique form f or NN and πN
interactions. To do so, the ∆-isobar was included besides the chiral-symmetry preserving pseu-
dovector meson-baryon coupling (PS πNN) instead of the previous pseudoscalar (PS) coupling
for πNN. Describing NN and πN interactions simultaneously was good as the previous results
for just NN int era ctio n.
In 20 03, von Geramb et al. propo sed another NN po tential based on Dirac equations
(two coupled Dirac equations with constraints from dynamics) combined with meson-exchange
picture (including the π, η, ρ, ω, σ exchanges) [1 55]. The resultant potentials, to use in partial-
wave Schrodinger-like equations, inspired by meson exchanges, fitted the Arndt et al. pp+np
phase shifts of T
lab
=0-3 GeV [156] as well as deuteron properties. The analyses showed
a universal core potential coming from relativistic meson-exchange dynamics, and that the
high-energy effects such as those of QCD and inelasticity were included. Besides the Dirac
meson-exchange potentials, they framed some local and no nlocal optical po t entials, which still
gave good agreement between theoretical and experimental data.
50

4.17 Moscow-G r ou p Potentials
Moscow-type (M-type) potentials are mainly a hybrid o f the quark-model and meson-exchange
picture. In general, in short distances, the quark and gluon degrees of freedom are used;
whereas for the LR and MR regions, OBEP’s+TPEP’s ar e often used. The first major version
was presented in 1997 [59] and latter improved in [15 7]. The main features of these potentials
are the emphasis on the deep substructures from QCD. One main difference of the M-type
potentials fro m the other NN potentials is that the common SR local repulsive core fro m
the ω (and also ρ) meson is strongly reduced, and is indeed replaced by t hat of a suitable
orthogonality condition plus a deep attractive potential. The orthogonality condition may be
interpreted as projecting the compo und six-quark states (ϕ
0
), with the maximal permutational
symmetry, into asymptotic NN channels. As a result, one now has a node around ∼ 0.6f m
that plays the role of the repulsive cor e and provides NN phase shifts. Meanwhile, the potential
also has strong attraction in intermediate parts commonly assigned to the pseudoscalar-meson
exchanges of π, η and the scalar-meson exchange of σ. Still, for the SR repulsion of the ω
meson, a repulsive core with a Gaussian form-factor with a positive finite coupling constant is
included. In other wo r ds, in the short distances (r . 1fm), the nonlocal and energy-dependent
terms, which are in turn coming from the retardation effects and six-quark bags (6 q-bags), are
replaced by a separable potential.
By the way, the main M-type potential in [59], with two major parts, reads
V = V
(0)
loc.
+ V
sep.
, (4.95)
where the local potential (V
loc.
) is ℓ- independent and includes a n OPEP and an attractive well,
and is depended on the spin and parity of NN system. The separable potential (V
sep.
) is a state-
dependent (depended on the ℓ, J of NN system) repulsive core with a Gaussian form-factor.
They are
V
loc.
= V
c
(r) + V
t
(r)S
12
= V
0
e
−ηr
2
+ V
(6)
OP EP
(m
pi
) + λ|ϕ >< ϕ|, (4.96)
in which
V
(6)
OP EP
(m
pi
) = −
f
2
πNN
4π
µ
3
4M
2
[f
tr
Y
c
(x)~σ
1
.~σ
2
+ (f
tr
)
n
Y
t
(x)S
12
] , (4.97)
and
ϕ = Nr
ℓ+1
e
−
1
2

r
r
0

2
,
Z
ϕ
2
dr = 1, f
tr
=

1 − e
−αr

, (4.98)
where x = µr as usual with µ for the average pion-mass, n as the power of the cutoff f actor
of f
tr
is different for the diff erent versions of the po t ential, r
0
is the radius of the repulsive
core (different slightly for t he different states), α is the cutoff radius of the OPEP, and λ is
different for different ℓ, J’s. Because of the freedom to choose the parameters of η, α, V
0
, one
can set η = α
2
; and then the width of η and the depth of V
0
are fitted to scattering data, which
are the scattering length and the effective range of the
1
S
0
wave here. It is mentionable that
the repulsive core is absent for ℓ ≤ 4 (T
lab
< 400 MeV) because of the second term in (4.97)
or the presence of Y
t
(x). Anyway, this M-type potential describes well deuteron properties
51

and NN scattering da ta up to 500 MeV with 6 free para meters, which are in turn physically
meaningful. Also, the off-shell behavior of the potential can be checked in NN bremsstrahlung.
It should also be mentioned that, contra ry to the other quark-meson hybrid models that use
the mixtures of both and so lead to energy-dependent nonlocal potentials, here the quark and
meson exchanges are orthogonal besides giving a microscopic description of NN interaction.
In the complete version presented then, the Gaussian form function was replaced by an
exp onential f orm function to describe better the phase shifts especially
3
S
1
−
3
D
1
phase-shift.
The potential so is written as
V = V
(M)
loc.
+ V
(7)
OP EP
+ V
sep.
, (4.99)
in which
V
(M)
loc.
= V
0
e
−βr
+ V
0
e
−βr
~
L.
~
S, (4.100)
and
V
(7)
OP EP
(µ) = −
f
2
πNN
4π
µ
3
(~τ
1
.~τ
2
)
h
Y
(2)
c
(x)(~σ
1
.~σ
2
) + Y
(2)
t
(x)
ˆ
S
12
i
, (4.101)
where this OPEP is written with a soft dipole form-factor, and now the tensor potential
becomes zero at the origin as it must be;
ˆ
S
12
= S
12
/3 and µ = (m
π
0
+ 2m
π
±
)/3; and now, the
same as in Arg94 po t ential [31] and Nijmegen HQ potentials [28], f
2
πNN
/4π = 0.075; α = Λ/µ,
and
Y
(2)
c
(x) = Y
c
(x) − αY
c
(αx) −

α
2
− 1

α
2
2
x Y
c
(αx),
Y
(2)
t
(x) = Y
t
(x) − α
3
Y
t
(αx),
(4.102)
with the notations in (4.23). Here just V
0
, α, β are free parameters in the local part of the
potential, which are in turn different for each spin and parity combination. In addition, the
parameters of λ and r
0
are independent for D- and F-waves; whereas for S- and P- waves, λ
goes to infinity and r
0
values are depended on the depth of the attractive local potential. In
general, with 32 parameters (a similar number to the so far mentioned HQ potentials) and the
πNN coupling constant, describing deuteron properties and partial waves in the energy range
of 0-400 MeV wa s very good (except for few higher ℓ’s).
Next, they developed a new mechanism to describe NN interaction in MR and SR par ts
[158]. In fact, instead of the oldest Yukawa formalism for SR interaction, a 6q-bag model,
dressed because of the π, ρ, σ mesons, was used there. That in turn produced an MR attractio n
that replaced the conventional σ-meson exchange. On the other hand, the ρ meson, produced
in the intermediate six-quark state, caused a nonlocal spin-orbit interaction in the SR par t. As
a result, the MR attraction and a part of the SR repulsion were described excellently, whereas
the SR repulsion was mainly because of the orthogonality of NN- and 6q-channels.
In other words, in the common OBE models, there are still many problems. For instance, the
cutoff parameters Λ
m
λ
NN
are oft en larger than the experimental values got by fitting the da ta;
the phenomenological Yukawa functions have also at least the base theoretical problems; and
52

discussing the σ meson, as a 2π resonance in S-state, is also controversial. Further, describing
3N and 4N systems with the settled OBEP’s is not addressed satisfactory yet. Therefore, the
new M-type potentials try to address some of the existing problems.
Here, with the dressed 6q-bag, the σ- (a nd even ρ-) meson exchange between nucleons is
considered because of the transitions between the p-shells of the excited quarks. In other
words, each p-shell quark emits a pion and during t he transition from the p-shell to s-shell,
the pio ns are absorbed by the di-quark pairs in the intermediate 6q bag-like states (suppose
as qq → σ + qq). Further, the σ meson, as a scalar-isoscalar excitation of the QCD vacuum,
is considered as a quasiparticle inside the hadrons (especially in a multi-quark bag) and not
as a real particle in the f ree space. Therefore, the scalar-isoscalar σ meson exists just in a
high-density medium and not in the vacuum (contrary to the ρ, ω mesons).
One can show the main features of the model by a simple phenomenological potential as
V = V
or th.
+ V
NqN
+ V
(7)
OP EP
, (4.103)
where
V
or th
= λ
0
0|ϕ
0
>< ϕ
0
|, (λ
0
→ ∞), (4.104)
as the orthogonality potential, provides the orthogonality condition between the intermediate
6q-bag and the especial NN channel fo r S- and P-waves. V
NqN
is the separable potential
attributable to the virtual transition of NN → (6q + 2π) + NN as
V
NqN
=
E
2
0
E
2
− E
2
0
λ|ϕ >< ϕ|, (4.105)
for the single channels and
V
NqN
=
E
2
0
E
2
− E
2
0

λ
11
|ϕ
1
>< ϕ
1
| λ
12
|ϕ
1
>< ϕ
2
|
λ
21
|ϕ
2
>< ϕ
1
| λ
22
|ϕ
2
>< ϕ
2
|

, ( 4.106)
for the coupled channels. E
0
≈ 600 −100 MeV is a sum of the 6q-bag energy and the σ-meson
mass inside the 6q-bag, and is same for all partia l waves with definite parities. The expression
for ϕ
i
is that in (4.98) with ϕ
i
, ℓ
i
here instead of ϕ, ℓ there. The potential parameters o f
λ
jk
(= λ
kj
), r
0
, E
0
, the phase shifts and the mixing parameters ε
1
are determined by fitting
the data with the cutoff parameter of Λ = Λ
dipole
= 0.50 − 0.75 MeV. The resultant potential
describes the partia l waves of ℓ < 2, which is in t urn equivalent to describing the phase shifts
in the energy range of 0-600 MeV and S-waves for an energy about 1200 MeV. Further, the
weak contributions because of the vector mesons in the baryon spectra and the strong spin-
orbit splitting are explainable by t his constituent-quark model. Still, the model leads to some
esp ecial 3N and 4N forces because of 2π and ρ exchanges.
In 2005, the group described elastic and inelastic NN scattering’s in the energy ra nge o f 1-2
GeV by a special EFT [1 59]. The previous approach, to describe MR and SR interactions
in [15 8], which made use of six-quark bags a nd the intermediate mesons of π, σ, ρ, ω, were
employed t here a nd improved as well.
53

The predictions of t hese M-type potentials for 3N systems were a lso analyzed in detail with
good results. Large deviations from the conventional NN potentials were established for the
momentum-distribution in the high-momentum region. In par t icular, the coulomb displace-
ment energy for the nuclei of
3
He−
3
H displayed a promising agreement with experiment when
the binding energy of
3
H was extrapolated to the exp erimental value [59], [157]. Further, in
[158], by using a new MR NN interaction model, based on the QCD bag model, an effective
energy-dependent NN interaction was constructed. The new potential described experimental
data up to 1 GeV and deuteron parameters well. Generalizations of the model to three-nucleon
force (3NF) and other related issues were also discussed [159]. Other mechanisms for the MR
and SR parts o f NN interaction are addressed by the group members in the references of [159].
It is remarkable that the M-type po t entials, and also the following two potentials/models, do
not have necessarily similar structures as the other standard ones and are almost special.
4.18 Budapest(IS)-Group Potential
Doleschall et al. set up a set of NN interactions in r-space to get the correct binding en-
ergy of trit on [160]. The potentials are nonrelativistic and almost phenomenological, nonlocal
and energy-independent. Nucleons are discussed as point-like objects and effects from their
structure is supposed to come from effective NN potentia ls. In some specific short regions,
the potentials are considered to be nonlocal, and in the outside regions as some local Yukawa
tails. The first aim was to find a nonlocal potential form to describe triton (
3
H as a 3N bound
system) binding energy as well as describ e exp erimental phase shifts and deuteron properties.
Later, they used those nonlocal NN potentials to describe some other 3N bound states [161]. In
fact, they modeled an NN potential respecting the well-known local behaviors in long ranges,
whereas it showed a nonlocality at the shorter ranges. The resultant potential provided a
satisfactory fit to NN scattering data while including CI and CS. The nonlocality in t he NN
potential guaranteed that no 3N forces were required to describe 3N bound-states.
4.19 MIK-Group Potential
The J-matrix inverse scattering approach to make NN potentials was started by Zaitsev et al.
in [162], [20], and was developed by Shirokov et al. in [163]. Indeed, t he nonlocal interactions
gained in this approach are in the forms of some matrices in oscillator basis in each NN
partial-wave separately. In other words, in the approach NN interaction is as a set of potential
matrices for various partial waves. However, a main aim to make the potentials was to earn
some satisfactory results in nuclear calculations of 3N systems and other light nuclei.
In the first serious tr y [163], based on [162], [20], they held the inverse scattering tridiagonal
potentials (ISTP), which are tridiagonal (quasi-tridiagonal) in the uncoupled (coupled) partial
waves. The dimension of the potentia l matrix was determined by the maximum value of N =
2n + ℓ (note that t he common nonrelativistic Schrodinger equation is used in the approach),
and was refereed as a N~ ω potential. We should note that the resultant interactions are
somehow effective and are not related to the usual meson-exchange theories though the main
54

features and results of NN interaction are common. It is also notable that, to describe a wider
energy range, the size of the potential matrix, or the oscillator basis par ameter ~ω, must be
increased. The potential [163] was used in nuclear calculations of
3
H,
3
He with giving good
results also in describing NN scattering data.
In [164], similar to that in [163], another class o f the J-matrix inverse scattering potentials
(JISP), called JISP6, was constructed. The resultant potentials described well NN scattering
data as well as the bound- and resonance-states of the light nuclei up to A = 6. A remarkable
feature of t he potentials was that by using the off-shell degrees of freedom, there was not
any need to include 3N potentials to describe well the light nuclei. The results to evaluate
binding energies of the nuclei
3
H,
3
He,
4
He,
6
He,
6
Li were as well as the results of the ot her
HQ potentials such as NijmI, NijmII [28], Arg94 [31], and CD-Bonn [9 9] potentials. In [164],
the base parameter was ~ω = 40 MeV, and that the potential described the Nijmegen PWA93
data [121] with χ
2
/N
data
= 1.0 3; look also in [21].
Next, they set up a JISP16 version and then further developed it a s a ISP16
2010
version [165].
The latter potentials were used to evaluate binding energies and spectra of the light nuclei in
No-Core-Shell-Model (NCSM) calculations. In a recent study [166], the progress in developing
the JISP NN interactions and other related issues are discussed as well.
4.20 Imaginary Potentials
As we know, above the pio n production threshold, the inelasticity’s and other high-energy
effects become important, and then one way to incorporate them is to consider optical or imag-
inary potentials suitable also to earn high quality descriptions of scattering data in medium
and high energies. Among a few imaginar y NN potentials, we discussed briefly the Nijmegen
ones in subsection of (4.15.7). In [167], some NN potentials such as Paris, Nijmegen a nd Ar-
gonne potentials, and those tr aced by quantum inversion, which describ e NN interaction for
the energies below 30 0 MeV, are extended to NN optical potentials in r-space. The up-to-date
phase-shift analyses, from 300 MeV to 3 GeV, are used to settle the extensions. The imagi-
nary parts of optical potentials account for the flux losing into direct or resonant production
processes. The optical potential approach is interesting as it allows one to imagine fusion and
resulting fission of nucleus when T-lab energies are above 2 GeV.
Discussions about optical potential from quantum inverse-scattering and scattering data as
well as modeling an optical potential are also given in [168]. There is also a r elat ivistic optical
NN potential, based on some idea of M-type potentials, in [169].
4.21 QCD-Inspired Potentials
The QCD- inspired models always use the fundamental quark and gluon degrees of freedom
instead of mesons. Indeed, because the SR part of NN intera ctio n is more related to quarks
and gluons, so the QCD-inspired potentials a r e used mo r e to describe this pa r t. Often one
uses the hybrids of quarks and mesons to describe t he interaction. There, the SR interaction
is always attributable to gluon exchanges and the MR and LR parts come from scalar- a nd
55

pseudoscalar- meson exchanges. One, of course, usually uses O PEP’s for the LR part, while the
MR part is handled by phenomenological or TPEP’s, as we described some potentials briefly.
Nevertheless, one should no t e that by applying the spo ntaneous chiral symmetry breaking to
QCD Lagrangian, one may be able to set up an NN interaction fully based on QCD . Then,
the common picture of the nucleon is a quark core surrounded by a pion cloud. So, in large
distances, one may use an effective meson-nucleon theory to describe nucleon interactions. In
the case, the form factors with free parameters in short distances are often used. For a typical
review up to 1988 see [48], and [49] for a study on NN QCD models up to 1 998, and [51] for a
historical and technical review on QCD-inspired models up to 2002. Discussing various aspects
of the QCD-inspired NN potentials needs another opportunity and is not in the level and aim
of the current note. Nevertheless, we try to a ddress some pro gresses and more plain potentials.
In f act, a pioneer study of the SR repulsion of NN interaction in the framework of quark
model was done in [170]. In the first studies, the quark and gluon dynamics (especially one-
gluon-exchanges (O GE’s) were included), to give quantitative description of SR pa r t , were
employed. Then, especially in 1980’s, the hybrid quark models were constructed (among the
first samples are in [171]), where for the LR and MR par t s they always used the potentials from
other phenomenological and boson-exchange models; look also at [172]. Describing scattering
data and deuteron properties with the earlier quark potential models was not so satisfactory.
In [173], nonrelativistic quark-cluster models were used to describe BB, NN and YN interac-
tions especially in the MR’s and SR’s with some good descriptions. After that, the hybrid
models (including quark, gluon and meson (especially pion) exchanges) were developed and
more improved. Among the earlier hybrid models, t hat is an NN potential [100] made from
Paris potential [44] for long- and intermediate-distances with the quark-cluster model (QCM)
for short distances. There, the effects of the quark degrees of freedom on NN observables
were surveyed. But, describing its time pp data was not good except if one used some other
adjustable potentials in the LR and MR part s.
In 1990’s, chiral constituent quark models (CCQM) were framed, which were always consid-
ered as a result of the spontaneous symmetry breaking of QCD Lagrangia n. There, OPE’s,
OSE’s (S for sigma) and OGE’s were included besides a phenomenological confining potential.
The resultant potentials describ ed NN scattering data and deuteron properties better than
any o t her QCD-inspired potential at that time. They are many of the CCQM potentials that
we mention just some studies. In [50], the SR NN interaction is described by a CCQM model
as well, where the constituent quarks interact through pseudoscalar meson exchanges. There,
projecting the six-quark wave-function into NN channel produces an SR node f or S-waves, like
M-type potentials [157]. So, the short distances are described microscopically, whereas the
medium and large distances are described through the Yukawa pion and sigma meson between
the quarks belong to t he nucleons. The CCQM models are further addressed in [51], where
it is discussed that they describe well the LR attractive and SR repulsive features in addition
that they are universal in describing a ll baryons on equal foot ings–Look also at [52].
Among other QCD-inspired NN potentials, t he hybrid quark-meson models in [174] and [175]
are nota ble, where the former was to apply to finite nuclei calculations and is ba sed on a rel-
56

ativistic quark model. The latter [175], which we call it Japan-group potential, is a unified
model to describe NN and YN interaction and, with few parameters, gives a good fit of its
time scattering data. That is another potential model for NN, ΛN and ΣN interactions in [53],
which we call it China-group potentia l, which is completely based on QCD ingredients with-
out any use of meson exchanges. The China-group potential is based on a quark delocalization
color screening model (QDCSM) and describes the SR and MR interactions simultaneously
besides a claim that it describes well NB (NN, NΛ, NΣ) scattering data.
It is also good to mention an extension of the chiral SU(3) quark model (CSQM) to describe BB
interactions. Indeed, in CSQM, a nonet of pseudoscalar and a nonet o f scalar meson exchanges
are used to describe the LR and MR pa rts of interactions, while the SR par t is described by
OGEP’s and a lso quark-exchange effects. The model gives a good description for NN and
YN systems. In [54], to study whether OGE’s or vector-meson exchang es could describe the
SR part of the interaction, the CSQM was extended t o include vector-meson exchanges as
well. In the resultant extended chiral SU(3) quark model (ECSQM), the strength of OGE
was largely reduced and the SR repulsion was owned to a combined effect of pseudoscalar and
scalar mesons, and was better described with a good fitting of scattering data.
The effects of the quark model calculations in the SR part on phenomenological and meson-
exchange calculations in the MR and LR parts are studied further in [176]. It is settled that
the Q CD quark models cannot describe the higher partial waves though they could describe
the lower partial waves well. Therefore, the hybrid models are indeed essential to describe well
scattering data. Then, one should employ LR and MR potentials from the other high-quality
meson-exchange or phenomenological potentials next t o quark-model potentials to describe the
SR part. The potentials so describe experimental NN data and bound states fairly but they
are not still as good as the fully phenomenological and meson-exchange HQ potentials.
It should be mentioned that, for the QCD-inspired models, some criteria are more im-
portant. Choosing a proper quark model, selecting suitable six-quark ground states, a nd the
methods t o evaluate phase shifts are important. It is also notable that the M-type NN poten-
tials [59] are other clear QCD-inspired pot entials, and also the Oxford potential [60] that we
discuss below briefly.
4.22 The Oxford Potential
The Oxford potential is among QCD-inspired pot entials. The group has applied nonrelativis-
tic constituent-quark models to low-energy NN interaction. They have shown [60] t hat the
potential reproduces well NN scattering data and deuteron properties as the high-precession
potentials such as CD-Bonn [99], Nijmegen I [28], ArgonneV18 [31] and NNNLO [80] po ten-
tials. Indeed, in the Oxford potential, a combination of one-pion-exchange (OPE), one-sigma-
exchange (OSE) and one-gluon-exchange (OGE), next to using t he charge dependence from
CD-Bonn potential, and some other subtleties are involved.
57

4.23 The First CHPT NN Potentials
Although some of the mentioned phenomenological and boson-exchange potentials are related
to QCD especially in the SR part, the relation is not systematic and consistent. The works
by Weinberg [68] and considering a Lagrangian, which includes the chiral symmetry of QCD,
written in terms of pions and nucleons a nd their covar iant deriva t ives, was a starting point to
build new generations of NN interactions. In the case, the mesons and hig her degrees of freedom
could be integrated out as their effects might be considered as some undetermined coefficients
and higher order terms. From the resultant effective Lagrangian, the potential so is expanded
systematically in the powers of ( Q/Λ
QCD
), where Q is a typical involved momentum. Therefore,
the resultant potential is consistent with QCD symmetries and is a logical and systematic way
to describe NN interaction and relate it to QCD. Detailed studies in the case need more space
and time and are not aim of this concise study. Nevertheless, for more preliminary details and
references, look at the subsections of (3.3), (3.4), (4.2).
First, Ordonez, Ray and van Kolck (Texas-group) [77], in 1993, proposed an exact two-
nucleon po tential based on an effective chiral Lagrangian. For intermediate states, they consid-
ered at least one pion (π(1 40)) and one isobar (∆(1232)), and that the resultant NN potential
was a sum of the involved irreducible diagrams. NN scattering amplitudes were then evaluated
by inserting the potential into LS or modified Schrodinger equation. The lowest order of that
perturbative expansion was because of tree graphs which resulted in an LR part O PEP. Still,
other diagrams up to the third order of chiral expansion, up to one-loop diagrams, reproduced
the other known features of NN interaction such as SR repulsion, MR attraction, spin-orbit
force, and many others. The potentials were written in p-space first, and in terms of some
operators in (4.20) and more dependencies of the functions on k
2
, q
2
. The dependence on k
2
was common, whereas the q
2
dependence was not usual. In addition, the energy dependence
in t he stat ic OPEP, which was in t urn more improved than the previous ones, was because of
the recoil effect of pion emission from nucleon. The MR part was because of TPEP with many
parameters, while the form f actors were used to regula rize the potential at the origin. Indeed,
for the Fourier tr ansform into r-space, the Ga ussian form factors with the cutoffs Λ as e
k
2
Λ
2
,
as in Nijmegen potentials [28], were used.
The general form of the potentia l in r-space can be written as (4.34) with n = 20, where the
functions V
i
here a r e in term of the radial coordinate r and its first and second derivative as
well as energy as
V
i
= V
(0)
i
(r, E) + V
(1)
i
(r, E)
∂
∂r
+ V
(2)
i
(r, E)
∂
2
∂r
2
, (4.107)
and 14 out of the 20 operators are those in (4.35) of the Urb81 potential and the 6 remaining
ones are
O
i=15,...,20
= S
12

~
L.
~
S

, S
12

~
L.
~
S

~τ
1
.~τ
2

, S
12
L
2
, S
12
L
2

~τ
1
.~τ
2

, S
12

~
L.
~
S

2
, S
12

~
L.
~
S

2

~τ
1
.~τ
2

,
(4.108)
abbreviated as tls, tlsτ, tll, tllτ, tls2, tls2τ. The first eight operators exist in almost all poten-
tials with only the ra dial functions of V
i
without derivatives, while the eight functions here are
58

depended on the first and second derivatives of r. For the next six operators, V
(1)
i
= V
(2)
i
= 0;
and the remaining six operators are amo ng special characterizes here. All extra terms come
from the q
2
dependence and recoil effects included in the potential.
Next, by having the potential, one may solve the Schrodinger equation numerically. The eval-
uated phase shifts and deuteron properties are depended on the undetermined parameters of
the Lagrangian. These parameters are fitted to the Nijmegen PWA93 database [28] and errors
from Arndt et al. [1 77], whereas the cutoff para meter is fixed to Λ = 3.90fm
−1
, which is in
turn equal to the ρ-meson mass. In general, with 26 para meters, the potential is fitted to the
pp+n p data up to T
lab
=100 MeV for J ≤ 2. The phase shifts for J > 2 are determined from
OPEP in low-energies, a nd are not used in the fitting process. The result is a qualita tive fitting
of deuteron properties and a quantitative fitting of the phase shifts. This means that this new
NN potential type can describe well the basic properties from a more fundamental and tight
theoretical ground. By including higher orders of the chiral perturbative expansion, one may
cover the higher energy ranges as well.
In summary, an advantage of these NN potentials is the systematic expansion of interaction
in terms of chiral power counting. Indeed, the Texas-group has earned an NN potential in a
certain order o f the chiral perturbation expansion in both p- and r-space [77]. The gro up has
only used the chiral Lagrangian of QCD in low energies and the resultant potential is f ree
of meson theories. The agreements with deuteron properties and experimental data below
100 MeV are satisfactory. The model has some likenesses with the Paris-group (because of
the pion dynamics on the LR part), Bonn-group (likenesses in low energies) and Nijmegen-
group potentials (relations to QCD) in some parts of interaction; but, here EFT is used in
general. In is mentionable that describing experimental data, by this first CHPT potential,
were not good as the phenomenological potentials. It is also notable that the SR nuclear forces
from χEFT were then surveyed by van Kolck in [178], and also in [70] as a related general
review. Meanwhile, look at [179] to study few-nucleon forces with this type potential, where
interactions arise in chiral perturbative expansion naturally.
4.24 Sao Paulo-Group CHPT Potentials
Robilotta and da Rocha, have tried to estimate tow-pion-exchange contribution to NN inter-
action based on chiral symmetry with resolving the problem met in the previous TPEP’s and
including just pions and nucleons [78]. In fact, by making use of the similar methods as those in
Partovi-Lomon potential [35], and by employing a chiral model, they framed some 2π-exchange
potentials. The model produced the central, spin-spin, spin-orbit and tensor components of
the potential with and without isospin dependence. From their view, NN potential r eads
V = V
core
+ V
S
+ V
P S
+ V
OP EP
, ( 4.109)
where V
core
stands for the SR core potential, V
S
stands for the contribution from the box
and crossed box diagrams, V
P S
stands for the cont ribution from chiral triangle and bubble
interactions, and V
OP EP
, as usual, is for OPEP tail. A problem with the first approach was
59

that it could not reproduce experimenta l data well for the related intermediate region. One
might improve the results by including further degrees of freedom such as ∆ resonances as
was done in [40], and the results were compared with those from parameterized Paris potential
[98], Arg94 potential [30], dTRS potential [108] and Bonn87 po tential [45].
Then, a relat ivistic chiral expansion up to O(k
4
) for the TPEP’s in p- space and further,
its contents and features in r-space, was given in [180] by Hig a and the former members of
this called Sao Paulo- group. One should note that k < 1 GeV here is for the pion four-
momentum and nucleon three-momentum, and is a typical scale for chiral perturbatio n theory.
The resultant potential in r-space reads
V = V
+
+ V
−
(~τ
1
.~τ
2
), (4.11 0)
where
V
±
= V
±
c
(r) + V
±
σ
(r)(~σ
1
.~σ
2
) + V
±
t
(r)S
12
+ V
±
ls
(r)
~
L.
~
S + V
±
q
(r)Q
12
. (4.111)
These r-space potential functions are in terms of some numerical coefficients (relat ed to pion-
and nucleon masses and involved coupling constants) that multiply some dimensionless func-
tions, where the latter a re in turn come from the Fourier transforms of the Feynman loop
integrals. It is notable that this parameterization is valid to describe NN interaction in the
range of about 0.8f m ≤ r ≤ 10fm. It is also notable that the TPE contribution for 3N force
in the same order O(k
4
) is presented as well in [181]; and a review on the subject is given in
[61]. The differences between t he formalism here, to discuss chiral TPEP contributions to NN
interactions, and heavy-baryon (HB) formalism in the next subsections, are discussed in [1 82].
Indeed, in HB formalism of chiral perturbative expansion, relativistic Lag rangian is expanded
in 1/M powers, which is in turn a kind of nonrelativistic expansion; for more details look also
at [183].
4.25 Munich-Group CHPT Potentials
The Munich-group, by using a similar CHPT Lagrangian as [77], and employing a covariant
perturbation theory and dimensional regularization, estimated the chiral two-pion-exchange
NN potential as well as the usual one-pion-exchange part [79]. The calculations were up to
the third order in low external momenta and one-loop order (or NLO). As a result, the phases
shifts with ℓ ≥ 2 and the mixing angles with J ≥ 2 were determined as free parameters,
and could be used as input in the next NN phase-shift a na lyses. By increasing the or bit al
angular-momentum, a close and better agreement with the usual OPEP became obvious. In
other wo rds, the study was to describe NN int eractio n in terms of OPE’s and TPE’s for
the LR and MR parts in a model independent ma nner. The potential was composed of the
central, spin-spin, tensor, spin-orbit and quadratic spin-orbit terms with and without isospin
dependence such as those in Sao Paulo-group potentials–Note that the involved pion-nucleon
Lagrangian here, similar t o those in the latter group, have the dimension 2 and a r e based
60

on dimensionally regularized Feynman diagrams; and because the pot entials are evaluated
perturbatively, the bound-states ar e not described well! Resultant expressions for the potentials
in r-space, coming fro m irreducible chiral 2π exchanges, ar e of the van-der-Waals type with
the asymptotic exponential behavior e
−2m
pi
r
r
n
valid at least for the ra nge about 1fm < r <
2fm. There is not any pion-nucleon form function in that for ℓ ≥ 2, J ≥ 2, the problematic
singularities in the Fourier transforms are not so impor t ant. Agreement with the phase shifts
up to D- wave up to T
lab
= 150 MeV are good, and for the higher waves agreements become
better and better up to the pion-production threshold in almo st 280 MeV. Fo r the lower partial
waves, the SR effects become important and so, just TPE is not enough to reproduce the phase
shifts. It is also notable that relevant p otentials are compared with Paris79 [98] and Full-Bonn
(Bonn87) [45] potentials.
Soon later, they also used two-pion exchange diagrams with virtual ∆(1232)-isobar degrees
of freedom and correlated 2π exchange as well as the ρ, ω vector-meson exchanges in [184].
As a result, they reproduced the experimental data up to 3 50 MeV for ℓ ≥ 3 and up to 80
MeV for D -waves, without any adjustable parameter. So, this is chiral symmetry that has
opened a nice window to NN interaction. It is mentionable that, to describe the lower partial
waves, no nperturbative methods and other SR parameterizations are still needed–It is good
to mention that the importance of the chiral TPEP’s was more confirmed in [185] (by some
members of the Nij megen-g roup and others), when they saw that the chiral TPE loops were
important in the LR part of pp interaction as they improved the results of just OPEP’s. In
other words, the group noted that by including both OPE and χTPE contributions, they
could find a good fit of data up to 35 0 MeV for r ≥ 1.4fm. The range below the mentioned
one was then parameterized by 23 boundary condition parameters in t he energy-dependent
partial-wave-analysis.
Furt her efforts, by K aiser, have taken to include chiral uncorrelated three-pion exchanges,
higher-loop and relativistic corrections to NN interactions [186]. Indeed, it was shown that
the uncorrelated 3π exchanges have negligible effects on NN interactions in r ≥ 0.8fm. The
local potentials produced by 2π-exchange diagrams in two-loop order of the heavy-baryon
chiral perturbation theory, besides including the second order ππN vertexes, and t he first
relativistic 1/M corrections in one-loop 2π-exchange diagrams, were discussed a s well. The
latter were the components for the chiral NN potential in the next-to-next-to-next-to-leading-
order (NNNLO). It should be mentioned that these two-loop diagra ms lead to contributions
about O(k
4
) in chiral expansion and so N
3
LO. By including 1/M
2
corrections to 2π-exchange
diagrams and their effects on various parts of interaction and va r ious states, the chiral NN
potential in this N
3
LO order is complete. We should remember that the potential structures
and operators here are almost the same as those of the Sao Paulo -group; a nd t ha t in the third
reference of [186], an explicit analytical expression for the potentia l in r-space fro m the p-space
one is presented. Next, he studied the spin-orbit coupling produced from 2π exchange in 3N
interaction by including the virtual ∆-isobar in [187].
It is notable that in [188], by another group, there are also a complete set of 2π-excha ng e
diagrams in the same fourth-order (N
3
LO) in chiral perturbative expansion. One could see
61

that the fourth-order contribution is less than the third-order one; and this in turn signals the
converging of chiral expansion. By employing the analytical expressions in [186], t hey applied
the methods to NN scattering to calculate scattering amplitudes; and then they compared
predictions with experimental phase shifts and tho se from the usual meson-exchange theories.
To make a more sensible comparison, they included OPE and iterated O PE contributions as
well, and next showed the phase shifts for ℓ ≥ 3 below the energy of 300 MeV. The agreement
between Full-Bonn potent ial and this N
3
LO potential was good.
By the way, many other studies are done by the group members. For instance, in [189],
chiral four-nucleon interactions in this framewo r k are studied. A microscopic optical potential
from two- and three-body chiral nuclear forces is constructed in [190]. Some members of the
group, next to others, have modeled YN potentials in NLO of chiral effective field theory in
[191]. In the latter, contributions from the one and two pseudoscalar-meson diagrams as well
as four-baryon contact terms are included. The SU(3) flavor symmetry was used to set up
potentials while its breaking by the physical masses of the pseudoscalar mesons (π, K, η) was
considered as well. Excellent results, compared with the counterpart HQ phenomenological
potentials, were gained. That is also a relativistic chiral SU(3)-invariant La grangian up to
O(q
2
) order to describe BB interaction in [192].
4.26 Idaho-Group CHPT Potentials
Along with va rious efforts after the first CHPT potential by Texas-group in [77], a better NN
potential based on chiral EFT appeared by Entem and Machleidt [80] in 2001. In the potential
both meson and quark degrees of freedom a r e included, while [77] is a meson-free potentia l.
Indeed, that is an NN potential, based on HB formalism of chiral perturbative expansion
that includes one-pion and two-pion exchanges up to the third order of chiral expansion.
The short-range force in the fourth order of expansion is involved because of good fitting of
the D-wave phase shifts. There, a two-pion exchange potential in the fourth-order of chiral
expansion is also presented. The potential has almost the same quality as the HQ Nijmegen
potentials [28], CD-Bonn [99 ] and Arg94 [31] potentials. The phase shifts below T
lab
=300
MeV, deuteron properties and low-energy np scattering parameters as well as Triton binding-
energy are described well with this potential [80].
Later, the authors modeled, in fact, the first accurate NN potential in N
3
LO (fourth-or der)
of chiral perturbat ive expansion [193]. The new pot ential, in reproducing its time pp and np
data below 290 MeV, is comparable with the best high-precession phenomenological potentials.
After mentioning main features of the previous HQ phenomenological and meson-exchange
potentials, it is also argued in [194] t ha t EFT approach to nuclear forces is better than all earlier
efforts in that it produces a wished precession, gives satisfactory results in nuclear calculations
as well as dealing with few-nucleon interactions on an equal footing as NN interaction. There
are also some reviews and many other related issues and progress presented in [72] and [55].
62

4.27 Bochum-Julich-Group CHPT Potentials
Bochum-Julich-group potentials are also ba sed on chiral EFT, similar to the other CHPT
potentials mentioned above, except that t hey extracted the Lagrangian’s by using a ”unitary
transformation” method. In fact, they have studied many NN (also 3N and few-nucleon) forces
besides various related aspects in LO, NLO, NNLO and NNNLO of CHPT by taking the most
general chiral Hamiltonian with pions and nucleon fields as we describe below concisely.
But before that, we note that in the standard method, such as that of Texas-group [77],
the most general Lagrangian including all symmetries such as chiral symmetry of QCD was
first written with an infinite number of terms including nucleon and pion fields and their
derivat ives. The breaking of chiral symmetry was clear in smallness of the pion mass, and
then the external momenta of the pion and nucleon should not exceed the scale of Q. As a
result, t he expansion parameter was Q/Λ
QCD
and nucleons were treated nonrelativistically,
where Λ
QCD
≈ 1 GeV that is almost the ρ-meson mass. The other degrees of freedom, such as
heavy mesons and other baryons which were then less important, were integrated out (except
maybe ∆ isobars) as their information was so included in the Lagrangian’s parameters. In
the process, a finite set of tree and loop diagrams were included. But a problem was that
due to the presence of low energy bound states, perturbative theory fa iled actually; or in
other words, infrared divergences with the few included nucleons disturbed the power counting
of chiral expansion. A way to solve the problem was to use the old-f ashioned time-ordered
perturbation theory by Weinberg [68], where the expansion parameter was Q/M, instead of
the covariant method. Still, in the latter method, the effective potential was not Hermitian as
it was depended on the incoming-nucleon energies, and t hat the nucleon wave functions were
not orthogonal there. So the unitary transformations here resolve the problems, where the
expansion parameter is now the small momenta of external particles. It is also notable that
resultant potentials are energy-independent, which makes the applications to few-body and
nuclear-structure calculations simpler.
4.27.1 LO, NLO and NNLO Potentials
In general, these potentials include contributions from one- and two-pion exchanges to simulate
LR and MR int era ctio ns besides contact terms to simulat e SR interactions. The resultant
interactions, from LO, NLO and NNLO of CHPT by considering t he most general chiral
Hamiltonian in terms of pions and nucleon fields, are given in [74]. The LO intera ctio n includes
two fo ur -nucleon contact terms and an OPE potential as
V
(0)
cont.
= C
s
+ C
t
(~σ
1
.~σ
2
), (4.112)
and
V
(0)
1P EP
= −

g
A
2f
π

2
(~τ
1
.~τ
2
)
(~σ
1
.
~
k)(~σ
2
.
~
k)
k
2
+ m
2
pi
, (4.113)
where the low-energy constants (L ECs) of C
s
, C
t
, C
1
, D
1
, ... are to be determined by fit t ing
some data, g
A
is the axial-vector coupling, f
π
is the pion decay-constant, and other symbols
63

are the same as used before. In NLO, the potential is a renormalized sum of one- and tow-pion
exchanges and contact interactions. This means t ha t next to above contributions, it includes
a TPEP contribution (V
(2)
T P EP
) and seven four-nucleon conta ct terms where the latter reads
V
(2)
cont.
= C
1
k
2
+ C
2
q
2
+ (C
3
k
2
+ C
4
q
2
)(~σ
1
.~σ
2
) + C
5
˜
LS
1
+ C
6
˜
S
(0)
12
+ C
7
˜
`
S
(0)
12
, (4.114)
where
˜
`
S
(0)
12
= (~σ
1
.~q)(~σ
2
.~q), the nine LECs are determined by fitting to the np S and P and
3
S
1
−
3
D
1
phase shifts, and the mixing parameter ε
1
for the laboratory energies below 100
MeV. TPEP in this NLO includes k, k
2
, q
2
dependence as well as the operators I,
˜
S
12
, isospin
dependence and some constants [74]. On the other hand, in NNLO, another TPEP (V
(3)
T P EP
) is
also included, which in turn includes some special combinatio ns of k, k
2
, q
2
with the operators
I,
˜
S
12
,
˜
LS
1
without and with isospin dependence and some constants. It is mentionable that if
one includes the contribution from ∆(1232)-isobar, t he resultant NNLO-∆ pot ential is almo st
same as NNLO one especially for low momenta.
We should also no t e that the pion-exchange NN potentials could be written generally, in p-
space, as
˜
V =
˜
V
+
+
˜
V
−
(~τ
1
.~τ
2
), (4.11 5)
where
˜
V
±
=
˜
V
±
c
+
˜
V
±
σ
(~σ
1
.~σ
2
) +
˜
V
±
ls
˜
LS
1
+
˜
V
±
q
˜
Q
12
+
˜
V
±
σk
˜
S
(0)
12
+
˜
V
±
σq
˜
`
S
(0)
12
, (4.116)
and to adjust more with the record in (4.19), we set SS
0
= (~σ
1
.~σ
2
); and that the f unctions of
˜
V
±
c
, ... are in terms of ~p
i
, ~p
f
, z with z = cos(~p
i
, ~p
f
), included masses and coupling constants.
To regularize or have right behavior for the potentials in large momenta (short distances), the
sharp and exponential for m factors are used as
F (k
2
)
sharp
= θ

Λ
2
− k
2

, F (k
2
)
exp.
= e
−k
2n
/Λ
2n
, (4.117)
where the sharp cutoff is proper here with Λ = 500 MeV for NLO and Λ = 8 75 MeV for
NNLO; and that in exponential form factors, n = 2, 3, ... with often n = 2 here, where the
latter is used especially to evaluate some deuteron properties with good results. In addition,
phase shifts and mixing parameters for hig h energies and angular momentums are described
well for the energies below 300 MeV, with a note that the partial waves higher than P are
free of adjustable parameters. Also, var ious properties of nuclei with A > 2 and especially
the binding energies of
3
H a nd
4
He are evaluated by these NLO and NNLO potentials with
an almost same quality as the standar d high-precision phenomenological and boson-exchange
potentials [74], [195].
4.27.2 NNNLO Potentials and More
Next development of the model was to NNNLO of chiral expansion [81]. The new potential
includes one-, two- a nd three-pion exchanges as well as the contact terms with zero, two and
four derivatives. Relativistic corrections and isospin-breaking mechanisms are also included.
64

In fact, next to the previous contact terms of (4.114) and (4.114), the new included contact
terms are
V
(4)
cont.
=D
1
k
4
+ D
2
q
4
+ D
3
k
2
q
2
+ D
4
n
2
+ (D
5
k
4
+ D
6
q
4
+ D
7
k
2
q
2
+ D
8
n
2
)(~σ
1
.~σ
2
)
+(D
9
k
2
+ D
10
q
2
)
˜
LS
1
+ (D
11
k
2
+ D
12
q
2
)
˜
S
(0)
12
+ (D
13
k
2
+ D
14
q
2
)
˜
`
S
(0)
12
+ D
15
˜
Q
12
,
(4.118)
where one could also include another 24 t erms which contain the isospin factor of (~τ
1
.~τ
2
). Now,
all 26 four-nucleon LECs are determined by fitting the pp+np Nijmegen-gr oup da t abase [121]
(the releva nt S, P, D phase shifts and mixing parameters) and nn scattering-length.
On the other hand, for pion-exchange parts, a new three-pion exchange contribution (V
(4)
3P EP
)
is considered t hough its effect is negligible (note that the n-pion-excha nge diagrams become
important aro und Q
2n−2
). These pion-exchange contributions can again be written as (4.115)
with (4.116), where for instance the lowest order of the scalar function of
˜
V
−
σk
is indeed (4.113)
without (~τ
1
.~τ
2
) factor. We remember that LS equations and a relativistic form f or kinetic
energy ar e employed to iterate the potential here. Reducing to a nonrelativistic form is more
useful in r eal calculations. The exponential f orm-factor of (4.117) with n = 3 is used to regu-
larize LS equations with the cutoffs of Λ = 450 − 600 MeV.
The isospin-breaking of strong interactions because of different masses of up and down quarks,
and from electromagnetic interactions because of different charges of up and down quarks are
also included. Indeed, the potentials fo r different NN systems and isospins are different such
that, for instance, V
1P EP
(pp) 6= V
1P EP
(np, T = 1) 6= V
1P EP
(np, T = 0) and so on. This is
finite-range isospin-breaking, while the long-range isospin-breaking is because of different elec-
tromagnetic interactions such that V
EM
(pp) 6= V
EM
(np) 6= V
EM
6= (nn). In other wor ds, the
quark mass splitting causes isospin-breaking in short distances, whereas the contact electro-
magnetic terms cause isospin-breaking in long distances–Look a lso at the discussion on Arg94
potential, Nijmegen HQ potentials and CD-Bonn potential in subsections of (4.12.2), (4.15.4)
and (4.13.2), respectively.
In summary, the group has set up some NN potentials by using the unitary transformation
method applied to the most general chiral invariant Hamiltonia n in terms of pion and nu-
cleon fields from LO up to N
3
LO. In the lat ter, CIB and CSB in leading order, the pion mass
differences in OPEP’s, kinematic effects because of the nucleon mass splitting, and electromag-
netic corrections such as those in Nijmegen PWA’s, and many other subtleties are included.
Deuteron properties and the low phase shifts of S, P, D are described excellently, whereas
the high partial waves o f F, G, H,... are parameter free and are well described depended on
the doubts in the cutoffs. In general, several improvements with respect to the lower or der
expansions and also to the previous CHPT potentials are notable.
Among many other studies by the group members, improvements to the Weinberg approach
to arrive at the effective potential and the renormalization problem there, a new appro ach
based on an effective Lagrangian with exact L orentz invariance and by using time-ordered
perturbation theory, without using HB expansion, were presented and a nalyzed in [196]. In-
deed, they improved the heavy chiral perturbation theory for NN interaction a nd analyzed
the OPEP iterations. As a result, it was shown that the used renormalization, for one-and
65

two-loop diagrams of OPEP iterations, removes all nucleon-mass dependencies that disturb
the power counting–It is good here to mention a pioneer work to resolve inconsistencies in the
Weinberg’s chiral expansion. Indeed, in [69], Kaplan et al. used a dimensional regularization
scheme with a novel subtraction (renormalization-group techniques) to get a consistent chiral
expansion and dissolve the failure of the Weinberg’s power counting scheme. They applied the
method in the order O(Q
0
) to
1
S
0
and
3
S
1
−
3
D
1
NN scattering channels, and t hen compared
the results with Nijmegen PWA93 [121] with satisfactory agreements. For some other old, and
of course related, typical studies in the phase look at [197], [198].
By the way, for a recent review on NN, 3N and few-nucleon intera ctio ns especially in the
framework of χEFT, advanta ges and disadvantag es of this approach to nuclear forces, look at
[76] by Epelbaum and references therein. To end the discussion in the phase, we cite [73] as
the last constructed optimized potential at NNLO by other people.
5 Some Other Model s and Pot entials
In general, almost all potentials are belong to one of the four main models. These are, the
almost full phenomenological model; the model based on field-theoretical methods, inverse-
scattering, quantum-dispersion relations and boson-exchange pictures; the model based on
QCD and constituent quark methods (the QCD-inspired model); the model based on CHPT
and EFT and their various extensions.
We have tried to include and study almost all models and po tentials to describe two-nucleon
interactions with an emphasis on some in more details as samples of well-known and hig h-
precession NN potentials. Technical studies o f some potentials need more space and time next
to many physical and mathematical backgrounds that is not the aim of this concise pedagogical
review. Nevertheless, there are still some other special NN interaction models and potentials,
and related topics, to be addressed. We mention some in what follows.
Among the standard and more theoretical potentials is theVirginia-group potential [96],
which is a special relativistic OBEP based on field-theoretical and dispersion-relation tech-
niques. In fact, they have framed a few potentials by taking various meson exchanges. The
Bochum-group potential [97] is another fundamental NN potential based on field-theoretical
and dispersion-relat ion methods that also uses various meson exchanges in long distances a nd
QCD effects; meanwhile the direct NN int eractio ns coming from t he intrinsic structure of nu-
cleon are considered in short distances. By including some two- and three-pion correlations,
they have claimed to hold good description of NN scattering data. The Seattle-group stud-
ies on NN interaction are also notable. Indeed, they have studied low-energy NN interactions
based on EFT, by using some simple models for interactions, up to NNLO in chiral expansion,
next to some other related topics during their study period in 1990’s [199].
There are the potentials based on Mean Field Theory (MFT), which are o f particular
interest in many- body calculatio ns in nuclear physics especially–Loo k, for instance, at [200]
and [201] for the first NN interaction made of relativistic mean field theory.
Renormalization Group (RG) approaches to NN interaction are other serious efforts.
66

In an RG flow viewpoint, a model-independent low-momentum interaction is obtained by
integrat ing out high-momentum compo nents ( cutting out problematic high-momentum modes)
of various potential models [101]. Indeed, the model independence of resulting potentials
shows that the physics of the nucleons interacting at low momenta does not depend on the
details of the high-momentum dynamics assumed in conventional potential models. Further
developments such as incorpora t ing the method into the Fermi liquid theory are also made [202],
[203]. In [204 ], detailed results for the model-independent low-momentum NN potent ial V
low k
are shown. There, they have a pplied the approach to some commonly used high-precession
NN potentials, and then compared resultant potent ials in various ways such as comparing
matrix elements of the potentials and various resulting phase shifts in p-space. In Figure 4
and Figure 5 are two such sample comparisons of some high-precession NN potentials together
and with two simple RG models, respectively. For a newer ”similarity renormalization group”
Figure 4: Diagonal matrix elements of some high-quality NN potentials (V
NN
) versus relative-
momentum (k) for
1
S
0
and
3
S
1
partial-wave, in momentum-space [204].
approach, see [205] and [206], and for a r ecent review and study of the subject look at [207].
Lattice QCD approach to NN interaction is another impo rtant way; look, for instance, at
[63, 64] and [65]. Among some typical studies, see [208], where a spin-dependent potential in
lattice QCD is presented; [209], a nd [210], where nonlocality of NN potentials, and deuteron
and some other two-body bound states in la t tice QCD are discussed–Look also at [67], where
QCD sum rules are used f or NN intera ctio ns. Altogether, this phase o f study is still improving
with giving better quantitative results as the previous good qualitative ones.
Tubingen-group has applied projection techniques on some former NN potentials among
the boson-exchange, phenomenological, RG flow and EFT ones to map them over the operat or
basis of relativistic field theory [211]. Indeed, they have presented a model-independent study
of NN interaction from its Dirac structure. That is a special way to compare various potentials,
where a nice agreement is found as well.
67

Figure 5: Diagonal matrix elemen ts of V
low k
(V
bare
in figure) for two simple RG potentials are
compared with V
low k
derived f rom some high-quality NN potentials [204].
They have also built a new energy-independent nonlocal potential above inelastic thresholds in
quantum field theories that satisfies a suitable Schrodinger equation at low energies [212]. The
potential is indeed composed of a set of Nambu- Bethe-Sa lpeter wave functions. By applying
the same method, one could set up three-nucleon potentials as well.
By the way, there may be other models and potentials not covered in this note and so, it
would be pleasure to hear mor e about ot her NN potentials. Meanwhile, there ar e still many
studies on various aspects of NN interactio n which need addressing. For examples, nonlocal
and local terms and their impact on NN interactions and their r oles in some NN pot entials are
studied, for instance, in [213]; a nd nonlocality of NN potentials in latt ice QCD is discussed,
for instance, in [209]. For a study o f CIB and CSB of NN interaction, look at [214] and for
parity violation in NN interaction, see, f or instance, [215].
We should also mention that Thr ee- and few-nucleon interactions are also interesting
to which less efforts t ha n two-nucleon interactions are a llocated. For there-nucleon force, look
at a recent review of [216]; and for a view to few-nucleon forces, look at [76, 217].
6 Outlook
Nowadays the theory of strong nuclear force is well experienced both quantitatively and qual-
itatively. The best qualitative results are obtained by using phenomenological and boson-
exchange potentials based on quantum field theory and dispersion relation techniques, and
even new potentials based on chiral perturbation theory. Indeed, more qualitative r esults are
of the QCD-inspired models and the models based on chiral EFT.
NN int era ctio n is now under contr ol f or t he energies below almost T
lab
=500 MeV well. Because
of the high-precession experimental NN data, describing the long- and intermediate-range parts
of the interaction based on various meson exchanges are quantitatively good and the hybrid
68

models of quark and gluon exchanges for the short-ranges seem to be more suitable.
Although we have now many high-precision NN potentials applying to nuclear-structure cal-
culations with satisfactory results, still some questions are remained to be answered. I think
the main pro blem is that we don’t have still a unique comprehensive model for including all
well-known features of NN interaction. Obviously, chiral EFT methods and models are better
in describing nuclear forces in general. They have a standard formalism a pplicable t o few-
nucleon systems with including many fundamental physics and mathematics of the problem.
But, there are still some problems and limits; look at [4], [5]. Among the issues with EFT
potentials, which one may ask, are the proper renormalizat ion of the chiral nuclear potentials
and sub-leading chir al few-nucleon forces; few- and multi-nucleon potentia ls in higher orders
of chiral expansion. Meanwhile, lattice QCD models for nuclear forces are still improving and,
in some recent studies, a lattice version of chiral EFT is also applied to nuclear forces [76].
On the ot her hand, after the well-conjectured string/gauge, AdS/CFT, duality and there-
after Holographic QCD studies, it seems that the NN interaction issue is faced with another
revolution. So, we should be wait for mo r e sophisticated models for two- and many- body
nuclear intera ctio ns in this language–Look, for instances, at [218] and [219].
Altogether, it seems that the nuclear force issue is still improving. I think tha t we may
someday have a unified scheme for NN interaction and link various known NN models and
potentials. Nevertheless, it will also be interesting to compare various NN potentials via some
suitable ways and try to understand more nucleon-nucleon interaction subtleties.
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