Non-renormalization Theorems without Supersymmetry

We derive a new class of one-loop non-renormalization theorems that strongly constrain the running of higher dimension operators in a general four-dimensional quantum field theory. Our logic follows from unitarity: cuts of one-loop amplitudes are products of tree amplitudes, so if the latter vanish then so too will the associated divergences. Finiteness is then ensured by simple selection rules that zero out tree amplitudes for certain helicity configurations. For each operator we define holomorphic and anti-holomorphic weights, $(w,\overline w) =(n - h,n+h)$, where $n$ and $h$ are the number and sum over helicities of the particles created by that operator. We argue that an operator ${\cal O}_i$ can only be renormalized by an operator ${\cal O}_j$ if $w_i \geq w_j$ and $\overline w_i \geq \overline w_j$, absent non-holomorphic Yukawa couplings. These results explain and generalize the surprising cancellations discovered in the renormalization of dimension six operators in the standard model. Since our claims rely on unitarity and helicity rather than an explicit symmetry, they apply quite generally.