Four-fermion interactions and scale invariance

Four-fermion interactions of the current-current type with $\mathrm{U}\mathrm{}\left(n\right)\mathrm{}$ 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 $\mathrm{SU}\mathrm{}\left(n\right)\mathrm{}$ currents, namely $g_v=0\mathrm{}$ and $g_v=\frac{4\pi }{\mathrm{}(n+1\mathrm{})\mathrm{}}$. 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 $\mathrm{SU}\mathrm{}\left(n\right)\mathrm{}$ coupling is zero for both solutions. However, the solution for $g_v=\frac{4\pi }{\mathrm{}(n+1\mathrm{})\mathrm{}}$ exhibits nontrivial four-point functions also for $g_B=0\mathrm{}$. It is shown, in an expansion around $g_v=0\mathrm{}$, that there is only one Callan-Symanzik function $\beta$ which depends only on $g_v$ and that $g_v=0\mathrm{}$ is relevant to the ultraviolet limit of the $g_v>0\mathrm{}$ 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.