Renormalization of the Standard Model effective field theory to dimension eight

TL;DR

This work develops a comprehensive framework for renormalizing the Standard Model EFT up to dimension eight, focusing on a Green’s basis for bosonic operators and complete one-loop RGEs. It systematises operator classification, removes redundancies via off-shell momentum-space tests, and derives on-shell relations to connect Green’s and physical bases. The study demonstrates large anomalous dimensions for Higgs-rich operator classes and analyzes applications to positivity bounds and oblique parameters, illustrating the phenomenological impact of higher-dimensional running. By detailing the algorithmic procedure and providing explicit results for bosonic (and partial fermionic) sectors, the work offers essential tools for consistent SMEFT analyses across energy scales and sets the stage for further automation and full fermionic results. The findings strengthen the SMEFT framework as a precise, model-independent bridge between high-scale physics and low-energy observables, with broad implications for precision Higgs and electroweak phenomenology.

Abstract

The Standard Model Effective Field Theory (SMEFT) provides a powerful, model-independent framework to explore deviations from the Standard Model (SM) by parametrising potential new physics through higher-dimensional operators. This thesis investigates the renormalisation structure of SMEFT, focusing on dimension-eight operators, which are increasingly relevant in precision analyses and in models where dimension-six effects are suppressed. We review renormalisation in quantum field theory, emphasising dimensional regularisation and the $\overline{\text{MS}}$ scheme, and outline the conceptual foundations of EFTs. One of the central results of this work is the systematic construction and classification of bosonic operators in SMEFT at dimension eight, employing Group Theory techniques and removing redundancies by working in momentum space. Building on this operator basis, we compute the complete one-loop renormalisation group equations (RGEs) involving insertions of dimension-eight-or-lower operators. This includes pure dimension-eight effects, pairs of dimension-six operators and lepton-number-violating sectors. Our calculations use an off-shell Green's function basis and leverage algebraic simplifications derived from symmetry and gauge invariance. These results are applied to positivity bounds and oblique parameters, providing essential tools for consistent SMEFT analyses across energy scales. The findings extend SMEFT's theoretical reach and support its use in high-precision phenomenology.

Figures (11)

• Figure 1: One-loop self-energy diagrams contributing to the propagator of the right-handed electron $e$ in the unbroken Standard Model. Each diagram involves a fermion-boson loop and is labelled according to the hypercharge of the internal fermion line: Left: $\Sigma_{-\frac{1}{2}}$ (left-handed electron and charged Higgs). Center: $\Sigma_0$ (right-handed electron and $B$ gauge boson). Right: $\Sigma_{\frac{1}{2}}$ (left-handed neutrino and neutral Higgs). These diagrams yield logarithmic divergences and motivate the introduction of regularisation and renormalisation.
• Figure 2: The resummation of 1PI diagrams leads to a diagram formally similar to the tree-level expression: the dressed propagator.
• Figure 3: Left: a Green's Function with a positive superficial degree of divergence. Centre: A Green's Function with zero superficial degree of divergence (in this case, it is divergent). Right: A Green's Function with a negative superficial degree of divergence.
• Figure 4: Feynman diagrams illustrating the contributions of the operators $\mathcal{O}_{8;BH^4D^2}^{(3)}$ (left) and $\mathcal{O}_{6;BDH}$ (right) to the on-shell realization of $\mathcal{O}_{8;H^6D^2}^{(2)}$. On the left, the $\Lambda^0$ is applied to the gauge boson leg via the gauge coupling $g_1$. On the right, the at order $\Lambda^{-2}$ is used, which requires the insertion of $\mathcal{O}_{6;HD}$. Black dots represent SM vertices, Green boxes represent dimension-six interactions and orange boxes represent dimension-eight interactions.
• Figure 5: Diagrams contributing to the amplitude of $H^+H^0\rightarrow H^+H^0$ at tree-level and one-loop. Only diagrams proportional to $\lambda$ are included here. The black dots represent SM vertices, and the black cross represents the counterterm.