A simplified 1D (one-dimensional) model for a generalized 3N system is
considered, as a testing ground for the explicit treatment of the meson
dynamics in this system. We focus attention on the irreducible diagrams
generated by the pion dynamics in the 3N system, and in particular to a new
type of three-nucleon force discussed recently in the literature, and generated
by the one-pion-exchange mechanism in presence of a nucleon-nucleon
correlation. It is found that these new terms in the simplified model have an
approximately 30% effect compared to the standard three-nucleon force terms in
a `Triton' binding energy calculation. It is suggested that this effect should
also not be ignored in realistic calculations.Comment: 4 pages, Contribution to "Mesons and Light Nuclei 2001", Prague 2-6,
  200

Canton, L.

Melde, T.

Svenne, J. P.

English

arXiv

L. Canton

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Meson dynamics and the resulting “3-nucleon-force” diagrams: Results from a simplified test case

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Meson Dynamics and the resulting
“3-Nucleon-Force” diagrams:
Results from a simplified test case
L. Canton∗, T. Melde† and J.P. Svenne∗∗
∗INFN and University of Padova, Padova, Italy, I-35131
†Institute for Theoretical Physics, University of Graz, 8010 Graz, Austria
∗∗WITP and University of Manitoba, Winnipeg, MB, Canada R3T 2N2
Abstract. A simplified 1D (one-dimensional) model for a generalized 3N system is considered, as
a testing ground for the explicit treatment of the meson dynamics in this system. We focus attention
on the irreducible diagrams generated by the pion dynamics in the 3N system, and in particular to a
new type of three-nucleon force discussed recently in the lierature, and generated by the one-pion-
exchange mechanism in presence of a nucleon-nucleon correlation. It is found that these new terms
in the simplified model have an approximately 30% effect compared to the standard three-nucleon
force terms in a ‘Triton’ binding energy calculation. It is suggested that this effect should also not
be ignored in realistic calculations.
INTRODUCTION
An important aspect in modern three-nucleon dynamics is theexplicit inclusion of
mesonic degrees of freedom that can not be described by conventional two-nucleon po-
tentials. These additional effects can be described by three-nucleon forces (3NF), which
describe the addition of possible irreducible mesonic contributions. The construction
and implementation of these terms is not trivial and care hasbeen taken to avoid double-
contributions. The standard 3NF approaches turned out to beable to account for the
underbinding of the three-nucleon bound state. However, sofar they could not fully ex-
plain the puzzle of the vector analyzing powers in nucleon-deuteron scattering at low
energies.
Recently, starting from the rigorous four-body theory of Grassberger-Sandhas [1] and
Yakubovsky [2], a new method to describe the coupledNNN−πNNN system has been
developed [3]. In a subsequent paper, an approximation scheme for this new method was
derived by the authors [4]. It is based on reasonable physical and mathematical approx-
imations and a procedure to freeze-out the pion channel.
After the approximations were performed and the pionic channel was projected out, it
was shown that the residual pion-dynamics produced correction terms to the standard
Faddeev-Alt-Grassberger-Sandhas 3N equation. These correction terms can be inter-
preted as three-nucleon force diagrams (3NF). The approachdeveloped has the distinc-
tive feature that these irreducible diagrams naturally appe r in the Faddeev equation, and
the corresponding 3NF is generated consistently with the 3Ndynamical equation used
for the actual calculations. To our knowledge, this level ofc nsistency has here been
obtained for the first time.
A simplified one-dimensional model was developed, which describes in first approx-
imation the dynamics of the three-nucleon system [5]. It is ba ed on a spinless one-
dimensional scattering theory with a potential which is the1D analogue of the standard
Malfliet-Tjon potential. The simplicity of the model allowed a straightforward investi-
gation of the effects due to the residual pion dynamics.
THE SIMPLIFIED TEST CASE
An approximation scheme for reducing theπ−NNN system to a tractable set of equa-
tions has been recently developed [4]. At the lowest order, th residual effects of the
pion dynamics result in irreducible corrections to the driving term of the Lovelace type
3N equation. The driving term now reads
Zab = Z
AGS
ab +Z
π
ab (1)
wherea,b are the standard Faddeev labels. The first term has the usual structure of an
AGS-type driving term, namely it is vanishing fora= b, while for a 6= b it is
ZAGS12 = 〈N1(N2N3)|g0 |N2(N1N3)〉 (2)
and the other term corresponds to the correction that comes from the residual pion dy-
namics. Another type of correction term results from projecting out the pion channel. It
describes the intermediate propagation of a pion under the presence of a correlated three
nucleon cluster. In this study such an additional correction is not taken into account.
To the lowest order, the correction termZπab contains two topologically different contri-
butions, fora 6= b,
Zπ12 = 〈N1(N2N3π) ;N1N2(N3π)|τ(N3π) |N2(N3N1π) ;N1N2(N3π)〉 (3)
and, fora= b,
Zπ11 = 〈N1(N2N3π) ;N1(N2N3)π|τ(N2N3) |(N1π)(N2N3) ;N1N3(N1π)〉 (4)
+〈(N1π)(N2N3) ;N1(N2N3)π|τ(N2N3) |N1(N2N3π) ;N1N3(N1π)〉
The static approximation of the first term corresponds to a corre tion term usually
attributed to a three-nucleon force diagram of the Fujita-Miyazawa type [6]. The second
term corresponds to a topologically different type of correction term that describes the
propagation of an intermediate pion under the presence of a crrelated two nucleon
cluster. Furthermore, it should be noted that the first corretion term has contributions to
the off-diagonal elements of the driving term only. On the other hand, the second type
has contributions to the diagonal elements of the driving term only, which in the standard
AGS driving term are always zero.
A simple test model was designed from a one-dimensional scattering theory describing
symmetric spinless particles. The strength parameters of the Malfliet-Tjon type potential
were chosen in a way to reproduce the deuteron binding energy. The potential also shows
a similar spatial behaviour as observed in more realistic nuclear potentials, which makes
it a good candidate for our test-calculation. The t-matrices are described by the Unitary
Pole Approximation (UPA), which is expected to be a good approximation due to the
simplicity of the toy-model. The ‘deuteron’ binding energies are found using a sturmian
procedure. Subsequently we calculated the ‘triton’ binding e ergy of this system without
the correction terms and found a value of−7.28MeV.
In the calculation the first type of correction term was included in a static approximation
which allowed the interpretation of the dynamical terms as 3NF. They were described
by the one-dimensional equivalent of the Tucson-Melbourneor Brasil type contact
term [7, 8]. The absence of spin in our model reduces all otherterms in the Tucson-
Melbourne/Brasil 3NF to identically zero. The contact term of the TM/B-3NF depends
on a freely variable parameter and theπN scattering length. The coefficienta1 in the
contact term, which depends on theπN scattering length is defined in our simplified
model by the expression
a1 =−
2πµ
h̄2
tπN
(
p= 0, p′ = 0,E = 0
)
(5)
where the thresholdπN t-matrix is approximated by the correspondingNN t-matrix
times an adjustable parametercπN. This parametercπN can be choosen freely over a
certain range to recover the triton binding energy. However, we also know that the
corresponding threshold scattering lengths differ by a factor of 0.01 and we choose
the parametercπN = 0.01 as a first guess. With this value we find a ‘triton’ binding
energy of−8.17MeV. We then varied the parametercπN and found that the correction
term depends strongly on this adjustable parameter. It is observed that it is in principle
possible to adjust the parametercπN in order to recover the experimental ‘triton’ binding
energy.
When calculating the effect of the second type of correctionterms a delicate cancellation
has to be observed in order to avoid overcounting. This cancell tion is described by
Canton and Schadow [9] and is of the typet − v. Namely, we have to subtract the
potential from the full t-matrix and it turns out that this subtraction does not result in
a complete cancellation at low energies. For more details onthe cancellation we refer to
the original publications [9, 10]. We then calculated the ‘triton’ energy including both
types of correction terms, where for the adjustable parameter cπN a value of 0.01 was
chosen, thereby obtaining−8.48MeV for the model ‘triton’ binding energy. It should
be noted that the effect of the ‘new’ correction terms in respect to the Fujita-Miyazawa
type terms is approximately 30%. If we use the parametercπN that was already adjusted
to recover the ‘triton’ binding energy, then the new type of crrection term leads to an
over binding. It is therefore important to fit the adjustableparametercπN with both types
of correction terms present.
CONCLUSIONS
We developed a model that included one pion degree of freedomexplicitly in an AGS
type three body equation. It was shown that the effect of the pion dynamics resulted
in correction terms in the driving terms of the two-cluster equations. We calculated the
explicit effects of these terms, which could be interpretedas 3NF terms, on the ‘triton’
binding energy. In a simplified model calculation we show that e correction terms can
account for an underbinding of the ‘three-nucleon’ bound state. However, a new type of
correction term also has an approximately 30% effect on the ‘triton’ binding energy. A
third topological type of correction term still needs to be investigated, but it is expected
that the effect of this term is not as important.
The calculations we presented were done only with a rather simple model. However,
one might expect that the new type of correction terms could have an effect also in
more realistic calculations. This suggestion has been recently confirmed by Canton and
Schadow in a more realistic study [9, 10] on thend vector analyzing powers, where
it was shown that a new 3NF term of tensor structure can be generated by these new
correction terms, and could indeed hold the key to solve theAy-puzzle. In conclusion,
a new way of handling the pion dynamics in 3N calculations has been developed.
Calculations suggest that a newly developed type of 3NF should also be included in
realistic three-nucleon calculations.
ACKNOWLEDGMENTS
L.C. thanks for support from the Italian MURST-PRIN Project“Fisica Teorica del Nu-
cleo e dei Sistemi a Piu Corpi”. T.M acknowledges support from the University of Man-
itoba and the conference organizing committee for waiving the conference fee. J.P.S.
acknowledges support from the Natural Science and Engineering Research Council of
Canada. The authors also would like to thank the Institutions of INFN (Padova), Univer-
sity of Manitoba, and Universita di Padova for hospitality during reciprocal visits.
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3. L. Canton, Phys. Rev. C58, 3121 (1998)
4. L. Canton, T. Melde and J. P. Svenne, Phys. Rev. C63, 034004 (2001)
5. T. Melde, PhD Thesis, University of Manitoba (2001)
6. J. Fujita and H. Miyazawa, Prog. Theor. Phys.17, 360 (1957)
7. S. A. Coon, M. D. Scadron, P. C. McNamee, B. R. Barrett, D. W.E. Blatt and B. H. J. Kellar, Nucl.
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Meson Dynamics and the resulting "3-Nucleon-Force" diagrams: Results from a simplified test case

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