Singular Potentials and the Z = 0 Condition

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Abstract
We consider an interaction Lagrangian consisting of renormalizable and nonrenormalizable terms. After a brief discussion of the vertex function and the boundary conditions determining the renormalization constant corresponding to a pseudoscalar bound state, we derive an explicit expression for the nonrelativistic S‐matrix for scattering by the potential 1/r 4. These results are then used to evaluate Z. The explicit expressions obtained contain a strong cut‐off dependence which cannot be factorized out. However, a specific value of the cut‐off reduces the equation to a well‐known eigenvalue problem, so that in this case a discrete spectrum of bound states may be obtained.

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