JLAB-THY-07-632
High-precision covariant one-boson-exchange potentials
for np scattering below 350 MeV
Franz Gross
Thomas Jefferson National Accelerator Facility, Newport News, VA 23606
College of William and Mary, Williamsburg, VA 23187
Alfred Stadler
Centro de F́ısica Nuclear da Universidade de Lisboa, 1649-003 Lisboa, Portugal
and Departamento de F́ısica da Universidade de Évora, 7000-671 Évora, Portugal
All realistic potential models for the two-nucleon interaction are to some extent based on boson
exchange. However, in order to achieve an essentially perfect fit to the scattering data, characterized
by a χ2/Ndata ∼ 1, previous potentials have abandoned a pure one boson-exchange mechanism
(OBE). Using a covariant theory, we have found a OBE potential that fits the 2006 world np data
below 350 MeV with a χ2/Ndata = 1.06 for 3788 data. Our potential has fewer adjustable parameters
than previous high-precision potentials, and also reproduces the experimental triton binding energy
without introducing additional irreducible three-nucleon forces.
A good understanding of the interaction between two
nucleons is essential for the study of nuclear structure and
nuclear reactions. In the long history of theoretical mod-
els of the NN interaction, One-Boson-Exchange (OBE)
models played a role of special importance. Yukawa’s [1]
insight that a short-range force can be generated through
the exchange of particles of finite mass led to the dis-
covery of the pion, and later it was found that the ex-
change of a pion can quantitatively describe the longer-
range part of the NN interaction. Since the range of the
force is inversely proportional to the exchanged mass, the
exchange of heavier mass bosons generates NN forces of
intermediate to short range. It was found that the vector
bosons ω and ρ contribute to the observed spin-orbit force
and strong repulsion at short internucleon distances [2],
and that scalar bosons provide intermediate attraction.
Today, with the development of potentials based on chi-
ral perturbation theory (ChPT) [3], we understand that
these scalar bosons are an approximate representation of
the two-pion exchange mechanism [4], which gives strong
attraction even if there were no two-pion resonances at
masses of around 500 MeV [5].
It is possible, of course, to construct phenomenolog-
ical NN potentials that, with a sufficently large num-
ber of parameters, give an accurate description of the
NN scattering data. However, OBE potentials have sev-
eral important advantages. First, they provide a phys-
ical mechanism for the interaction between nucleons.
This implies that the parameters in these models have
a physical meaning, and that, at least in principle, they
can be related to, or even be determined through other
physical processes. Second, it is possible to construct
consistent electroweak currents for systems interacting
through OBE, since the underlying microscopic processes
are known [6]. With phenomenological potentials this
construction is less straightforward because there is no
implied microscopic description of the flow of electroweak
charges through a nuclear system. Third, when OBE
is used in a covariant formalism without time ordering,
effective three- and many-body forces are automatically
generated from the off-shell couplings of purely two-body
OBE [7, 8]. With phenomenological potentials three-
body forces must be independently constructed. Finally,
OBE models are relatively simple, and depend only on a
moderate number of parameters. A quantitatively accu-
rate OBE model represents a very economical description
of the NN interaction.
OBE models also have their limitations. Since they
are not fundamental interactions, their validity does not
extend to very short distances where QCD should pro-
vide the correct description. In potential models, this
unknown short-distance part of the interaction is usually
parameterized phenomenologically through vertex form
factors with adjustable parameters. These form factors
also serve to regularize otherwise divergent loop integrals
that appear when the kernel is iterated. But parameters
that describe the unknown short distance physics cannot
be avoided; even more fundamental potential models de-
rived from ChPT require subtraction constants to renor-
malize and absorb infinities arising from the unknown
short range physics. At fourth order, a potential based
on ChPT will have at least 24 unknown subtraction con-
stants (parameters) [9].
After early phase shift analyses by the VPI group [10],
both the VPI [11] and Nijmegen [12] groups obtained op-
timal values of χ2/Ndata ≈ 1 after eliminating data sets
from their analyses, based on statistical arguments about
their incompatibility with other data sets [13]. The Ni-
jmegen group also updated their OBE potential (Nijm78)
to the new phase shift analysis, but they were unable to
get the χ2/Ndata of this 15 parameter model (now called
Nijm93) below 1.87 [14]. In order to construct very accu-
rate NN potentials they abandoned a pure OBE struc-
ture and made several boson parameters dependent on in-
http://arxiv.org/abs/0704.1229v2
dividual partial-waves. Similarly, the (almost) pure OBE
potentials of the Bonn family, such as Bonn A, B, and C,
were superseded by the realistic CD-Bonn, which also
incorporates partial-wave dependent boson parameters
[15]. The Argonne group also motivated their construc-
tion of largely phenomenological potentials like AV18 by
the apparent failure of the OBE mechanism (apart from
the pion-exchange tail) to allow a perfect fit to the data
[16].
The main objective of this letter is to show that within
the Covariant Spectator Theory (CST) it is, in fact, pos-
sible to derive realistic OBE potentials, and that these
require comparatively few parameters. This somewhat
surprising finding contradicts the earlier conclusion and
common belief that the OBE mechanism is missing some
important feature of the NN interaction. Accurate OBE
models may provide a useful intermediate step between
fundamental physics and experiment.
In CST [17, 18], the scattering amplitude M is the so-
lution of a covariant integral equation derived from field
theory (sometimes referred to as the “Gross equation”).
In common with many other equations, it has the form
M = V − V GM (1)
where V is the irreducible kernel (playing the role of a
potential) and G is the intermediate state propagator.
As with the Bethe-Salpeter (BS) equation [19], if the
kernel is exact and nucleon self energies are included in
the propagator, iteration of the CST equation generates
the full Feynman series. In cases where this series does
not converge (nearly always!) the equation solves the
problem nonperturbatively. With the BS equation the
four-momenta of all A intermediate particles are sub-
ject only to the conservation of total four-momentum
pi, so the integration is over 4(A − 1) vari-
ables. In the CST equation, all but one of the intermedi-
ate particles are restricted to their positive-energy mass
shell, constraining A−1 energies (they become functions
of the three-momenta) and leaving only 3(A−1) internal
variables, the same number of variables as in nonrelativis-
tic theory. Since the on-shell constraints are covariant,
the resulting equations remain manifestly covariant even
though all intermediate loop integrations reduce to three
dimensions, which greatly simplifies their numerical so-
lution and physical interpretation. This framework has
been applied successfully to many problems, in particular
also to the two- and three-nucleon system [7, 8, 20].
The specific form of the CST equation for the two-
nucleon scattering amplitude M , with particle 1 on-shell
in both the initial and final state, is [20]
M12(p, p
′;P ) = V 12(p, p
′;P )
(2π)3
V 12(p, k;P )G2(k, P )M12(k, p
′;P ) , (2)
where P is the conserved total four-momentum, and p, p′,
and k are relative four-momenta related to the momenta
of particles 1 and 2 by p1 =
P + p, p2 =
P − p, and
M12(p, p
′;P ) ≡ Mλλ′,ββ′(p, p
′;P )
= ūα(p, λ)Mαα′ ;ββ′(p, p
′;P )uα′(p
′, λ′) (3)
is the matrix element of the Feynman scattering ampli-
tude M between positive energy Dirac spinors of particle
1. The propagator for the off-shell particle 2 is
G2(k, P ) ≡ Gββ′ (k2) =
(m+ /k2)
m2 − k22 − iǫ
h4(k22) (4)
with k2 = P − k1, k
1 = m
2, and h the form factor of the
off-shell nucleon (related to its self energy), normalized
to unity when k22 = m
2. In this paper we use
h(p2) =
(Λ2N −m
−m2)2 + (m2 − p2)2
, (5)
where ΛN is an adjustable cutoff parameter. The indices
1 and 2 refer collectively to the two helicity or Dirac
indices of particle 1, either {λλ′} or {αα′}, and particle
2, {ββ′}.
The covariant kernel V is explicitly antisymmetrized.
In its Dirac form it is
V αα′;ββ′(p, k;P )
Vαα′;ββ′(p, k;P ) + (−)
IVβα′;αβ′(−p, k;P )
, (6)
where the isospin indices have been suppressed, so that
the factor of (−)I (with I=0 or 1 the isospin of the NN
state) insures that the remaining amplitude has the sym-
metry (−)I under particle interchange {p1, α} ↔ {p2, β}
as required by the generalized Pauli principle. This sym-
metry insures that identical results emerge if a different
particle is chosen to be on-shell in either the initial or
final state.
Next we assume that the kernel can be written as a
sum of OBE contributions
V b12(p, k;P ) = ǫbδ
Λb1(p1, k1)⊗ Λ
2(p2, k2)
+ |q2|
f(Λb, q) (7)
with b = {s, p, v, a} denoting the boson type, q = p1 −
k1 = k2 − p2 = p − k the momentum transfer, mb the
boson mass, ǫb a phase, and δ = 1 for isoscalar bosons
and δ = τ1 · τ2 = −1 − 2(−)
I for isovector bosons. All
boson form factors, f , have the simple form
f(Λb, q) =
+ |q2|
with Λb the boson form factor mass. The use of the
absolute value |q2| amounts to a covariant redefinition of
the propagators and form factors in the region q2 > 0. It
is a significant new theoretical improvement that removes
all singularities and can be justified by a detailed study
of the structure of the exchange diagrams. The axial
TABLE I: Mathematical forms of the bNN vertex func-
tions, with Θ(p) ≡ (m − /p)/2m. The vector propaga-
tor is ∆µν = gµν − qµqν/m
v with the boson momentum
q = p1 − k1 = k2 − p2.
JP (b) ǫb Λ1 ⊗ Λ2 Λ(p, k) or Λ
µ(p, k)
0+(s) − Λ1Λ2 gs − νs [Θ(p) + Θ(k)]
0−(p) + Λ1Λ2 gpγ
−gp(1− λp)
Θ(p)γ5 + γ5Θ(k)
1−(v) + Λ
Λν2∆µν gv
γµ + κv
iσµν(p− k)ν
+gvνv [Θ(p)γ
µ + γµΘ(k)]
1+(a) + Λ
Λν2gµν gaγ
vector bosons are treated as contact interactions, with
the structure as in (7) but with the propagator replaced
by a constant, m2a+ |q
2| → m2 with a nucleon mass scale.
The explicit forms of the numerator functions Λb1⊗Λ
2 can
be inferred from Table I. Note that λp = 0 corresponds
to pure pseudovector coupling, and that the definitions
of the off-shell coupling parameters λ or ν differ for each
boson.
In the most general case the kernel is the sum of the
exchange of pairs of pseudoscalar, scalar, vector, and ax-
ial vector bosons, with one isoscalar and one isovector
meson in each pair. If the external particles are all on-
shell, it can be shown that these 8 bosons give the most
general spin-isospin structure possible (because the vec-
tor mesons have both Dirac and Pauli couplings, the re-
quired 10 invariants can be expanded in terms of only 8
boson exchanges), explaining why bosons with more com-
plicated quantum numbers are not required. By allowing
boson masses (except the pion) to vary we let the data fix
the best mass for each boson in each exchange channel.
Finally, we break charge symmetry by treating charged
and neutral pions independently, and by adding a one-
photon exchange interaction, simplified by assuming the
neutron coupling is purely magnetic, iσµνqν , and that
all electromagnetic form factors have the dipole form. To
solve the CST NN equation numerically, it was expanded
in a basis of partial wave helicity states as described in
[20].
Previous models of the kernel, such as models IA, IB,
IIA, and IIB of [20] and the updated, ν-dependent ver-
sions such as W16 used in [7], had been obtained by fit-
ting the potential parameters to the Nijmegen or VPI
phase shifts. In a second step the χ2 to the observables
was determined. The models presented in this paper were
fit directly to the data, using a minimization program
that can constrain two of the low-energy parameters (the
deuteron binding energy, Ed = −2.2246 MeV, and the
1S0 scattering length, a0 = −23.749 fm, chosen to fit the
very precise cross sections at near zero lab energy). This
was a significant improvement, both because the best fit
to the 1993 phase shifts did not guarantee a best fit to the
2006 data base, and because the low-energy constraints
stabilized the fits. After the first fit was found, it would
TABLE II: Values of the 27 parameters for WJC-1 with 7
bosons and 2 axial vector contact interactions. All masses
and energies are in MeV; other couplings are dimensionless;
Gb = g
b/(4π). Parameters in bold were varied during the fit;
those labeled with an ∗ were constrained to equal the one above.
The deuteron D/S ratio is ηD, and the triton binding energy
is Et. Experimental values are in parentheses.
b I Gb mb λb or νb κv Λb
π0 1 14.608 134.9766 0.153 — 4400
π± 1 13.703 139.5702 −0.312 — 4400∗
η 0 10.684 604 0.622 — 4400∗
σ0 0 2.307 429 −6.500 — 1435
σ1 1 0.539 515 0.987 — 1435
ω 0 3.456 657 0.843 0.048 1376
ρ 1 0.327 787 −1.263 6.536 1376∗
h1 0 0.0026 — — — 1376
a1 1 −0.436 — — — 1376
ΛN = 1656; ηD = 0.0256(1) (0.0256(4)); Et = −8.48 (−8.48)
TABLE III: Values of the 15 parameters for WJC-2 with 7
bosons. See the caption to Table II for further explanation.
b I Gb mb λb or νb κv Λb
π0 1 14.038 134.9766 0.0 — 3661
π± 1 14.038∗ 139.5702 0.0 — 3661∗
η 0 4.386 547.51 0.0 — 3661∗
σ0 0 4.486 478 −1.550 — 3661
σ1 1 0.477 454 1.924 — 3661
ω 0 8.711 782.65 0.0 0.0 1591
ρ 1 0.626 775.50 −2.787 5.099 1591∗
ΛN = 1739; ηD = 0.0257(1) (0.0256(4)); Et = −8.50 (−8.48)
then be possible to vary the off-shell sigma coupling, νσ,
to give essentially a perfect fit to the triton binding en-
ergy. However, the binding energies we report here were
obtained from the best fit without any adjustment , con-
firming the results reported in Fig. 1 of Ref. [7].
TABLE IV: Comparison of precision np models and the 1993
Nijmegen phase shift analysis. Our calculations are in bold
face.
models χ2/Ndata
Reference #a yearb 1993 2000 2007
PWA93[12] 39c 1993 0.99 — —
1.11 1.12
Nijm I[14] 41c 1993 1.03c — —
AV18[16] 40c 1995 1.06 — —
CD-Bonn[15] 43c 2000 — 1.02 —
WJC-1 27 2007 1.03 1.05 1.06
WJC-2 15 2007 1.09 1.11 1.12
aNumber of parameters
bIncludes all data prior to this year.
cFor a fit to both pp and np data.
dOur fitting procedure uses the effective range expansion. The Ni-
jmegen 3S1 parameters were taken from Ref. [21], but as no
parameters are available we used those of WJC-1.
The parameters obtained in the fits are shown in Ta-
bles II and III. The χ2/Ndata resulting from the fits are
compared with results obtained from earlier fits in Table
IV. The data base used in the fits is derived from the
previous SAID and Nijmegen analyses with new data af-
ter 2000 added. The current data set includes a total of
3788 data, 3336 of which are prior to 2000 and 3010 prior
to 1993. For comparison, the PWA93 was fit to 2514,
AV18 to 2526, and CD-Bonn to 3058 np data. We re-
stored some data sets previously discarded because their
χ2 were no longer outside of statistically acceptable lim-
its, and this increased the χ2 slightly. Phase shifts and
a full discussion of the data and theory will be published
elsewhere.
In both of our models the high momentum cutoff is
provided by the nucleon form factor and not the me-
son form factors. Hence the very hard pion form fac-
tors merely reflect the fact that the nucleon form fac-
tors are sufficient to model the short range physics in the
pion exchange channel. The off-shell scalar couplings are
perhaps the most uncommon features of these models.
They are clearly essential for the accurate prediction of
three-body binding energies [7]. It is gratifying to see
that the pseudoscalar components of the pion couplings
(proportional to λp) remain close to zero, even when un-
constrained, and that effective masses of all the bosons
remain in the expected range of 400-800 MeV.
Aside from this, the parameters of WJC-2 are quite
close to values expected from older OBE models of nu-
clear forces. A possible exception is the pion coupling
constant, somewhat larger than the g2/(4π) = 13.567
found by the Nijmegen group. The high-precision model
WJC-1 shows some novel features: (a) gπ0 > gπ± , (b)
large gη, and (c) small gω.
Why do these OBE models work so well? We are re-
minded of the Dirac equation; it automatically includes
the p4/(8m3) energy correction that contributes to fine
structure, the Darwin term (including the Thomas pre-
cession), the spin-orbit interaction, and the anomalous
gyromagnetic ratio. Similarly, the CST automatically
generates relativistic structures hard to identify, and im-
possible to add to a nonrelativistic model without new
parameters.
We draw the following major conclusions from this
work: (1) The reproduction of the np data by the WJC-
1 kernel is essentially as accurate as any other np phase
shift analysis or any other model. This surprising result
is achieved with only 27 parameters, fewer than used by
previous high precision fits to np data. It remains to be
seen whether the results will be equally successful once
the pp data are included. (2) Model WJC-1 gives us
a new phase shift analysis, updated for all data until
2006, which is useful even if one does not work within
the CST. (3) The larger number of parameters of WJC-
1 is not necessary unless one wants very high precision;
model WJC-2 with only 15 parameters is also excellent
and comparable to previously published high precision
fits. (4) The OBE concept, at least in the context of
the CST where it can be comparatively easily extended
to the treatment of electromagnetic interactions [6] and
systems with A > 2, can be a very effective description
of the nuclear force.
Acknowledgements: This work is the conclusion
of an effort extending over more than a decade, sup-
ported initially by the DOE through grant No. DE-FG02-
97ER41032, and recently supported by Jefferson Science
Associates, LLC under U.S. DOE Contract No. DE-
AC05-06OR23177. A. S. was supported by FCT un-
der grant No. POCTI/ISFL/2/275. We also acknowledge
prior work by R. Machleidt and J.W. Van Orden, who
wrote some earlier versions of parts of the NN code. The
data analysis used parts of the SAID code supplied to us
by R. A. Arndt. Helpful conversations with the the Ni-
jmegen group (J. J. de Swart, M. C. M. Rentmeester, and
R.G.E. Timmermans) and with R. Schiavilla are grate-
fully acknowledged.
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Swart, Phys. Rev. C 67, 044001 (2003).
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Spring Seminar on Nuclear Physics, Maiori, Italy, May
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and J. J. de Swart, Phys. Rev. C 48, 792 (1993).
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http://arxiv.org/abs/nucl-th/0107057
http://arxiv.org/abs/nucl-th/9509032

  All realistic potential models for the two-nucleon interaction are to some
extent based on boson exchange. However, in order to achieve an essentially
perfect fit to the scattering data, characterized by a chi2/Ndata ~ 1, previous
potentials have abandoned a pure one boson-exchange mechanism (OBE). Using a
covariant theory, we have found a OBE potential that fits the 2006 world np
data below 350 MeV with a chi2/Ndata = 1.06 for 3788 data. Our potential has
fewer adjustable parameters than previous high-precision potentials, and also
reproduces the experimental triton binding energy without introducing
additional irreducible three-nucleon forces.

JLAB-THY-07-632
High-precision covariant one-boson-exchange potentials
for np scattering below 350 MeV
Franz Gross
Thomas Jefferson National Accelerator Facility, Newport News, VA 23606
College of William and Mary, Williamsburg, VA 23187
Alfred Stadler
Centro de F́ısica Nuclear da Universidade de Lisboa, 1649-003 Lisboa, Portugal
and Departamento de F́ısica da Universidade de Évora, 7000-671 Évora, Portugal
All realistic potential models for the two-nucleon interaction are to some extent based on boson
exchange. However, in order to achieve an essentially perfect fit to the scattering data, characterized
by a χ2/Ndata ∼ 1, previous potentials have abandoned a pure one boson-exchange mechanism
(OBE). Using a covariant theory, we have found a OBE potential that fits the 2006 world np data
below 350 MeV with a χ2/Ndata = 1.06 for 3788 data. Our potential has fewer adjustable parameters
than previous high-precision potentials, and also reproduces the experimental triton binding energy
without introducing additional irreducible three-nucleon forces.
A good understanding of the interaction between two
nucleons is essential for the study of nuclear structure and
nuclear reactions. In the long history of theoretical mod-
els of the NN interaction, One-Boson-Exchange (OBE)
models played a role of special importance. Yukawa’s [1]
insight that a short-range force can be generated through
the exchange of particles of finite mass led to the dis-
covery of the pion, and later it was found that the ex-
change of a pion can quantitatively describe the longer-
range part of the NN interaction. Since the range of the
force is inversely proportional to the exchanged mass, the
exchange of heavier mass bosons generates NN forces of
intermediate to short range. It was found that the vector
bosons ω and ρ contribute to the observed spin-orbit force
and strong repulsion at short internucleon distances [2],
and that scalar bosons provide intermediate attraction.
Today, with the development of potentials based on chi-
ral perturbation theory (ChPT) [3], we understand that
these scalar bosons are an approximate representation of
the two-pion exchange mechanism [4], which gives strong
attraction even if there were no two-pion resonances at
masses of around 500 MeV [5].
It is possible, of course, to construct phenomenolog-
ical NN potentials that, with a sufficently large num-
ber of parameters, give an accurate description of the
NN scattering data. However, OBE potentials have sev-
eral important advantages. First, they provide a phys-
ical mechanism for the interaction between nucleons.
This implies that the parameters in these models have
a physical meaning, and that, at least in principle, they
can be related to, or even be determined through other
physical processes. Second, it is possible to construct
consistent electroweak currents for systems interacting
through OBE, since the underlying microscopic processes
are known [6]. With phenomenological potentials this
construction is less straightforward because there is no
implied microscopic description of the flow of electroweak
charges through a nuclear system. Third, when OBE
is used in a covariant formalism without time ordering,
effective three- and many-body forces are automatically
generated from the off-shell couplings of purely two-body
OBE [7, 8]. With phenomenological potentials three-
body forces must be independently constructed. Finally,
OBE models are relatively simple, and depend only on a
moderate number of parameters. A quantitatively accu-
rate OBE model represents a very economical description
of the NN interaction.
OBE models also have their limitations. Since they
are not fundamental interactions, their validity does not
extend to very short distances where QCD should pro-
vide the correct description. In potential models, this
unknown short-distance part of the interaction is usually
parameterized phenomenologically through vertex form
factors with adjustable parameters. These form factors
also serve to regularize otherwise divergent loop integrals
that appear when the kernel is iterated. But parameters
that describe the unknown short distance physics cannot
be avoided; even more fundamental potential models de-
rived from ChPT require subtraction constants to renor-
malize and absorb infinities arising from the unknown
short range physics. At fourth order, a potential based
on ChPT will have at least 24 unknown subtraction con-
stants (parameters) [9].
After early phase shift analyses by the VPI group [10],
both the VPI [11] and Nijmegen [12] groups obtained op-
timal values of χ2/Ndata ≈ 1 after eliminating data sets
from their analyses, based on statistical arguments about
their incompatibility with other data sets [13]. The Ni-
jmegen group also updated their OBE potential (Nijm78)
to the new phase shift analysis, but they were unable to
get the χ2/Ndata of this 15 parameter model (now called
Nijm93) below 1.87 [14]. In order to construct very accu-
rate NN potentials they abandoned a pure OBE struc-
ture and made several boson parameters dependent on in-
http://arxiv.org/abs/0704.1229v2
dividual partial-waves. Similarly, the (almost) pure OBE
potentials of the Bonn family, such as Bonn A, B, and C,
were superseded by the realistic CD-Bonn, which also
incorporates partial-wave dependent boson parameters
[15]. The Argonne group also motivated their construc-
tion of largely phenomenological potentials like AV18 by
the apparent failure of the OBE mechanism (apart from
the pion-exchange tail) to allow a perfect fit to the data
[16].
The main objective of this letter is to show that within
the Covariant Spectator Theory (CST) it is, in fact, pos-
sible to derive realistic OBE potentials, and that these
require comparatively few parameters. This somewhat
surprising finding contradicts the earlier conclusion and
common belief that the OBE mechanism is missing some
important feature of the NN interaction. Accurate OBE
models may provide a useful intermediate step between
fundamental physics and experiment.
In CST [17, 18], the scattering amplitude M is the so-
lution of a covariant integral equation derived from field
theory (sometimes referred to as the “Gross equation”).
In common with many other equations, it has the form
M = V − V GM (1)
where V is the irreducible kernel (playing the role of a
potential) and G is the intermediate state propagator.
As with the Bethe-Salpeter (BS) equation [19], if the
kernel is exact and nucleon self energies are included in
the propagator, iteration of the CST equation generates
the full Feynman series. In cases where this series does
not converge (nearly always!) the equation solves the
problem nonperturbatively. With the BS equation the
four-momenta of all A intermediate particles are sub-
ject only to the conservation of total four-momentum
pi, so the integration is over 4(A − 1) vari-
ables. In the CST equation, all but one of the intermedi-
ate particles are restricted to their positive-energy mass
shell, constraining A−1 energies (they become functions
of the three-momenta) and leaving only 3(A−1) internal
variables, the same number of variables as in nonrelativis-
tic theory. Since the on-shell constraints are covariant,
the resulting equations remain manifestly covariant even
though all intermediate loop integrations reduce to three
dimensions, which greatly simplifies their numerical so-
lution and physical interpretation. This framework has
been applied successfully to many problems, in particular
also to the two- and three-nucleon system [7, 8, 20].
The specific form of the CST equation for the two-
nucleon scattering amplitude M , with particle 1 on-shell
in both the initial and final state, is [20]
M12(p, p
′;P ) = V 12(p, p
′;P )
(2π)3
V 12(p, k;P )G2(k, P )M12(k, p
′;P ) , (2)
where P is the conserved total four-momentum, and p, p′,
and k are relative four-momenta related to the momenta
of particles 1 and 2 by p1 =
P + p, p2 =
P − p, and
M12(p, p
′;P ) ≡ Mλλ′,ββ′(p, p
′;P )
= ūα(p, λ)Mαα′ ;ββ′(p, p
′;P )uα′(p
′, λ′) (3)
is the matrix element of the Feynman scattering ampli-
tude M between positive energy Dirac spinors of particle
1. The propagator for the off-shell particle 2 is
G2(k, P ) ≡ Gββ′ (k2) =
(m+ /k2)
m2 − k22 − iǫ
h4(k22) (4)
with k2 = P − k1, k
1 = m
2, and h the form factor of the
off-shell nucleon (related to its self energy), normalized
to unity when k22 = m
2. In this paper we use
h(p2) =
(Λ2N −m
−m2)2 + (m2 − p2)2
, (5)
where ΛN is an adjustable cutoff parameter. The indices
1 and 2 refer collectively to the two helicity or Dirac
indices of particle 1, either {λλ′} or {αα′}, and particle
2, {ββ′}.
The covariant kernel V is explicitly antisymmetrized.
In its Dirac form it is
V αα′;ββ′(p, k;P )
Vαα′;ββ′(p, k;P ) + (−)
IVβα′;αβ′(−p, k;P )
, (6)
where the isospin indices have been suppressed, so that
the factor of (−)I (with I=0 or 1 the isospin of the NN
state) insures that the remaining amplitude has the sym-
metry (−)I under particle interchange {p1, α} ↔ {p2, β}
as required by the generalized Pauli principle. This sym-
metry insures that identical results emerge if a different
particle is chosen to be on-shell in either the initial or
final state.
Next we assume that the kernel can be written as a
sum of OBE contributions
V b12(p, k;P ) = ǫbδ
Λb1(p1, k1)⊗ Λ
2(p2, k2)
+ |q2|
f(Λb, q) (7)
with b = {s, p, v, a} denoting the boson type, q = p1 −
k1 = k2 − p2 = p − k the momentum transfer, mb the
boson mass, ǫb a phase, and δ = 1 for isoscalar bosons
and δ = τ1 · τ2 = −1 − 2(−)
I for isovector bosons. All
boson form factors, f , have the simple form
f(Λb, q) =
+ |q2|
with Λb the boson form factor mass. The use of the
absolute value |q2| amounts to a covariant redefinition of
the propagators and form factors in the region q2 > 0. It
is a significant new theoretical improvement that removes
all singularities and can be justified by a detailed study
of the structure of the exchange diagrams. The axial
TABLE I: Mathematical forms of the bNN vertex func-
tions, with Θ(p) ≡ (m − /p)/2m. The vector propaga-
tor is ∆µν = gµν − qµqν/m
v with the boson momentum
q = p1 − k1 = k2 − p2.
JP (b) ǫb Λ1 ⊗ Λ2 Λ(p, k) or Λ
µ(p, k)
0+(s) − Λ1Λ2 gs − νs [Θ(p) + Θ(k)]
0−(p) + Λ1Λ2 gpγ
−gp(1− λp)
Θ(p)γ5 + γ5Θ(k)
1−(v) + Λ
Λν2∆µν gv
γµ + κv
iσµν(p− k)ν
+gvνv [Θ(p)γ
µ + γµΘ(k)]
1+(a) + Λ
Λν2gµν gaγ
vector bosons are treated as contact interactions, with
the structure as in (7) but with the propagator replaced
by a constant, m2a+ |q
2| → m2 with a nucleon mass scale.
The explicit forms of the numerator functions Λb1⊗Λ
2 can
be inferred from Table I. Note that λp = 0 corresponds
to pure pseudovector coupling, and that the definitions
of the off-shell coupling parameters λ or ν differ for each
boson.
In the most general case the kernel is the sum of the
exchange of pairs of pseudoscalar, scalar, vector, and ax-
ial vector bosons, with one isoscalar and one isovector
meson in each pair. If the external particles are all on-
shell, it can be shown that these 8 bosons give the most
general spin-isospin structure possible (because the vec-
tor mesons have both Dirac and Pauli couplings, the re-
quired 10 invariants can be expanded in terms of only 8
boson exchanges), explaining why bosons with more com-
plicated quantum numbers are not required. By allowing
boson masses (except the pion) to vary we let the data fix
the best mass for each boson in each exchange channel.
Finally, we break charge symmetry by treating charged
and neutral pions independently, and by adding a one-
photon exchange interaction, simplified by assuming the
neutron coupling is purely magnetic, iσµνqν , and that
all electromagnetic form factors have the dipole form. To
solve the CST NN equation numerically, it was expanded
in a basis of partial wave helicity states as described in
[20].
Previous models of the kernel, such as models IA, IB,
IIA, and IIB of [20] and the updated, ν-dependent ver-
sions such as W16 used in [7], had been obtained by fit-
ting the potential parameters to the Nijmegen or VPI
phase shifts. In a second step the χ2 to the observables
was determined. The models presented in this paper were
fit directly to the data, using a minimization program
that can constrain two of the low-energy parameters (the
deuteron binding energy, Ed = −2.2246 MeV, and the
1S0 scattering length, a0 = −23.749 fm, chosen to fit the
very precise cross sections at near zero lab energy). This
was a significant improvement, both because the best fit
to the 1993 phase shifts did not guarantee a best fit to the
2006 data base, and because the low-energy constraints
stabilized the fits. After the first fit was found, it would
TABLE II: Values of the 27 parameters for WJC-1 with 7
bosons and 2 axial vector contact interactions. All masses
and energies are in MeV; other couplings are dimensionless;
Gb = g
b/(4π). Parameters in bold were varied during the fit;
those labeled with an ∗ were constrained to equal the one above.
The deuteron D/S ratio is ηD, and the triton binding energy
is Et. Experimental values are in parentheses.
b I Gb mb λb or νb κv Λb
π0 1 14.608 134.9766 0.153 — 4400
π± 1 13.703 139.5702 −0.312 — 4400∗
η 0 10.684 604 0.622 — 4400∗
σ0 0 2.307 429 −6.500 — 1435
σ1 1 0.539 515 0.987 — 1435
ω 0 3.456 657 0.843 0.048 1376
ρ 1 0.327 787 −1.263 6.536 1376∗
h1 0 0.0026 — — — 1376
a1 1 −0.436 — — — 1376
ΛN = 1656; ηD = 0.0256(1) (0.0256(4)); Et = −8.48 (−8.48)
TABLE III: Values of the 15 parameters for WJC-2 with 7
bosons. See the caption to Table II for further explanation.
b I Gb mb λb or νb κv Λb
π0 1 14.038 134.9766 0.0 — 3661
π± 1 14.038∗ 139.5702 0.0 — 3661∗
η 0 4.386 547.51 0.0 — 3661∗
σ0 0 4.486 478 −1.550 — 3661
σ1 1 0.477 454 1.924 — 3661
ω 0 8.711 782.65 0.0 0.0 1591
ρ 1 0.626 775.50 −2.787 5.099 1591∗
ΛN = 1739; ηD = 0.0257(1) (0.0256(4)); Et = −8.50 (−8.48)
then be possible to vary the off-shell sigma coupling, νσ,
to give essentially a perfect fit to the triton binding en-
ergy. However, the binding energies we report here were
obtained from the best fit without any adjustment , con-
firming the results reported in Fig. 1 of Ref. [7].
TABLE IV: Comparison of precision np models and the 1993
Nijmegen phase shift analysis. Our calculations are in bold
face.
models χ2/Ndata
Reference #a yearb 1993 2000 2007
PWA93[12] 39c 1993 0.99 — —
1.11 1.12
Nijm I[14] 41c 1993 1.03c — —
AV18[16] 40c 1995 1.06 — —
CD-Bonn[15] 43c 2000 — 1.02 —
WJC-1 27 2007 1.03 1.05 1.06
WJC-2 15 2007 1.09 1.11 1.12
aNumber of parameters
bIncludes all data prior to this year.
cFor a fit to both pp and np data.
dOur fitting procedure uses the effective range expansion. The Ni-
jmegen 3S1 parameters were taken from Ref. [21], but as no
parameters are available we used those of WJC-1.
The parameters obtained in the fits are shown in Ta-
bles II and III. The χ2/Ndata resulting from the fits are
compared with results obtained from earlier fits in Table
IV. The data base used in the fits is derived from the
previous SAID and Nijmegen analyses with new data af-
ter 2000 added. The current data set includes a total of
3788 data, 3336 of which are prior to 2000 and 3010 prior
to 1993. For comparison, the PWA93 was fit to 2514,
AV18 to 2526, and CD-Bonn to 3058 np data. We re-
stored some data sets previously discarded because their
χ2 were no longer outside of statistically acceptable lim-
its, and this increased the χ2 slightly. Phase shifts and
a full discussion of the data and theory will be published
elsewhere.
In both of our models the high momentum cutoff is
provided by the nucleon form factor and not the me-
son form factors. Hence the very hard pion form fac-
tors merely reflect the fact that the nucleon form fac-
tors are sufficient to model the short range physics in the
pion exchange channel. The off-shell scalar couplings are
perhaps the most uncommon features of these models.
They are clearly essential for the accurate prediction of
three-body binding energies [7]. It is gratifying to see
that the pseudoscalar components of the pion couplings
(proportional to λp) remain close to zero, even when un-
constrained, and that effective masses of all the bosons
remain in the expected range of 400-800 MeV.
Aside from this, the parameters of WJC-2 are quite
close to values expected from older OBE models of nu-
clear forces. A possible exception is the pion coupling
constant, somewhat larger than the g2/(4π) = 13.567
found by the Nijmegen group. The high-precision model
WJC-1 shows some novel features: (a) gπ0 > gπ± , (b)
large gη, and (c) small gω.
Why do these OBE models work so well? We are re-
minded of the Dirac equation; it automatically includes
the p4/(8m3) energy correction that contributes to fine
structure, the Darwin term (including the Thomas pre-
cession), the spin-orbit interaction, and the anomalous
gyromagnetic ratio. Similarly, the CST automatically
generates relativistic structures hard to identify, and im-
possible to add to a nonrelativistic model without new
parameters.
We draw the following major conclusions from this
work: (1) The reproduction of the np data by the WJC-
1 kernel is essentially as accurate as any other np phase
shift analysis or any other model. This surprising result
is achieved with only 27 parameters, fewer than used by
previous high precision fits to np data. It remains to be
seen whether the results will be equally successful once
the pp data are included. (2) Model WJC-1 gives us
a new phase shift analysis, updated for all data until
2006, which is useful even if one does not work within
the CST. (3) The larger number of parameters of WJC-
1 is not necessary unless one wants very high precision;
model WJC-2 with only 15 parameters is also excellent
and comparable to previously published high precision
fits. (4) The OBE concept, at least in the context of
the CST where it can be comparatively easily extended
to the treatment of electromagnetic interactions [6] and
systems with A > 2, can be a very effective description
of the nuclear force.
Acknowledgements: This work is the conclusion
of an effort extending over more than a decade, sup-
ported initially by the DOE through grant No. DE-FG02-
97ER41032, and recently supported by Jefferson Science
Associates, LLC under U.S. DOE Contract No. DE-
AC05-06OR23177. A. S. was supported by FCT un-
der grant No. POCTI/ISFL/2/275. We also acknowledge
prior work by R. Machleidt and J.W. Van Orden, who
wrote some earlier versions of parts of the NN code. The
data analysis used parts of the SAID code supplied to us
by R. A. Arndt. Helpful conversations with the the Ni-
jmegen group (J. J. de Swart, M. C. M. Rentmeester, and
R.G.E. Timmermans) and with R. Schiavilla are grate-
fully acknowledged.
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