Reido Kobayashi

Generalization of the Calogero–Cohn bound on the number of bound states

It is shown that for the Calogero–Cohn‐type upper bounds on the number of bound states of a negative spherically symmetric potential V(r), in each angular momentum state, that is, bounds containing only the integral ∫∞0‖V(r)‖1/2 dr, the condition V′(r)≥0 is not necessary, and can be replaced by the less stringent condition (d/dr)[r1−2p(−V)1−p]≤0, 1/2≤p<1, which allows oscillations in the potential. The constants in the bounds are accordingly modified, depend on p and l, and tend to the standard value for p=1/2.

The absolute definition of the phase-shift in potential scattering

The variable phase approach to potential scattering with regular spherically symmetric potentials satisfying Eq. (1), and studied by Calogero in his book [Variable Phase Approach to Potential Scattering (Acadamic, New York, 1967)] is revisited, and we show directly that it gives the absolute definition of the phase-shifts, i.e., the one which defines δl(k) as a continuous function of k for all k⩾0, up to infinity, where δl(∞)=0 is automatically satisfied. This removes the usual ambiguity ±nπ, n integer, attached to the definition of the phase-shifts through the partial wave scattering amplitudes obtained from the Lippmann–Schwinger integral equation, or via the phase of the Jost functions. It is then shown rigorously, and also on several examples, that this definition of the phase-shifts is very general, and applies as well to all potentials which have a strong repulsive singularity at the origin, for instance those which behave like gr−m, g>0, m⩾2, etc. We also give an example of application to the low-e...

Generalization of the Birman–Schwinger method for the number of bound states

We generalize the Birman–Schwinger method, and derive a general upper bound on the number of bound states in the S wave for a spherically symmetric potential. This general bound includes, of course, the Bargmann bound, but also leads, for increasing (negative) potentials, to a Calogero–Cohn-type bound. Finally, we show that for a large class among these potentials, one can obtain further improvements.

A sufficient condition for the existence of bound states in a potential

For a wide class of purely attractive potentials, we obtain a new sufficient condition for the existence of bound states for any angular momentum. Applied to some exactly soluble cases, the condition gives good results as compared to exact results.

Gauge interaction of baryons in hidden local symmetry

SummaryThe hidden local symmetry of the non-linear sigma model is extended to include baryons. In two papers (M. Bando, T. Kugo and Y. Yamawaki:Phys. Rep.,164, 217 (1988);Prog. Theor. Phys.,73, 1541 (1985)), the low-energy theorems for mesons were successfully reproduced. It seems important to investigate whether some kind of the low-energy theorems for baryons holds in the same framework. Indeed, theS-wave pion-nucleon scattering lengths, the conserved vector currents (CVC) of β-decays of baryons and Sakurais scenario of the electromagnetic interactions of hadrons are all reproduced. Our interaction Lagrangians are stable under the small variation of each vertex. Thus, they deserve the effective one.

Chiral symmetry breaking in hidden local symmetry

SummaryChiral breaking for baryon and meson interaction is formulated in the framework of the hidden local symmetric theory of the non-linear sigma model on the manifoldU(3)L×U(3)R/U(3)V. The chiral symmetry-breaking term transforms as (3,3*) underU(3)L×U(3)R at the first order. Several low-energy theorems are derived for the electromagnetic interaction of baryon. The symmetry-breaking term preservesSU(2)×U(1) symmetry; we have confirmed that the Sakurai formula of the electromagnetic interaction of vector meson holds and directly shown that the photon mass vanishes. The fundamental gauge coupling constantg and the mixing angle θ of vector meson are fixed by independent phenomena of vector meson masses and eē decay of vector meson. Using these values ofg and θ, we extend our analysis to the nucleon form factor and indicate a breaking of the OZI rule and a sizable contents of s-quark pair in the nucleon.

The crossing matrix for SU 3 ⊗ SU 3

SummaryA general expression is given for the elements of the crossing matrix for theSU3⊗SU3 algebra in terms of Clebsch-Gordan coefficients for the chiralSU3⊗SU3 algebra. Relations between the crossing matrices are derived. The crossing matrices are given for several cases.RiassuntoSi dà un’espressione generale per gli elementi della matrice d’incrocio per l’algebraSU3⊗SU3 in termini dei coefficienti di Clebsch-Gordan per l’algebra chiraleSU3⊗SU3. Si derivano relazioni tra le matrici d’incrocio. Si formulano le matrici d’incrocio per parecchi casi.РеэюмеПриводится обшее выражение для злементов кроссинг-матрицы для алгебрыSU3⊗SU3 в терминах козффициентов Клебща-Гордона для киральнойSU3⊗SU3 алгебры. Выводятся соотнощения между кроссинг-матрицами. Для некоторых случаев выписываются кроссинг-матрицы.

The crossing matrix forSU3⊗SU3

SummaryA general expression is given for the elements of the crossing matrix for theSU3⊗SU3 algebra in terms of Clebsch-Gordan coefficients for the chiralSU3⊗SU3 algebra. Relations between the crossing matrices are derived. The crossing matrices are given for several cases.RiassuntoSi dà un’espressione generale per gli elementi della matrice d’incrocio per l’algebraSU3⊗SU3 in termini dei coefficienti di Clebsch-Gordan per l’algebra chiraleSU3⊗SU3. Si derivano relazioni tra le matrici d’incrocio. Si formulano le matrici d’incrocio per parecchi casi.РеэюмеПриводится обшее выражение для злементов кроссинг-матрицы для алгебрыSU3⊗SU3 в терминах козффициентов Клебща-Гордона для киральнойSU3⊗SU3 алгебры. Выводятся соотнощения между кроссинг-матрицами. Для некоторых случаев выписываются кроссинг-матрицы.

Nucleon electromagnetic form factors in hidden local symmetry theory

SummaryIn previous papers (Nuovo Cimento A,108 (1995) 241, 1051), the hidden local symmetry of the chiral theory based on the non-linear sigma-model on the manifoldU(3)L×U(3)R/U(3)V was extended to the baryon interaction and shown to reproduce well the low-energy theorems. Introducing anSU(3) breaking in the theory, the mass formula and the Sakurai formula of the electromagnetic interaction of vector mesons are obtained consistently. In this paper, by introducing the effective tensor interactions of orderO(p2), the magnetic form factor of the nucleon is analysed as well as the electric form factor, partly done in the previous papers, and a breaking of the OZI rule and a sizeable content of s-quark pair in both the electric and magnetic form factors of the nucleon are indicated again.