We provide the leading near conformal corrections on a cylinder to the scaling dimension $\Delta_Q^\ast$ of fixed isospin charge $Q$ operators defined at the lower boundary of the Quantum Chromodynamics conformal window: \begin{ equation} \Delta_Q = \Delta_Q^\ast +\left(\frac{ m_{ \sigma}}{ 4 \pi \nu}\right)^2 \, Q^{ \frac{ \Delta}{ 3}} B_1 + \left(\frac{ m_\pi(\theta)}{ 4\pi \nu} \right)^4\ Q^{ \frac{ 2}{ 3}(1-\gamma)} B_2 + \mathcal{ O}\left ( m_\sigma^4 , m_\pi^8, m_\sigma^2 m_\pi^4\right) \ . \nonumber \end{ equation} The results are expressed in powers of the dilaton and pion masses in units of the chiral symmetry breaking scale $4\pi \nu$ with the theta-angle dependence encoded directly in the pion mass. The characteristic $Q$-scaling is dictated by the quark mass operator anomalous dimension $\gamma$ and the one characterising the dilaton potential $\Delta$. The coefficients $B_i$ with $i=1,2$ depend on the geometry of the cylinder and properties of the nearby conformal field theory.
Categories: hep-th hep-lat hep-ph