These terms are of higher order in the power counting used in the mentioned references, but in other scenarios they must be considered and they can be only captured by field redefinitions~[300] or via our explicit matching of scattering amplitudes. Both of these approaches guarantee exactly the invariance of the S-matrix, while equations of motion do not.

As of now, we have successfully applied this method to the SMEFT and other EFTs up to dimension eight including (massless or massive) scalars and gauge bosons, and we are working on extending it to fermions as well~[301].

\section{Summary and outlook}

Effective field theories are basic tools in our description of nature. On the one hand, they let us calculate physical quantities without knowing the underling theory; e.g., the SMEFT provides an efficient way to characterize new physics at the EW scale in terms of coefficients of higher-dimension operators
without knowing the underlying UV completion. On the other hand, even if we want to consider specific NP models, EFTs offer a comprehensive approach to compare with data while, at the same time, providing a way to sum large logarithms via the use of the RG-improved perturbation theory.

However, implementing the EFT approach in an automated way suited for systematic phenomenological analyses is a formidable task. This approach includes, for instance, the identification of higher-dimensional operators appearing in an EFT, the extraction of Feynman rules, the calculation of the matching coefficients between EFTs valid at different energy scales, and the calculation of anomalous dimensions to evolve the WCs between different energy scales. Many of these tasks are not feasible in a reasonable amount of time without computer codes as they can involve hundreds or even thousands of WCs, like in the SMEFT and the LEFT.

The interpretation of experimental data in terms of constraints on effective couplings also requires automation, in all but a few restricted cases.
Extensions of the SM are typically parameterized in terms of the SMEFT. Each WC, however, can appears in many observables.
This is also a challenging task, as it involves computing predictions for a large number of observables and performing global fits with hundreds of experimental constraints.

With this report, we have documented the large efforts in the theory community in automating calculations within EFTs framework.
They have been presented at the SMEFT-Tools 2022 workshop held at the University of Zurich from 14th-16th September 2022.
The milestones reached so far can be summarized as follows:
\begin{itemize}
\item For the two major extensions of the SM, SMEFT and LEFT below the electroweak scale, the anomalous dimensions of dimension-six operators have been calculated to leading order and implemented in user-friendly programs such as \texttt{DsixTools} and \texttt{Wilson}.
They also provide the complete matching between SMEFT and LEFT at tree level and one-loop order.
Higher order operators (dimension sever or higher) can be studied with \texttt{Sym2Int}, which automatically build explicit bases of operators for EFTs, given their fields and symmetries.
Derivation of Feynman rules is enabled by \texttt{SmeftFR}, which also generate
\texttt{UFO} model files suitable for further symbolic or numerical calculation of matrix elements and cross-sections.
\item The viability study of specific BSM scenarios can be largely simplified
by first matching the NP models to the SMEFT and then determining the experimental constraints on the SMEFT WCs.
Such matching between any realization of NP can be performed in an automated way at tree level with tools like \texttt{Matchmakereft}, \texttt{Machete} and \texttt{CoDEx}.
These programs allow to get all effective operators at the EW scale.
Matching at one-loop is possible in many cases. A completely generic implementation at one-loop level in currently under development.
\item Several programs (e.g., \texttt{smelli}, \texttt{HighPT}, \texttt{HEPfit}, \texttt{EOS}) have been developed for phenomenological studies, implementing predictions in the presence of EFT WCs which are then used to constraints NP effects in global fits.
They include observables from a wide range of high-energy physics,
such as flavor physics, physics at hadron colliders,
electroweak precision tests, Higgs physics, and other precision tests of the SM.
\end{itemize}

Many issues and development directions were also addressed during the workshop.
Most probably, the next-to-leading logarithms in SMEFT and LEFT will have to be tackled.
Given the large number of operators, it is inconvenient to consider separately the two-loop running of
the various EFTs separately. In fact, it is more advantageous to calculate the anomalous dimensions for a generic
EFTs with an arbitrary number of real scalars and left-handed fermions, invariant under a generic gauge structure.
Result for NLO running of the SMEFT or LEFT WCs can be derived in a second step by specifying the
field content and the gauge group. An ongoing project aims at evaluating the one-loop RG equations for all the dimension-six operators in such generic EFT.

Another relevant development is the complete classification of contributions to dimension six operators in SMEFT which arise from BSM models only at the one-loop level. Such one-loop matching program is using on-shell matching to avoid the use of (redundant) Green's basis.

Also a consistent treatment of $\gamma_5$ in running of four-fermion operators is necessary since it gives rise to evanescent structures in dimensional regularization. Renormalization of chiral Abelian theory up to two loops has been developed recently in the so called BMHV scheme for $\gamma_5$ and it will be extended to the non-Abelian case in preparation for the application to the SM.

\section*{Acknowledgements}

The organizers are grateful for the generous support from the Pauli Center for Theoretical Studies, the UZH Alumni foundation and the ICCUB-Universitat de Barcelona through the Maria de Maeztu excellence program. Furthermore, we thank Denise Caneve and Monika Röllin for their help with the workshop organization.

\bigskip
Jason Aebischer acknowledges financial support from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement 833280 (FLAY), and from the Swiss National Science Foundation (SNF) under contract 200020-204428.

\bigskip
The work of Matteo Fael was supported by the European Union’s Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No.\ 101065445 -- PHOBIDE.

\bigskip
The work of Javier Fuentes-Mart\'in is supported by the Spanish Ministry of Science and Innovation (MCIN) and the European Union NextGenerationEU/PRTR under grant IJC2020-043549-I, by the MCIN grant PID2019-106087GB-C22, and by the Junta de Andaluc\'ia grants P21\_00199, P18-FR-4314 (FEDER) and FQM 101.

\bigskip
The work of Anders Eller Thomsen has received funding from the Swiss National Science Foundation (SNF) through the Eccellenza Professorial Fellowship “Flavor Physics at the High Energy Frontier” project number 186866.

\bigskip
Javier Virto acknowledges funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk\l odowska-Curie grant agreement No 700525 `NIOBE', from the Spanish MINECO through the ``Ram\'on y Cajal'' program RYC-2017-21870, the “Unit of Excellence Mar\'ia de Maeztu 2020-2023” award to the Institute of Cosmos Sciences (CEX2019-000918-M) and from the grants PID2019-105614GB-C21 and 2017-SGR-92, 2021-SGR-249 (Generalitat de Catalunya).

% Lukas Allwicher

\bigskip
Supratim Das Bakshi and Sunando Kumar Patra are thankful to Joydeep Chakrabortty for numerous suggestions and discussions. Their work is supported by the Department of Science and Technology, Government of India, under the Grant IFA12/PH/34 (INSPIRE Faculty Award);  the Science and Engineering Research Board, Government of India, under the agreement SERB/PHY/2016348 (Early Career Research Award), and Initiation Research Grant, agreement no. IITK/PHY/2015077, by IIT Kanpur. SDB is supported by SRA (Spain) under Grant No.\ PID2019-106087GB-C21 / 10.13039/501100011033 and PID2021-128396NB-100; by the Junta de Andalucía (Spain) under Grants No.\ FQM 101, A-FQM-467-UGR18, and P18-FR-4314 (FEDER).

\bigskip
The work of  Herm\`{e}s B\'{e}lusca-Ma\"{i}to and Paul K\"{u}hler is supported by the Croatian Science Foundation (HRZZ) under the project ``PRECIOUS'' \verb|HRZZ-IP-2016-06-7460|, the project ``Basic interactions and related systems in statistical physics'', as well as the German Science Foundation DFG, grant STO 876/8-1.

\bigskip
The work of Jorge de Blas has been supported by the FEDER/Junta de Andaluc\'ia project grant P18-FRJ-3735. The work of Angelica Goncalves and Laura Reina has been supported by the U.S. Department of Energy under grant DE-SC0010102. Additional support from Italian Ministry of Research (MIUR) under grant PRIN 20172LNEEZ is also acknowledged.

\bigskip
The work of Mikael Chala is supported by SRA under grants PID2019-106087GB-C21 and PID2021-128396NB-I00, by the Junta de Andaluc\'ia grants FQM 101, A-FQM-211-UGR18, P21-00199 and P18-FR-4314 (FEDER), as well as by the Spanish MINECO under the Ram\'on y Cajal programme.

% Juan Carlos Criado

% Athanasios Dedes

\bigskip
Renato Fonseca acknowledges the financial support from the Ministerio de Economía y Competitividad through grant number PID2019-106087GB-C22, and from the Junta de Andalucía through grants number P18-FR-4314 (FEDER) and P21\_00199.

% Matthias König

\bigskip
The contribution of Miko{\l}aj Misiak and Ignacy Na{\l\c{e}}cz is part of a project that is carried out together with Jason Aebischer, Patryk Mieszkalski and Nudzeim Selimovi\'c. Miko{\l}aj Misiak has been partially supported by the National Science Center, Poland, under the research project 2020/37/B/ST2/02746.

\bigskip
The work of Michal Ryczkowski was supported in part by Polish National Science Center under research grant DEC-2019/35/B/ST2/02008. The research of  Janusz Rosiek has received funding from the Norwegian Financial Mechanism for years 2014-2021, under the grant no 2019/34/H/ST2/00707.

\bigskip
The work of José Santiago is supported by grants FQM 101, P18-FR-4314 (Junta de Andalucía, Fondos FEDER), and PID2019-106087GB-C22 (AEI/10.13039/501100011033).

% Peter Stangl

\bigskip
Peter Stoffer gratefully acknowledges financial support by the Swiss National Science Foundation (Project No.~PCEFP2\_194272).

\bigskip
The work of Avelino Vicente is supported by the Spanish grants PID2020-113775GB-I00 (AEI/10.13039/501100011033) and CIPROM/2021/054 (Generalitat Valenciana), and from MINECO through the Ramón y Cajal contract RYC2018-025795-I.

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