1. Possibility of the Efimov-like structure in the hadron system
Shinsho Oryu
Department of Physics, Faculty of Science and Technology,
Tokyo University of Science,
2641 Yamazaki, Noda City, Chiba Japan
The Seventh Asia-Pacific Conference on Few-Body Problems in Physics
(APFB 2017) is to be held on Aug. 25 (Fri.) – 30 (Wed.), 2017 in Guilin

2. Introduction A lot of evidences for the Efimov states
were found in the cold atomic systems.
However, there are few discussions about the Efimov physics in hadron systems.

3. The most important requirements of Efimov physics is (1) the scattering length of the sub-system should be

4. (2) The result is that the three-body binding energies
The most important requirements of Efimov physics is
(1) the scattering length of the sub-system should be
(2) The result is that the three-body binding energies
condense on the three-body break-up threshold
(3BT) where the energy level structure is given by

5. The most important requirements of Efimov physics is
(1) the scattering length of the sub-system should be
(2) The result is that the three-body binding energies
condense on the three-body break-up threshold
where the energy level structure is given by
(3) It is known that such an energy level structure
can be obtained by

6. We would like to check the three criteria in the hadron
systems.
The first criterion is not satisfied in the hadron system:

7. 2) The second criterion:
We would like to check this three criteria in the hadron systems.
The first criterion is not satisfied in the hadron system:
2) The second criterion:
there are some instances that energy levels come
near the threshold region.
However, it is very hard to confirm whether they
are Efimov levels or not.

8. 3) The third criterion is that the nuclear potential
We would like to check this three criteria in the hadron systems.
The first criterion is not satisfied in the hadron system:
2) The second criterion is that there are some
instances that energy levels come near the threshold
region. However, it is very hard to confirm whether
they are Efimov levels or not.
3) The third criterion is that the nuclear potential
is usually a short-range where the longest-range is
one pion exchange Yukawa potential.
Therefore, it seems that such a potential does not
support Efimov physics in the hadron system.

9. However, we can reevaluate the Efimov physics from another vantage point.
First of all we would like to pay attention to the thresholds in the three-body reactions.
The 3BT appears in the reactions:

10. 2) Another threshold from the three-body bound
However, we can reevaluate the Efimov physics from another vantage point.
1) First of all we would like to pay attention to the thresholds in the three-body reactions.
The three-body break-up thresholds (3BT) appear in the reactions:
2) Another threshold from the three-body bound
state to the quasi two-body system which is
the quasi two-body threshold (Q2T) such as

11. At the 3BT,
the Born term Z of the Faddeev or the Alt-Grassberger-Sandhas (AGS) equation, and
the propagator have singularities;
The coincidence of both singularities causes a
serious problem.

12. This situation is very similar to the Coulomb
1) At the three-body threshold (3BT),
the Born term Z of the Faddeev or the Alt-Grassberger-Sandhas (AGS) equation and
the propagator have singularities
The coincidence of both singularities causes a serious problem.
This situation is very similar to the Coulomb
scattering on the Lippmann-Schwinger equation
where the Coulomb potential singularity
coincides with the Green's function singularity.

13. 2) For the quasi two-body equation (Q2E),
the multi-channel two-body Lippmann-Schwinger (MLS) equation should be
introduced from the AGS equation:
a+(bc) b+(ca)
At the quasi two-body threshold (Q2T),
the Green’s function of MLS equation
should diverge !

14. Q2T
3BT

24. The overlapping singularity p-e system, we have many energy levels in the Hydrogen atom by
For the three-body overlapping singularity at the 3BT, 　Efimov discovered a new potential which causes a lot of energy levels in the negative energy region.
Recently, for the overlapping singularity at the quasi-two-body threshold we found a local potential which
we call a general particle transfer (GPT) potential:

25. It is easily seen that the GPT potential includes

30. Conclusion
Contrary to the first Efimov criterion, the Q2T does not require:

31. Conclusion
Contrary to the first Efimov criterion, the Q2T
does not require:
The parameter depends on the mass-ratio between a transfer- particle and other masses. It is because corresponds to the mass-less particle.

32. Conclusion
Contrary to the first Efimov criterion, the Q2T
does not require
The parameter depends on the mass-ratio between a transfer particle and others.
It is because corresponds to the mass-less particle.
In this stage, whether the Efimov-like energy levels could appear or not is one side,
but another side is the existence of the long range interactions which we discover.

33. Besides the Yukawa-type short range
Conclusion
Contrary to the first Efimov criterion, the Q2T
does not require
The parameter depends on mass ratio between a transfer particle and others. It is because
corresponds to the mass-less particle.
In this stage, whether the Efimov-like energy levels could appear or not is one side,
but another side is the existence of the long range interactions which we discover.
Besides the Yukawa-type short range
potential, these long range interactions could
give rise to unusual phenomena in the hadron
systems.

34. If such a long range potential does not
produce additional bound states,
still our attractive long range part of GPT
potential could interfere with the Coulomb
potential and the centrifugal potential,
and affect the final-state interactions in the
break-up reaction.

35. 5) This kind of additional long range potential
4) If such a long range potential does not produce
additional bound states,
still our attractive long range part of GPT
potential could interfere with the Coulomb
potential and the centrifugal potential,
and affect the final-state interactions in the
break-up reaction.
5) This kind of additional long range potential
could universally occur, not only in three-
body systems, but also in many-body systems.

36. 6) Then the most popular benefit of the potential
4) If such a long range potential does not produce
additional bound states,
still our attractive long range part of GPT
potential could interfere with the Coulomb
potential and the centrifugal potential,
and affect the final-state interactions in the
break-up reaction.
5) This kind of additional long range potential
could universally occur, not only in three-
body systems, but also in many-body systems.
6) Then the most popular benefit of the potential
will be seen in the neutron-rich nucleus such
as a halo, or unstable nuclei, or the nuclear
fusion problems.

37. 7) The GPT potential predicts not only the Efimov-like systems, but also some other long range potential systems.

38. 7) The GPT potential predicts not only the Efimov-like systems, but also some other long range potential systems.
8) Some question arise:
a) Could not the Faddeev equation reduce
to the MLS equation?
b) Or, where the quasi two-body threshold
exists in the Faddeev equation?
c) Is a single pole or branch
point of Q2-cut?
d) Or, whether the applicability of the Faddeev
equation could be restricted or not?.

39. Thank you very much for your attention !

41. If qα ,qβ　are outer lines, intermediate energy is not E
but Ecm.

42. Part 1
Bound State Problem
by E2Q

43. E2Q Potential

45. The two-body potential reduction has the energy dependence.