A Possibility of a Long Range Three-Body Force in the Hadron System
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Shinsho Oryu
A Possibility of a Long Range Three-Body Force in the Hadron System
Received: 28 July 2020 / Accepted: 1 September 2020 © The Author(s) 2020
Abstract The existence of a kinematic long range component in the one particle transfer three-body potential or so-called “general particle transfer (GPT) potential” was proposed several years ago. In this investigation the mass dependence of the exchanged particle and the index number of the long range property are clarified. On the basis of the GPT potential, a new long range three-body force is proposed in the hadron system, which will be compared with the Efimov potential.
1 Introduction It has been found that a kinematic property in the three-body system generates a long range effect in a very low energy region. The effect is called a general particle transfer (GPT) potential, which becomes the Yukawatype potential for the shorter range but a 1/r n potential for the longer range [1–6]. We will review the GPT formalism in Sect. 2. It was found that the similar property can occur not only at the three-body threshold (3BT) but also at the quasi two-body threshold (Q2T) with two-body bound states in the three-body system. However, the GPT potential has two parameters: a damping range (a) and an index number (n) which shows the degree of the long range. We would like to investigate a relation between the two parameters by utilizing the NNπ system, where the familiar one pion exchange potential (OPEP) will be the example. Such a long range potential was first proposed by Efimov [7,8], which was not given by a single two-body potential but a three-body potential where a nonlinear form or an entangled two-body long range potential was proposed. We would like to generalize the three-body long range Efimov potential by using the GPT potential in Sect. 3. The conclusion and discussion will be given in Sect. 4.
2 The Quasi Two-Body Systems in the AGS Equation 2.1 Fourier Transformation of the One Particle Exchange Potential For the three-body free energy E and the two-body binding energy z = − B ≤ 0, we take into account E cm ≡ E + B < 0. The off-shell part of the two-body subsystem is written in terms of p j = (m i qk −m k qi )/(m k +m i ), and q j which is the relative momentum between the center of mass of the two-body subsystem and the spectator particle j with the reduced mass μ j = m j (m k + m i )/(m i + m j + m k ). The one particle transfer potential, or the AGS [9] Born term in the three-body Faddeev formalism [10,11], is represented by S. Oryu (B) Department of Physics, Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba 278-8510, Japan E-mail: [email protected]
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S. Oryu
Z in, jm (qi , q j ; −|E cm |) gin (pi )g jm (p j )(1 − δi, j ) = −|E cm | − q 2j /2μ j = −2μ j =−
(1)
gin (pi )g jm (p j )(1 − δi, j ) σ 2 + q 2j
Cin, jm (qi , q j ) σ 2 + q 2j
,
(2)
where σ and the numerator are defined by σ ≡ 2μ j |E cm |, Cin, jm (qi , q j ) ≡ 2μ j gin (pi )g jm (p j )(1 − δi, j ).
(3) (
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