Title: On the $ε=d-2$ expansion of the Ising model

Abstract: We study the Ising model in $d=2+\epsilon$ dimensions using the conformal bootstrap. As a minimal-model Conformal Field Theory (CFT), the critical Ising model is exactly solvable at $d=2$. The deformation to $d=2+\epsilon$ with $\epsilon\ll 1$ furnishes a relatively simple system at strong coupling outside of even dimensions. At $d=2+\epsilon$, the scaling dimensions and correlation function coefficients receive $\epsilon$-dependent corrections. Using numerical and analytical conformal bootstrap methods in Lorentzian signature, we rule out the possibility that the leading corrections are of order $\epsilon^{1}$. The essential conflict comes from the $d$-dependence of conformal symmetry, which implies the presence of new states. A resolution is that there exist corrections of order $\epsilon^{1/k}$ where $k>1$ is an integer. The linear independence of conformal blocks plays a central role in our analyses. Since our results are not derived from positivity constraints, this bootstrap approach can be extended to the rigorous studies of non-positive systems, such as non-unitary, defect/boundary and thermal CFTs.