Callan–Symanzik equation and asymptotic freedom in the Marr–Shimamoto model

Abstract

The exactly soluble nonrelativistic Marr–Shimamoto model was introduced in 1964 as an example of the Lee model with a propagator and a nontrivial vertex function. An exactly soluble relativistic version of this model, known as the Zachariasen model, has been found to be asymptotically free in terms of coupling constant renormalization at an arbitrary spacelike momentum and on the basis of exact solutions of the Gell–Mann–Low equations. This is accomplished with conventional cut-off regularization by setting up the Yukawa and Fermi coupling constants at Euclidean momenta in terms of on mass-shell couplings and then taking the asymptotic limit. In view of this background, it may be expected that an investigation of the nonrelativistic Marr–Shimamoto theory may also exhibit asymptotic freedom in view of its manifest mathematical similarity to that of the Zachariasen model. To prove this point, the present paper prefers to examine asymptotic freedom in the nonrelativistic Marr–Shimamoto theory using the powerful concepts of the renormalization group and the Callan–Symanzik equation, in conjunction with the specificity of dimensional regularization and on-shell renormalization. This approach is based on calculations of the Callan–Symanzik coefficients and determinations of the effective coupling constants. It is shown that the Marr–Shimamoto theory is asymptotically free for dimensions D<3 and for values of D>3 occurring in periodic intervals over the range of 0<D<27 of particular interest. This differs from the original Lee model which has been shown by several authors, using these same concepts, to be asymptotically free only for D<4.