Transcript The Hypernucleus - IFT

5
Hypernucleus He
In A Two Frequency Model

Yiharn Tzeng, S.Y.Tsay Tzeng,
T.T.S.Kuo
5
B.E of He - A challenging problem

Prescriptions
• the use of an earlier Λ N central force,
• the introduction of a repulsive Λ NN force
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the consideration of quark effects,
the use of a modified Nijmegen NSC89 potential in
the first order Bruckner-Hartree calculations,
• the destruction of He-4 core structure,
• the recent coherent
Λ -Σ coupling .
• Despite of all these efforts, it seems the end of the
case still distance away
Shell Model calculations
•
Validity of hypernucleus' Shell structure has been firmly
established in experiments. Hence shell model has been the
framework for exploring the structure of hypernuclei.
•
Most shell model calculations assume same frequencies for
nucleon's and hyperon's harmonic oscillator basis wave
functions, i.e.,   =   e.g., 21 MeV for 5 He , 11

N
Y
16
MeV for O
.

Our Treatment
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In this work, we set  N >  Y
Our justification:
Since the rms radius 1 / m
and
mY  mN
If the rms radii of N and Y about same
Or, if the hyperon is less bounded than
the nucleon in a hypernucleus as the
usual case, i.e., the rms radius for Y is
greater than that for N
Effective Hamiltonian
The full-space YN many-body problem H  E
very difficult to solve.
Choose a small model space, P, then the full-space
problem is formally reduced
to a model-space, or P-space problem, namely
H eff Pm  ( H 0  Veff ) Pm  Em m
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where the eigenvalues Em form a subset of the
eigenvalues of the original full-space equation.
• In the above, H eff is the effective Hamiltonian
which contains two parts, the unperturbed
Hamiltonian H 0
and the effective interaction Veff
• H 0  (mN  t N  uN )  (mY  tY  uY ) is defined with
the auxiliary single particle (sp) potentials,
• single particle w.f. and energy defined as
H 0n   nn
•
H  H 0  H int
H int  V  uN  uY
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;
• A central problem in this approach is of course
how to derive H eff , or how to derive the modelspace effective interaction Veff
• The accuracy of the YN G-matrix from the
realistic YN potentials is clearly crucial to our
investigation.
• A non-trivial problem is to treat the Pauli
• operators in the G-matrix equation.
• For instance, the N G-matrix may be
defined by the  coupled integral equation
G-Matrix Element
• Define
G  V  VQ
1
QG
  Q (mN  t N  mY t Y )Q
QYN
QH (YN ) 
t  (mN  tN  mY  tY )
  1 N 2 | G (t ) |  3 N 4   1 N 2 | VYN |  3 N 4 
   1 N 2 | VYN |  ' N '   ' N ' | QH N | " N "  " N " | G |  3 N 4 
   1 N 2 | VYN | ' N '  ' N ' | QH N | " N "  " N " | G |  3 N 4 
G-matrix
• Define G-matrix as
1
G  V  VQ
QG
  Q (mN  t N  mY t Y )Q
• In calculating the G-matrix elements, frequencies for a
hyperon's and a nucleon's basis wave functions are set
differently.
• There we expanded wavefunctions with Y in terms
of those with N so that the difficulty of
transformations between the CM-relative coordinates
and the two particle states with different oscillator length
can be removed.
• The frequency for a nucleon is taken from the empirical
formula N = 45 A1/ 3  25 A2 / 3 , and that for a
hyperon  is kept as a parameter.
Y
Folded-diagram series
• Using these G-matrix elements, we will be able
to calculate various diagrams to be included in
the  -box.
Q
• An energy
independent Veff can be obtained
from the Q -box
• folded-diagram series being summed up to all
orders
• using the Lee-Suzuki iteration method to find
Veff
Lambda’s binding energy
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Λ’s single-particel energy is then calculated as a
function of Y
from the Q box up to the

diagrams of second-order in G.
• The lambda's single particle energies are plotted as
functions of Y calculated via various YN
potentials. Note that each curve presents a
saturation minimum at a certain Y . The
positions of these saturation points vary with
potentials. For examples, the minimum appears
around Y = 8 MeV for the curve from NSC89, at
9.2 MeV for that from NSC97f, and at 10 MeV for
NSC97d.
• Although this is not a variation calculation,
these saturation minima assure the results
being stable with respect to a small
variation of Y . Hence we take these
minima as the lambda's binding energies.
Results
• the binding energies calculated from all
NSC97 potentials are between 3.8 MeV
and 4.6 MeV, all too large compared to the
experimental value of 3.12$\pm$ 0.02 MeV,
• while the one from NSC89 is only about
1.7 MeV, much smaller than the
experimental value.
• Result consistent with \cite{halderson} where the
binding energy calculated from several
potentials modified from the NSC89 one and
obtained the results ranging from 1.61 to 2.23
MeV. Haldeson \cite{halderson} pointed out the
underbinding mainly from that the N  N
tensor force still remains too strong in both the
original NSC89 and his modified potentials.
• Akaishi and collaborators \cite{akaishi} proposed
to split N  N
coupling into
"coherent" and "incoherent" parts and suggested
the problem could be solved by suppressing the
"incoherent" part but keeping the "coherent"
part. They obtained $B_{\Lambda}$ =2.38 MeV
and 3.57 MeV respectively from NSC97f(S) and
NSC97e(S), which are simplified potentials from
NSC97f and NSC97e by including both central
and tensor parts in N
and N channels.
• Comparison of our calculations by using
the original NSC97 full potentials with
theirs may provide some clues for further
studies on this problem.
Conclusion
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The binding energy $B_{\Lambda}$ is obtained from the
saturation minimum of the single particle energy versus
$ 
curve when keeping N constant.
Y


•
N
We obtain underbinding from NSC89 potential but have
overbinging from all NSC97 ones.
• The underbinding may come from too strong tensor force
in
coupling, as pointed in \cite{halserson}.
N  N
• On the overbinding side, results from different
NSC97 modes appear to have different binding
strengths, with the one from NSC97d the
strongest, and the one from the NSC97f the
least strong but the closest to the desired
experimental value. We also found that
decreasing the frequency N would be able
to make $B_{\Lambda}$ even closer to the
experimental one. This may suggest that we
need a little bit large He-4 core.
• However, this statement needs to be justified by
both accurate YN realistic interaction and
5
measured size of
though it may not
 He
be easy experimentally).
• Nevertheless, our work provides an alternative
line of thinking for solving this problem.
Improvements
• The fact of our energy levels' not fitting the
experimental data exactly may be explained as
that the $\Lambda$ particle-neutron hole
formalism may not be perfectly applied to such a
light hyprnucleus. The few -body effects very
likely have to be taken into accounts here.
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• Since our results are obtained directly from the
realistic Nijmegen potentials, one very possible
reason for the deviation from experimental data
is because of the potentials themselves.