Non-renormalization Theorems without Supersymmetry

TL;DR

This work addresses how higher-dimension operators in four-dimensional, massless quantum field theories renormalize at one loop without appealing to supersymmetry. It introduces holomorphic and anti-holomorphic weights $w$ and $wbar$ derived from on-shell amplitudes and leverages unitarity cuts to derive both tree-level and one-loop selection rules, notably $w_i >= w_j$ and $wbar_i >= wbar_j$ and, at one loop, $w_i = w_j + w_k -4$ and $wbar_i = wbar_j + wbar_k -4$. These weight-based non-renormalization theorems constrain operator mixing, explaining cancellations observed in Standard Model dim-6 renormalization and applying generally to any massless 4D QFT. The analysis shows that, except for exceptional amplitudes stemming from non-holomorphic Yukawas, operators can only mix into same or higher weight operators, providing symmetry-free constraints on EFT renormalization with potential implications for higher-loop and higher-dimensional theories.

Abstract

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.

Figures (2)

• Figure 1: Weight lattice for dimension five and six operators, suppressing flavor and Lorentz structures, e.g. on which fields derivatives act. The non-renormalization theorems in Eq. (\ref{['eq:nonrenorm']}) let operators to mix into operators of equal or greater weight. Pictorially, this forbids transitions down or to the left.
• Figure 2: Diagrams of tree factorization and one-loop unitarity, with the weight selection rules from Eqs. (\ref{['eq:treerelation']}) and (\ref{['eq:looprelation']}).