We present the complete 4-loop renormalisation of an Abelian chiral gauge theory in the Breitenlohner-Maison / ‘t Hooft-Veltman (BMHV) scheme. Employing a non-anticommuting $\textit{γ}$ in dimensional regularisation, we determine the full set of symmetry restoring counterterms from the quantum action principle. Our calculation represents the highest-order application of the BMHV framework so far, pushing the limits of a self-consistent treatment of $\textit{γ}$ at the multi-loop level. We describe the computational setup that we developed to perform the computations and discuss key implementation aspects, such as the BMHV algebra and tensor reduction. Our work demonstrates the feasibility of applying the BMHV scheme at high loop orders and establishes a solid foundation for future studies of high-precision electroweak physics.

Germany

Andreas Manteuffel

Dominik Stöckinger

Matthias Weißwange

Four-loop renormalisation of chiral gauge theories with non-anticommuting $\textit{γ}$

Higher order calculations in chiral gauge theories such as the Electroweak Standard Model require a sound treatment of the notoriously problematic $\textit{γ}$ -matrix in Dimensional Regularization (DReg). In the all-order consistent BMHV scheme anticommutativity has to be sacrificed, resulting in spurious breakings of BRST invariance, the restoration of which necessitates finite, symmetry-restoring counterterms. Following recent advances in successfully applying this scheme to multi-loop calculations for Abelian models, we shall here present the first complete non-Abelian two-loop result for the case of SU(2), which is of particular interest to the Standard Model. We provide the complete list of finite, two-loop symmetry-restoring counterterms and discuss intricacies of the non-Abelian implementation. Except for one novel term, the finite counterterm action exhibits the same structure as at one-loop order.

Paul Kühler

Two-loop renormalization of a chiral SU(2) gauge theory in dimensional regularization with non-anticommuting $\textit{γ}$

The mathematical consistency of the BMHV scheme of dimensional regularisation (DReg) comes at the cost of requiring symmetry-restoring counterterms to cancel the regularisation-induced breaking of gauge and BRST invariance. There is no unique way to extend a 4-dimensional theory to dimensions, and different choices can be made for the dimensionally regularised fermions, evanescent parts of their kinetic terms and evanescent gauge interactions. Here we present a detailed study of the impact of changing such evanescent details. We leverage this freedom to identify a particularly convenient formulation that simplifies practical calculations. In order to thoroughly study the available options, we focus on a general abelian chiral gauge theory including scalar fields and adopt a general approach to the BMHV implementation. This allows for specialisation to various models and different approaches, including those from the literature. Importantly, our model can be specialised to the abelian sector of the Standard Model (SM). Consequently, this article also serves as a roadmap for upcoming applications to the full SM.

Paul Ebert

Shedding light on evanescent shadows — Exploration of non-anticommuting

Models with vector-like leptons can strongly modify the lepton mass generation mechanism and lead to correlated effects in lepton–Higgs couplings and lepton dipole moments. Here we begin an analysis of higher-order corrections in such models by setting up a renormalization scheme with full on-shell conditions on the lepton self energies, masses and fields. A minimal set of fundamental parameters is renormalized in the   <math> <mover> <mi>MS</mi> <mo>¯</mo> </mover> </math>  $$ \overline{\textrm{MS}} $$  scheme. We provide a detailed discussion of lepton mixing and redundancies at higher orders, show how the relevant counterterms can be obtained from the renormalization conditions, and determine the β-functions corresponding to the scheme. As a first application we calculate the one-loop effective muon–Higgs coupling and analyse its correlation with the muon anomalous magnetic moment   <math> <mtext>∆</mtext> <msubsup> <mi>a</mi> <mi>μ</mi> <mi>VLL</mi> </msubsup> </math>  $$ \Delta {a}_{\mu}^{\textrm{VLL}} $$ . In the interesting case of large masses and opposite-sign coupling, the lowest-order correlation implies a fixed value of   <math> <mtext>∆</mtext> <msubsup> <mi>a</mi> <mi>μ</mi> <mi>VLL</mi> </msubsup> </math>  $$ \Delta {a}_{\mu}^{\textrm{VLL}} $$  around 22.5 × 10−10, while the higher-order corrections significantly reduce this value to the interval (10 . . . 18) × 10−10.

Kilian Möhling

Hyejung Stöckinger-Kim

On-shell renormalization with vector-like leptons, one-loop muon–Higgs coupling and muon g − 2

The Minimal R-symmetric Supersymmetric Standard Model (MRSSM) provides a realisation of supersymmetry in which the parameter space is less constrained by the current LHC data than in the simplest supersymmetric scenarios. In the present paper, we obtain the most precise theoretical predictions in the MRSSM for squark production at the LHC, enabling accurate interpretations of LHC data in terms of the MRSSM. We perform threshold resummation of soft gluon corrections to the total cross sections for the production of squark-(anti)squark pairs at the LHC in the MRSSM framework. The resummation is carried out using the direct QCD method and reaches the next-to-next-to-leading-logarithmic (NNLL) accuracy, which requires calculating the one-loop matching coefficients in the relevant production channels. The resummed cross sections are then matched to the available NLO results and evaluated for   <math> <msqrt> <mi>S</mi> </msqrt> </math>  $$ \sqrt{S} $$  = 13.6 TeV. Compared with the Minimal Supersymmetric Standard Model (MSSM), the cross sections in the MRSSM can be significantly reduced, implying less stringent limits on squark and gluino masses. Our results carry significant implications for exploring the viability of supersymmetry at the LHC. The results of our calculation are publicly available as a numerical package.

Christoph Borschensky

Fausto Frisenna

Poland

Wojciech Kotlarski

Anna Kulesza

Squark production with R-symmetry beyond NLO at the LHC

In this work we present a complete three-loop renormalisation of an abelian chiral gauge theory within the Breitenlohner-Maison/’t Hooft-Veltman (BMHV) scheme of dimensional regularisation (DReg). In this scheme the γ 5-matrix appearing in gauge interactions is a non-anticommuting object, leading to a breaking of gauge and BRST invariance. Employing an efficient method based on the quantum action principle, we obtain the complete three-loop counterterm action which serves both to render the theory finite and to restore gauge and BRST invariance. The UV singular counterterms involve not only higher order ϵ-poles but also new counterterm structures emerging at the three-loop level for the first time; the finite symmetry-restoring counterterms are restricted to the same structures as at lower loop orders, just with different coefficients, aligning with our expectations. Both the singular and the finite counterterms include structures which cannot be obtained by the standard multiplicative renormalisation. Our results demonstrate that a rigorous treatment of chiral gauge theories with γ 5 defined in the BMHV scheme at the multi-loop level is possible and that the obtained counterterm action is suitable for computer implementations, allowing automated calculations without ambiguities caused by γ 5.

Full three-loop renormalisation of an abelian chiral gauge theory with non-anticommuting γ 5 in the BMHV scheme

In recent years, theoretical and phenomenological studies with effective field theories have become a trending and prolific line of research in the field of high-energy physics. In order to discuss present and future prospects concerning automated tools in this field, the SMEFT-Tools 2022 workshop was held at the University of Zurich from 14th–16th September 2022. The current document collects and summarizes the content of this workshop.

Switzerland

Jason Aebischer

CERN

Matteo Fael

Spain

Javier Fuentes-Martìn

Anders Thomsen

Javier Virto

Lukas Allwicher

Supratim Bakshi

Croatia

Hermès Bélusca-Maïto

Jorge Blas

Mikael Chala

Juan Criado

Greece

Athanasios Dedes

Renato Fonseca

United States

Angelica Goncalves

Amon Ilakovac

Matthias König

India

Sunando Patra

Marija Mador-Božinović

Mikołaj Misiak

Italy

Víctor Miralles

Ignacy Nałȩcz

United Kingdom

Méril Reboud

Laura Reina

Janusz Rosiek

Michal Ryczkowski

José Santiago

Luca Silvestrini

Peter Stangl

Peter Stoffer

Avelino Vicente

Computing tools for effective field theories

We consider the flavor structure of the <math><msub><mi>S</mi><mn>1</mn></msub></math> leptoquark model and derive conservative constraints on the elements of the left- and right-handed coupling matrices in a number of representative scenarios. We focus on the cases where the muon <math><mi>g</mi><mo>−</mo><mn>2</mn></math> deviation is explained by real muon couplings to either the top-quark or to the charm-quark or to all up-type quarks. The most significant constraints arise from charged lepton flavor-violating decays of the muon and the <math><mi>τ</mi></math> lepton and from the <math><mi>μ</mi><mi>–</mi><mi>e</mi></math> conversion process. Kaon decays and perturbativity provide further constraints. We find strong constraints on almost all coupling matrix elements, implying a very hierarchical matrix structure, where individual entries must differ by at least four orders of magnitude. The flexiblesusy program was used with appropriate model files incorporating the parametrization of the couplings in the up-type mass diagonal basis. The expressions for the leptonic observables were generated and cross-checked with the help of the NPointFunctions extension of the flexiblesusy program.

Uladzimir Khasianevich

Johannes Wünsche

Constraint on scalar leptoquark from low-energy leptonic observables

Thomas Kwasnitza

Alexander Voigt

Addendum to: Improved MSSM Higgs mass calculation using the 3-loop FlexibleEFTHiggs approach including x  t -resummation

In the Grimus-Neufeld model (GNM) the neutrino mass generation from an extended Higgs sector leads to bounds for charged lepton flavor violating (cLFV) processes. Here we update bounds from a previous study by extending the parameter space to a nonvanishing Majorana phase of the Pontecorvo-Maki-Nakagawa-Sakata matrix and to heavier charged Higgs boson masses. Three-body cLFV decays are shown to contribute significantly in the large mass regions, as the boxes are enhanced relatively to photonic diagrams. This is in contrast to the smaller mass region studied before, in which the two-body decays tightly restrict the parameter space. The Majorana phase is shown to change limits within 1 order of magnitude. The tiny seesaw scale is assumed, which makes the cLFV decays in the GNM similar to the cLFV decays in the scotogenic model and the scoto-seesaw models.

Lithuania

Vytautas Dūdėnas

Thomas Gajdosik

Box-enhanced charged lepton flavor violation in the Grimus-Neufeld model