Four-fermion interactions and scale invariance

Abstract

Four-fermion interactions of the current-current type with $\mathrm{U}(n)$ symmetry, in one space and one time dimension, are investigated. It is shown that the equations of motion yield scale-invariant solutions only for two values of the coupling ${g}_{v}$ of the $\mathrm{SU}(n)$ currents, namely ${g}_{v}=0$ and ${g}_{v}=\frac{4\ensuremath{\pi}}{(n+1)}$. This holds for any value of the coupling ${g}_{B}$ of the U(1) currents. For the above two values of ${g}_{v}$ and any ${g}_{B}$ the theory is solved completely. Operator products of spinor fields are shown to be equal to $c$-number functions singular on the light cone times analytic bilocal operators expressed in terms of currents and free spinor fields. The currents are free for the above two values of ${g}_{v}$. The connection with the coupling as defined through four-point functions is discussed, and it turns out that the combination corresponding to $\mathrm{SU}(n)$ coupling is zero for both solutions. However, the solution for ${g}_{v}=\frac{4\ensuremath{\pi}}{(n+1)}$ exhibits nontrivial four-point functions also for ${g}_{B}=0$. It is shown, in an expansion around ${g}_{v}=0$, that there is only one Callan-Symanzik function $\ensuremath{\beta}$ which depends only on ${g}_{v}$ and that ${g}_{v}=0$ is relevant to the ultraviolet limit of the ${g}_{v}>0$ theories. When mass terms are introduced, this still holds in an infinite interval for ${g}_{B}$, which is bounded below by a certain negative value and in which the mass term is soft.