Low-Energy Effective Field Theory below the Electroweak Scale: Anomalous Dimensions (1711.05270v3)

Abstract: We compute the one-loop anomalous dimensions of the low-energy effective Lagrangian below the electroweak scale, up to terms of dimension six. The theory has 70 dimension-five and 3631 dimension-six Hermitian operators that preserve baryon and lepton number, as well as additional operators that violate baryon number and lepton number. The renormalization group equations for the quark and lepton masses and the QCD and QED gauge couplings are modified by dimension-five and dimension-six operator contributions. We compute the renormalization group equations from one insertion of dimension-five and dimension-six operators, as well as two insertions of dimension-five operators, to all terms of dimension less than or equal to six. The use of the equations of motion to eliminate operators can be ambiguous, and we show how to resolve this ambiguity by a careful use of field redefinitions.

Overview of the Paper on Low-Energy Effective Field Theory

The paper is focused on the detailed computations of one-loop anomalous dimensions within the low-energy effective field theory (LEFT) below the electroweak scale, with operators up to dimension six. This work is critical considering that experimental data from the Large Hadron Collider (LHC) have, so far, corroborated the Standard Model (SM) well among its energy ranges. Nonetheless, due to the lack of new identifiable particles up to about 1 TeV, this research advocates leveraging effective field theories (EFTs) to paper beyond-the-Standard Model (BSM) physics without committing to any specific model.

The primary framework under consideration is Standard Model Effective Field Theory (SMEFT), where all operators stem from standard model gauge transformations acting on SM particle components, including a single fundamental scalar (Higgs) doublet. Higgs Effective Field Theory (HEFT) is introduced as an alternative that relaxes the requirement of treating the Higgs boson as an integral part of the electroweak symmetry breaking doublet. The derivation of a low-energy effective field theory (LEFT) involves integrating out particles with mass near the electroweak scale, such as the $W^\pm$, the $Z$ bosons, the top quark, and the Higgs boson itself. This formulation is particularly useful in the paper of $B$-meson and kaon physics, providing essential insights into constraints on new physics.

The work revolves around systematically analyzing and calculating one-loop anomalous dimensions, pivotal to tracking how Wilson coefficients of effective operators evolve with the energy scale via renormalization group equations (RGEs). These calculations encapsulate one-loop effects involving 70 dimension-five and 3631 dimension-six operators when baryon and lepton number preservation is considered. The paper also extends the operator set to include those violating baryon and lepton numbers.

A notable complexity highlighted in the paper involves the equation of motion (EOM) and its role in operator redefinitions, given the potential for differing manipulations to influence the higher-order EFT expansion. This conceptual aspect incorporates meticulous precision in handling loops containing both singular and quadratic dimension-five operator insertions, which address uncertainties and redefinitions naturally introduced through perturbative expansions.

Critically, the research goes on to reveal intriguing cancellations due to what can be interpreted as partial holomorphy within the RGEs. Although the gauge coupling RGEs respect holomorphy by coupling only with self-dual interaction terms, the dipole component exhibits deviations due to mass term dependencies.

In terms of implications, this research fortifies the bridge between high-energy constraints provided by the LHC and the low-energy experimental constraints to predict and possibly guide future experimental phenomena. The contrasts and comparisons between SMEFT and HEFT underscore the subtleties and profound implications that operators at different dimensions could encapsulate, particularly in distinguishing between SMEFT-specific and more general HEFT predictions.

For future directions, refining and expanding the anomalous dimension computations to incorporate even higher-order corrections or utilizing more sophisticated mathematical methods to simplify calculations could bolster the understanding of LEFT. As technology and precision in experimental particle physics advance, a symbiotic development in the computational domain will likely provide deeper insights into uncharted territories of particle physics.

In conclusion, the paper presents a thorough theoretical framework for understanding low-energy operator effects below the electroweak scale, equipping researchers with tools to explore deviations from the Standard Model. The meticulous work done in categorizing and computing the anomalous dimensions represents an instrumental step in decoding the BSM physics landscape.