Functional methods can be applied to the quantum effective action to efficiently determine counterterms and matching conditions for effective field theories. We extend the toolbox to two-loop order and beyond and show how to evaluate the expansion of the path integral in a manifestly gauge-covariant manner. We also generalize the method to theories with mixed spin statistics and prove the validity of the hard-region matching formula to all loop orders. The methods are exemplified with a two-loop matching calculation of the Euler–Heisenberg Lagrangian resulting from decoupling the electron in QED.

Spain

Javier Fuentes-Martín

Adrián Moreno-Sánchez

Switzerland

Ajdin Palavrić

Anders Thomsen

A guide to functional methods beyond one-loop order

The next goalpost in precision calculations for physics beyond the Standard Model is determining the two-loop renormalization group (RG) equations in the Standard Model Effective Field Theory (SMEFT). We progress towards this goal by determining the RG equations for a simplified version of the SMEFT without any fermion fields. Our calculation relies on functional methods that are newly developed for multi-loop computations and adapted for the determination of RG equations here.

Lukas Born

Sandra Kvedaraite

Two-loop running in the bosonic SMEFT using functional methods

Flavor deconstruction refers to ultraviolet completions of the Standard Model where the gauge group is split into multiple factors under which fermions transform non-universally. We propose a mechanism for charging same-family fermions into different factors of a deconstructed gauge theory in a way that gauge anomalies are avoided. The mechanism relies in the inclusion of a strongly-coupled sector, responsible of both anomaly cancellation and the breaking of the non-universal gauge symmetry. As an application, we propose different flavor deconstructions of the Standard Model that, instead of complete families, uniquely identify specific third-family fermions. All these deconstructions allow for a new physics scale that can be as low as few TeV and provide an excellent starting point for the explanation of the Standard Model flavor hierarchies.

Javier Lizana

Deconstructing flavor anomalously

The recent <italic>B</italic>-meson anomalies are coherently explained at the TeV scale by <italic>4321 gauge models</italic> with hierarchical couplings reminiscent of the Standard Model Yukawas. We show that such models arise as the low-energy limit of a complete theory of flavor, based on a warped fifth dimension where each Standard Model family is quasi-localized on a different brane. The Higgs is identified as a pseudo-Nambu-Goldstone boson emerging from the same dynamics responsible for 4321 symmetry breaking. This novel construction unifies quarks and leptons in a flavor non-universal manner, provides a natural description of flavor hierarchies, and addresses the electroweak hierarchy problem.

Gino Isidori

Javier M. Lizana

Nudžeim Selimović

Ben A. Stefanek

Flavor hierarchies, flavor anomalies, and Higgs mass from a warped extra dimension

We present a systematic method for determining the two-loop effective Lagrangian resulting from integrating out a set of heavy particles in an ultraviolet scalar theory. We prove that the matching coefficients are entirely determined from the (double-)hard region of the loop integrals and present a master formula for matching, applicable to both diagrammatic and functional approaches. We further employ functional methods to determine compact expressions for the effective Lagrangian that do not rely on any previous knowledge of its structure or symmetries. The same methods are also applicable to the computation of renormalization group equations. We demonstrate the application of the functional approach by computing the two-loop matching coefficients and renormalization group equations in a scalar toy model.

Anders Eller Thomsen

Functional matching and renormalization group equations at two-loop order

In the presence of axions or axion-like particles (ALPs) that couple to the Standard Model via dimension-five interactions, dimension-six SMEFT interactions are generated via renormalization-group evolution. As many of these SMEFT contributions are experimentally tightly constrained, this “ALP-SMEFT interference” can be used to derive indirect bounds on the ALP couplings to the Standard Model particles. We present a global analysis of the Wilson coefficients of the ALP effective Lagrangian based on Higgs, top, and low-energy data. The obtained bounds are model independent and are competitive or even stronger than direct bounds in the GeV to TeV ALP-mass range.

Germany

Anke Biekötter

Anne Galda

United States

Matthias Neubert

A global analysis of axion-like particle interactions using SMEFT fits

Studying the impact of new-physics models on low-energy observables necessitates matching to effective field theories at the relevant mass thresholds. We introduce the first public version of Matchete, a computer tool for matching weakly-coupled models at one-loop order. It uses functional methods to directly compute all matching contributions in a manifestly gauge-covariant manner, while simplification methods eliminate redundant operators from the output. We sketch the workings of the program and provide examples of how to match simple Standard Model extensions. The package, documentation, and example notebooks are publicly available at  https://gitlab.com/matchete/matchete  .

Matthias König

Julie Pagès

Felix Wilsch

A proof of concept for matchete: an automated tool for matching effective theories

Effective Field Theory calculations used in countless phenomenological analyses employ dimensional regularization, and at intermediate stages of computations, the operator bases extend beyond the four-dimensional ones. The extra pieces — the evanescent operators — can ultimately be removed with a suitable renormalization scheme, resulting in a finite shift of the physical operators. Modern Effective Field Theory matching techniques relying on the method of expansion by regions have to be extended to account for this. After illustrating the importance of these shifts in two specific examples, we compute the finite shifts required to remove all evanescent operators appearing in the one-loop matching of generic ultraviolet theories to the Standard Model Effective Field Theory and elucidate the formalism for generic Effective Field Theory calculations.

Evanescent operators in one-loop matching computations

Motivated by the recent LHCb announcement of a 3.1σ violation of lepton- flavor universality in the ratio R  K  = Γ(B → Kμ + μ  − )/Γ(B → Ke + e  − ), we present an updated, comprehensive analysis of the flavor anomalies seen in both neutral-current (b → sℓ + ℓ  − ) and charged-current (b →   <math> <mi>cτ</mi> <mover> <mi>ν</mi> <mo>¯</mo> </mover> </math>  $$ c\tau \overline{\nu} $$ ) decays of B mesons. Our study starts from a model-independent effective field-theory approach and then considers both a simplified model and a UV-complete extension of the Standard Model featuring a vector leptoquark U 1 as the main mediator of the anomalies. We show that the new LHCb data corroborate the emerging pattern of a new, predominantly left-handed, semileptonic current-current interaction with a flavor structure respecting a (minimally) broken U(2)5 flavor symmetry. New aspects of our analysis include a combined analysis of the semileptonic operators involving tau leptons, including in particular the important constraint from B  s -  <math> <msub> <mover> <mi>B</mi> <mo>¯</mo> </mover> <mi>s</mi> </msub> </math>  $$ {\overline{B}}_s $$  mixing, a systematic study of the effects of right-handed leptoquark couplings and of deviations from minimal flavor-symmetry breaking, a detailed analysis of various rare B-decay modes which would provide smoking-gun signatures of this non-standard framework (LFV decays, di-tau modes, and B → K (*)   <math> <mi>ν</mi> <mover> <mi>ν</mi> <mo>¯</mo> </mover> </math>  $$ \nu \overline{\nu} $$ ), and finally an updated analysis of collider bounds on the leptoquark mass and couplings.

Claudia Cornella

Darius Faroughy

Reading the footprints of the B-meson flavor anomalies

We analyze the neutrino mass spectrum and discuss the extra-dimensional interpretation of a three-site Pati-Salam model which i) unifies all families of quark and leptons, ii) provides a natural description of the Standard Model Yukawa couplings, iii) could account for the recent B-physics anomalies. The key feature of the model is a breaking of the Pati-Salam and electroweak gauge symmetries localized on opposite sites, communicated to the other sites in an attenuated manner via nearest-neighbor interactions. We show that in this context gauge-singlet fermions localized on each site, receiving hierarchical Majorana masses, can allow the implementation of an inverse seesaw mechanism leading to light anarchic neutrino masses consistent with data. The continuum limit of this three-site setup has a natural interpretation in terms of a warped extra dimension with three defects, where the required exponential hierarchies can be achieved from <math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math> differences in the bulk field masses.