Noboru Nakanishi

Vector‐Scalar Sector Solutions to the Spinor‐Spinor Bethe‐Salpeter Equation

A new method is proposed to discuss the exact J = 0 vector‐scalar sector solutions to the equal‐mass spinor‐spinor Bethe‐Salpeter equation for the massless‐meson exchange ladder model at the vanishing total 4‐momentum. Under the assumption that all solutions belonging to a discrete spectrum have a discrete spectral representation in the relative 4‐momentum squared, it is proved that no discrete solutions other than the solution (in the vector‐coupling case) found by Bastai, Bertocchi, Furlan, and Tonin exist in any case of the scalar, pseudoscalar, and vector couplings. As for the case of the axialvector coupling, it is shown that possible eigenvalues have to belong to one of three exponentially increasing sequences; but the existence of any solution other than Keams one is quite unlikely. It is mathematically interesting that in the above analysis one encounters some Diophantine equations of the second degree.

Crossing-Symmetric Decomposition of the n-Point Veneziano Formula into Tree-Graph Integrals. I

Given a cyclic ordering of external particles and ann-point tree Feynman graph T, the tree-graph integral F T is defined in such a way that F T has only the poles relevant to T, that there is a birational transformation by which F T is transformed into an integral identical with the n-point Veneziano formula apart from its integration domain, and that the crossing-symmetry property and Chans bootstrap condition are manifest. It is proved that the n-point Veneziano formula is written as a sum of FT over all tree graphs T belonging to the given cyclic ordering of external particles.

Fundamental Properties of Perturbation‐Theoretical Integral Representations. III

In the first part of this paper, it is investigated, apart from the perturbation‐theoretical basis, under what conditions the perturbation‐theoretical integral representations can be derived, and two theorems are given concerning this problem. In the second part, the asymptotic behavior of the weight function in the integral representation is investigated in perturbation theory. It is proved that the weight function vanishes at infinity for an infinite sum over certain graphs which are much more general than the ladderlike graphs. This result gives the analyticity in the right half‐plane of complex angular momentum.

An Axiomatic Formulation of the Theory of Coinciding Simple Poles and Multiple Poles

In the Bethe‐Salpeter formalism, the scattering Greens function is known to have multiple poles synthesized out of coinciding simple poles. The present paper proposes an axiomatic approach to the problem of finding the residues of the multiple poles in terms of those of M coinciding simple poles. The latter residues are regarded as finite‐dimensional, mutually orthogonal projection operators on a reflexive Banach space and its dual. Then various properties of the residues of the multiple poles are derived without recourse to the original Bethe‐Salpeter equation, and especially it is shown mathematically that they can be decomposed into a direct sum of operators which commute with the Bethe‐Salpeter operator. The residues of multiple poles are explicitly determined in two particular cases, M = N + 1 and M = 2, where N denotes the highest order of the singularities (in a parameter) of the residues of the coinciding simple poles.

Poles of the Proper Vertex Function in the Bethe‐Salpeter Formalism

A formal general proof of the statement of Goebel and Sakita is presented on the basis of the Bethe‐Salpeter formalism; namely, it is shown that the poles of a proper vertex function cannot appear in the corresponding scattering amplitude. Some related conjectures are also verified. An exactly solvable example is presented and discussed in this connection.

Remarks on the Double Dispersion Approach to the Bethe‐Salpeter Equation

The following remarks are made on the applicability of the double dispersion approach to the Bethe‐Salpeter equation introduced previously. (1) Any invariant solution of the Bethe‐Salpeter equation in ladder approximation satisfies the double dispersion representation when the total energy‐momentum is spacelike. (2) There are some exceptional invariant solutions which are not given by the previous method in the equal‐mass case, but the existence of such solutions is very unlikely in the unequal‐mass case. (3) In the case of the general separated kernel the previous results give the correct solutions even if the kernel does not reproduce the double dispersion representation.