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

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Abstract
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 Keam's one is quite unlikely. It is mathematically interesting that in the above analysis one encounters some Diophantine equations of the second degree.