Automatic Fine-Tuning in the Two-Flavor Schwinger Model

I discuss the two-flavor Schwinger model both without and with fermion masses. I argue that the phenomenon of “conformal coalescence,” in unparticle physics in which linear combinations of short-distance operators can disappear from the long-distance theory, makes it easy to understand some puzzling features of the model with small fermion masses. In particular, I argue that for an average fermion mass $m_f$ and a mass difference $\delta m$, so long as both are small compared to the dynamical gauge boson mass $m=e\sqrt{2/\pi }$, isospin-breaking effects in the low-energy theory are exponentially suppressed by powers of $\exp [-(m/m_f{)}^{2/3}]$ even if $\delta m\approx m_f$. In the low-energy theory, this looks like exponential fine-tuning, but it is done automatically by conformal coalescence.