  A system of two-species, one-dimensional fermions, with an attractive
two-body interaction of the derivative-delta type, features a scale anomaly. In
contrast to the well-known two-dimensional case with contact interactions, and
its one-dimensional cousin with three-body interactions (studied recently by
some of us and others), the present case displays dimensional transmutation
featuring a power-law rather than a logarithmic behavior. We use both the
Schr\"{o}dinger equation and quantum field theory to study bound and scattering
states, showing consistency between both approaches. We show that the
expressions for the reflection $(R)$ and the transmission $(T)$ coefficients of
the renormalized, anomalous derivative-delta potential are identical to those
of the regular delta potential. The second-order virial coefficient is
calculated analytically using the Beth-Uhlenbeck formula, and we make comments
about the proper $\epsilon_B\rightarrow 0$ (where $\epsilon_B$ is the
bound-state energy) limit. We show the impact of the quantum anomaly (which
appears as the binding energy of the two-body problem, or equivalently as Tan's
contact) on the equation of state and on other universal relations. Our
emphasis throughout is on the conceptual and structural aspects of this
problem.
