We present the first part of a systematic calculation of the two-loop anomalous dimensions in the low-energy effective field theory (LEFT): the effects at dimension five in the power counting. Our calculation is performed in a basis with generic mass matrices. We employ the algebraically consistent ’t Hooft-Veltman scheme for $\textit{γ}$ and we correct for evanescent as well as chiral-symmetry-breaking effects by including the appropriate finite counterterms. We also provide results for the $\textit{CP}$-even sector in a scheme that coincides with naive dimensional regularization. We discuss two methods to avoid the explicit construction of gauge-variant operators, which in principle are needed for the cancellation of sub-divergences, even in the background-field method. The two methods are consistent with each other and with existing partial results. Our work is a further step towards a complete EFT framework for physics beyond the Standard Model at next-to-leading-logarithmic accuracy.

Switzerland

Luca Naterop

Peter Stoffer

Renormalization-group equations of the LEFT at two loops: dimension-five effects

<p>A precise evaluation of the electroweak contribution to the anomalous magnetic moment of the muon requires control over all aspects of the standard model, ranging from Higgs physics, over multiloop computations for bosonic and (heavy-)fermion diagrams, to nonperturbative effects in the presence of light quarks. Currently, the dominant uncertainties arise from such hadronic effects in the vector–vector–axial-vector three-point function, an improved understanding of which has recently emerged in the context of hadronic light-by-light scattering. Profiting from these developments as well as new perturbative and nonperturbative input for the charm contribution, we obtain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><msubsup><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow><mrow><mi>EW</mi></mrow></msubsup><mo>=</mo><mn>154.4</mn><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo><mo>×</mo><msup><mrow><mn>10</mn></mrow><mrow><mo>−</mo><mn>11</mn></mrow></msup></mrow></math>.</p>

Martin Hoferichter

Austria

Jan Lüdtke

Massimiliano Procura

Improved Evaluation of the Electroweak Contribution to Muon <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>g</mi><mo>−</mo><mn>2</mn></mrow></math>

In Refs. <cross-refs refid='br0010 br0020' id='crs0080'>[1,2]</cross-refs> we provided a complete dispersive evaluation of the hadronic light-by-light (HLbL) contribution to the anomalous magnetic moment of the muon. While the evaluation strategy was developed for the kinematic situation determined by the muon mass, a similar approach also applies to the HLbL corrections to the anomalous magnetic moments of the electron and <italic>&#964;</italic> lepton, shifting the sensitivity in the loop integrals to smaller and larger momenta, respectively. In this Letter, we propagate the corresponding uncertainties of the various contributions, obtaining <math altimg='si1.svg'><msubsup><mrow><mi>a</mi></mrow><mrow><mi>e</mi></mrow><mrow><mtext>HLbL</mtext></mrow></msubsup><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mn>3.51</mn><mo stretchy='false'>(</mo><mn>23</mn><mo stretchy='false'>)</mo><mo>&#215;</mo><msup><mrow><mn>10</mn></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>&#8722;</mo><mn>14</mn></mrow></msup></math> and <math altimg='si2.svg'><msubsup><mrow><mi>a</mi></mrow><mrow><mi>&#964;</mi></mrow><mrow><mtext>HLbL</mtext></mrow></msubsup><mo linebreak='goodbreak' linebreakstyle='after'>=</mo><mn>3.77</mn><mo stretchy='false'>(</mo><mn>29</mn><mo stretchy='false'>)</mo><mo>&#215;</mo><msup><mrow><mn>10</mn></mrow><mrow><mo linebreak='badbreak' linebreakstyle='after'>&#8722;</mo><mn>8</mn></mrow></msup></math>.

Maximilian Zillinger

Hadronic light-by-light scattering in the anomalous magnetic moments of electron and <italic>&#964;</italic>

We derive two types of dispersion relations for the hadronic vector-vector-axial-vector (VVA) correlator: one in generic three-point kinematics with fixed photon virtualities, the second in the kinematic limit of one soft photon. The VVA correlator enters in the electroweak contribution to the muon anomalous magnetic moment and it also emerges as the leading term in the operator-product expansion for hadronic light-by-light scattering in the limit of two large photon momenta. Previously, this correlator was only described in terms of hadronic models. Our new dispersive treatments are in analogy to the established and the newly proposed dispersive approaches to hadronic light-by-light. The VVA correlator allows us to investigate the relation between the two types of dispersion relations in a simpler example and to elucidate the reshuffling of different hadronic contributions in this comparison. As a byproduct, we reduce the theoretical uncertainties on the first-family VVA contribution to the muon anomalous magnetic moment by combining the dispersive representation with both asymptotic and low-energy constraints.

Dispersion relations for the hadronic VVA correlator

United States

Emanuele Mereghetti

Christopher Monahan

Matthew Rizik

Germany

Andrea Shindler

Erratum to: One-loop matching for quark dipole operators in a gradient-flow scheme

In this work, we present an evaluation of subleading effects in the hadronic light-by-light contribution to the anomalous magnetic moment of the muon. Using a recently derived optimized basis, we first study the matching of axial-vector contributions to short-distance constraints at the level of the scalar basis functions, finding that also the tails of the pseudoscalar poles and tensor mesons play a role. We then develop a matching strategy that allows for a combined evaluation of axial-vector and short-distance constraints, supplemented by an estimate of tensor-meson contributions based on simplified assumptions for their transition form factors. Uncertainties are primarily propagated from the axial-vector transition form factors and the variation of the matching scale, but we also consider estimates of the low-energy effect of hadronic states not explicitly included. In total, we obtain $$ {\left.{a}_{\mu}^{\textrm{HLbL}}\right|}_{\textrm{subleading}} $$ = 33 2(7 2) × 10 , which in combination with previously evaluated contributions in the dispersive approach leads to $$ {\left.{a}_{\mu}^{\textrm{HLbL}}\right|}_{\textrm{total}} $$ = 101 9(7 9) × 10

Dispersion relation for hadronic light-by-light scattering: subleading contributions

<p>Hadronic light-by-light (HLbL) scattering defines one of the critical contributions in the Standard Model prediction of the anomalous magnetic moment of the muon. In this Letter, we present a complete evaluation using a dispersive formalism, in which the HLbL tensor is reconstructed from its discontinuities, expressed in terms of simpler hadronic matrix elements that can be extracted from experiment. Profiting from recent developments in the determination of axial-vector transition form factors, short-distance constraints for the HLbL tensor, and the vector–vector–axial-vector correlator, we obtain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi>a</mi><mi>μ</mi><mi>HLbL</mi></msubsup><mo>=</mo><mn>101.9</mn><mo stretchy="false">(</mo><mn>7.9</mn><mo stretchy="false">)</mo><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>11</mn></mrow></msup></math>, which meets the precision requirements set by the final result of the Fermilab experiment.</p>

Complete Dispersive Evaluation of the Hadronic Light-by-Light Contribution to Muon <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>g</mi><mo>−</mo><mn>2</mn></mrow></math>

We present a new basis for the hadronic light-by-light (HLbL) tensor that is optimized for the evaluation of narrow-resonance contributions to HLbL scattering in the anomalous magnetic moment of the muon. As main advantage, kinematic singularities are manifestly absent for pseudoscalar, scalar, and axial-vector states, while the remaining singularities for tensor resonances are minimized, even avoided for special cases, and simple crossing relations among the scalar functions maintained. We scrutinize the properties of this new basis for the scalar-QED pion box, demonstrating that the partial-wave convergence even slightly improves compared to our previous work, and discuss the physical sum rules that ensure basis independence of the HLbL contribution. Finally, we provide explicit expressions for narrow (pseudo-)scalar, axial-vector, and tensor intermediate states in terms of their respective transition form factors.

An optimized basis for hadronic light-by-light scattering

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.

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

Angelica Goncalves

Amon Ilakovac

Matthias König

India

Sunando Patra

Paul Kühler

Marija Mador-Božinović

Poland

Mikołaj Misiak

Italy

Víctor Miralles

Ignacy Nałȩcz

United Kingdom

Méril Reboud

Laura Reina

Janusz Rosiek

Michal Ryczkowski

José Santiago

Luca Silvestrini

Peter Stangl

Dominik Stöckinger

Avelino Vicente

Matthias Weißwange

Computing tools for effective field theories

The low-energy effective field theory below the electroweak scale (LEFT) describes the effects at low energies of both the weak interaction and physics beyond the Standard Model. We study the one-loop renormalization of the LEFT in the ’t Hooft-Veltman scheme, which offers an algebraically consistent definition of the Levi-Civita symbol and γ 5 in dimensional regularization. However, in connection with minimal subtraction this scheme leads to a spurious breaking of chiral symmetry in intermediate steps of the calculation. Based on the ’t Hooft-Veltman prescription, we define a renormalization scheme that restores chiral symmetry by including appropriate finite counterterms. To this end, we extend the physical LEFT operator basis by a complete set of off-shell and one-loop-evanescent operators and we perform the renormalization at one loop. We determine the finite counterterms to the physical parameters that compensate both the insertions of evanescent operators, as well as the chiral-symmetry-breaking terms from the renormalizable part of the Lagrangian in D dimensions. Our results can be applied in next-to-leading-log calculations in the ’t Hooft-Veltman scheme: using our renormalization scheme instead of pure minimal subtraction separates the physical sector from the unphysical evanescent sector and leads to results that are manifestly free of spurious chiral-symmetry-breaking terms.

Low-energy effective field theory below the electroweak scale: one-loop renormalization in the ’t Hooft-Veltman scheme