Representation of the β-function and anomalous dimensions by nonsingular integrals: Proof of the main relation

Abstract We describe the on-shell method to derive the Renormalization Group (RG) evolution of Wilson coefficients of high dimensional operators at one loop, which is a necessary part in the on-shell construction of the Standard Model Effective Field Theory (SMEFT), and exceptionally efficient based on the amplitude basis in hand. The UV divergence is obtained by firstly calculating the coefficients of scalar bubble integrals by unitary cuts, then subtracting the IR divergence in the massless bubbles, which can be easily read from the collinear factors we obtained for the Standard Model fields. Examples of deriving the anomalous dimensions at dimension six are presented in a pedagogical manner. We also give the results of contributions from the dimension-8 H4D4 operators to the running of V+V−H2 operators, as well as the running of B+B−H2D2n from H4D2n+4 for general n.

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The low-energy effective theory of axions and ALPs

Journal of High Energy Physics ◽

10.1007/jhep04(2021)063 ◽

2021 ◽

Vol 2021 (4) ◽

Author(s):

Martin Bauer ◽

Matthias Neubert ◽

Sophie Renner ◽

Marvin Schnubel ◽

Andrea Thamm

Keyword(s):

High Energy ◽

New Physics ◽

Mass Scale ◽

Low Energy ◽

Effective Theory ◽

Loop Order ◽

Flavor Changing ◽

Effective Lagrangian ◽

Anomalous Dimensions ◽

Pseudoscalar Mesons

Abstract Axions and axion-like particles (ALPs) are well-motivated low-energy relics of high-energy extensions of the Standard Model, which interact with the known particles through higher-dimensional operators suppressed by the mass scale Λ of the new-physics sector. Starting from the most general dimension-5 interactions, we discuss in detail the evolution of the ALP couplings from the new-physics scale to energies at and below the scale of electroweak symmetry breaking. We derive the relevant anomalous dimensions at two-loop order in gauge couplings and one-loop order in Yukawa interactions, carefully considering the treatment of a redundant operator involving an ALP coupling to the Higgs current. We account for one-loop (and partially two-loop) matching contributions at the weak scale, including in particular flavor-changing effects. The relations between different equivalent forms of the effective Lagrangian are discussed in detail. We also construct the effective chiral Lagrangian for an ALP interacting with photons and light pseudoscalar mesons, pointing out important differences with the corresponding Lagrangian for the QCD axion.

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