Renormalization of effective field theories via on-shell methods: the case of axion-like particles

Problem Statement
The Standard Model (SM) of particle physics, while successful, fails to address several observational and theoretical open questions, such as the strong CP problem, flavor hierarchies, and the stability of the electroweak scale. Axion-Like Particles (ALPs) are a prominent class of extensions involving light pseudoscalars that can address these issues and serve as dark matter candidates. The interactions of ALPs with SM fields are typically described by an Effective Field Theory (EFT) involving dimension-5 operators. A crucial step in making theoretical predictions for experimental observables is running the ALP Lagrangian from the high-energy ultraviolet (UV) scale $\Lambda$ down to the experimental energy scale $E$. This requires the evaluation of the anomalous dimension matrix of the effective operators. While Renormalization Group Equations (RGEs) for CP-odd ALPs have been computed previously using diagrammatic methods, the case involving both CP-even and CP-odd components (leading to CP-violating effects) and the inclusion of dimension-6 operators generated at one-loop order required further investigation.

Methodology
This paper employs on-shell and unitarity-based methods to compute the one-loop renormalization of ALP effective field theories. The core of the approach relies on the relation between the action of the dilatation operator and the S-matrix on form factors. Specifically, the anomalous dimensions are extracted from the discontinuities of form factors via unitarity cuts, evaluated through phase-space integrals.

The authors utilize two distinct parameterization techniques for these phase-space integrals to cross-check results and highlight computational efficiencies:

1. Angular Variables: A parameterization based on spinor rotation matrices and angular integration.
2. Stokes' Theorem: A method utilizing complex analysis and Stokes' theorem to project the rational coefficients of 2-point functions directly out of double-cuts. This approach is noted for being technically simpler and more direct for isolating UV singularities.

To handle mass effects within a massless formalism, the authors apply the Higgs low-energy theorem, effectively treating mass insertions as interactions with extra massless Higgs fields. The calculations are performed for both the CP-even and CP-odd sectors of the ALP Lagrangian, covering operators up to dimension-6.

Key Contributions and Results
The paper provides a comprehensive derivation of the anomalous dimension matrix for ALP couplings and the resulting SM effective operators, extending previous results in the literature.

• Renormalization of ALP Couplings: The authors compute the UV anomalous dimensions for the ALP couplings to fermions ($\phi \bar{f}f$ and $\phi \bar{f}i\gamma_5 f$), photons ($\phi F\tilde{F}$ and $\phi FF$), and gluons ($\phi G\tilde{G}$ and $\phi GG$).

  • They confirm that the operators $\phi FF$ and $\phi G\tilde{G}$ renormalize precisely like the gauge kinetic terms ($FF$ and $GG$), a property that is manifest in the on-shell method but obscured in standard Feynman diagram approaches.
  • The mixing between CP-even and CP-odd sectors is explicitly calculated, providing the RGEs for the Wilson coefficients $C_\gamma, C_g, \tilde{C}_\gamma, \tilde{C}_g$ and the fermion couplings $Y^S_{ij}, Y^P_{ij}$.

• Renormalization of Induced SM Operators (Below EW Scale): By integrating out the ALP at the one-loop level, the authors derive the RGEs for the induced dimension-6 SM effective operators. These include:

  • The Weinberg operator ($GG\tilde{G}$) and its CP-even counterpart ($GGG$).
  • Electric and chromoelectric dipole moments ($\bar{f}\sigma \cdot F i\gamma_5 f$, $\bar{f}\sigma \cdot G i\gamma_5 f$) and their magnetic counterparts.
  • Four-fermion operators of various chiralities ($LL, RR, LR$).
  • The results generalize previous findings for CP-odd ALPs to the CP-violating case, reproducing known limits while providing new mixing terms.

• Renormalization Above EW Scale: The study extends to the unbroken SM phase, calculating RGEs for SMEFT operators in the Warsaw basis. This includes classes $X^3$, $X^2H^2$, $H^6$, $H^4D^2$, $\psi^2HX$, $\psi^2H^2D$, $\psi^2H^3$, and four-fermion operators. The authors identify that only specific unitarity cuts contribute to these renormalizations due to helicity selection rules.

Significance and Comparison with Standard Methods
The paper explicitly compares the on-shell method with standard Feynman diagrammatic techniques. The authors claim that the on-shell approach offers significant computational advantages:

• Reduced Complexity: The on-shell method drastically reduces the number of diagrams and calculations required. For instance, the renormalization of the $\phi GG$ vertex, which involves multiple non-trivial Feynman diagrams with gauge-dependent structures in the standard approach, reduces to a single convolution of a tree-level amplitude and a form factor in the on-shell method.
• Gauge Invariance: The on-shell method is inherently gauge-invariant, avoiding the need for explicit checks of gauge invariance and the cancellation of gauge-dependent terms that plague standard calculations, particularly those involving non-Abelian gauge bosons.
• Manifest Symmetries: The method makes hidden structures and symmetries more apparent. Specifically, it directly reveals the relationship between the anomalous dimensions of CP-dual operators (e.g., $\phi FF$ vs. $\phi F\tilde{F}$), which appear as entirely different Lorentz structures in the standard approach.
• Technical Efficiency: The application of Stokes' theorem for phase-space integration is highlighted as a particularly efficient tool for isolating UV divergences, simplifying the extraction of anomalous dimensions compared to traditional angular integration.

In conclusion, the paper demonstrates that on-shell methods are not only a viable alternative but a superior tool for computing the renormalization group equations of CP-violating ALP theories, offering a more elegant and efficient path to results that are consistent with, and extend, existing literature.