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 278-8510 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

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.

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 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

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

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

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.

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.

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:

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

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.

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.

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 !

Q2T 3BT

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:

It is easily seen that the GPT potential includes

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

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.

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.

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.

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 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 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.

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

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?.

Thank you very much for your attention !

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

Part 1 Bound State Problem by E2Q

E2Q Potential

The two-body potential reduction has the energy dependence.