Super-renormalizable Theory Research Articles

O 4 symmetry, light-cone expansion and the electroproduction structure function

The electroproduction structure function is examined by formulating the method of the light-cone expansion via theO4 symmetry advocated by Nachtman. The hadron matrix elements of the expansion operators are expressed in terms of theO4 spherical harmonics. The structure functions, for the elementary and composite particles the wave functions of which are given by the Bethe-Salpeter amplitudes, are studied by taking account of the dynamical effects for the superrenormalizable and renormalizable interacting spinless particles. For the composite particles, by considering the high-rank behaviour of the matrix elements, it is shown that the Drell-Yan-West threshold relation holds for the renormalizable interaction if the anomalous dimension is neglected. In the case of the superrenormalizable theory, the relation holds exactly. There exist possible Regge poles in the complex angular-momentum plane ofO4 symmetry for the matrix elements due to the Regge behaviour in the scattering process for the constituent particles. In the case of the superrenormalizable theory, we get the Regge behaviour for the structure function, but, for the renormalizable theory, the structure function may inevitably have the non-Regge behaviour indicated by De Rujulaet al.

Off-shell and on-shell behavior of vertex functions

The off-shell and on-shell asymptotic behaviors of the three-point vertex function are examined for renormalizable theories with anomalous dimensions. A general argument based on the conformal-operator expansion is given, according to which the off-shell vertex function has the same asymptotic behavior as the on-shell vertex function if the field of minimal dimension related to the particle has dimension less than 2. If the dimension is larger than 2 the two asymptotic behaviors differ. This result is verified by an explicit calculation on a model with anomalous dimensions, and it is shown to be related to the occurrence of shadow singularities in a three-point function. A comparison with the different situation in superrenormalizable theories is given.

Renormalization and second sheet poles in unstable particle theory

The renormalization of an unstable field is studied in the perturbative framework of Matthews and Salam. With the chosen coupling one has to face overlapping divergencies: thus it is possible to generalize a result already obtained in super-renormalizable theories. In addition the existence of second sheet poles is studied. In perturbation theory Lagrange's theorem is used to prove their existence to all orders and it also gives an explicit formula to compute them.

An attempt to evaluate renormalized radiative corrections by computer

An attempt is made to apply algebraic computational techniques to renormalization theory. Starting from the functional formalism for a scalar theory as well as for quantum electrodynamics we describe the Lisp and Fortran programs which generate Feynman diagrams. Within the gΘ 3 scalar model, we integrate the Feynman integrals over the internal momenta and remove the divergences of this superrenormalizable theory. A Lisp program performs these operations. The numerical evaluation of the multidimensional integrals is briefly discussed in the last part of this paper.

Convergence of regularized, renormalized perturbation series for super-renormalizable field theories

It is shown for the two-dimensional scalar Yukawa interaction, that the renormalized perturbation series has at least a finite radius of convergencewhen regularized (cut-off in space-time and momentum space). This is accomplished by writing down the explicit renormalized series, studying some of its associated combinatorics, and applying Caianiello’s standard arguments on the unrenormalized series (16). The result extends to any super-renormalizable theory of the\(\varphi \bar \psi \psi \) form.