In this work we discuss the connection between Feynman integrals and Fox functions. Illustrative examples are given.

This work is based on a bottom{-}up approach to the standard{-}model effective field theory (SMEFT), resulting in an equiprobable space of Wilson coefficients. The randomly generated Wilson coefficients of the SMEFT (in the Warsaw basis) are treated as pseudo-data and, for each observable, the corresponding probability density function is computed. The goal has been to understand how large are the deviations from the SM once the SMEFT scale and the range of the Wilson coefficients are selected. Correlations between different observables are also discussed.

Dedicated to the memory of Prof. Veltman, one of the founding fathers of our discipline: his legacy lives on. Many times we have to turn back and follow his footprints to find the right path. After reviewing general aspects of high energy physics where he gave a seminal contribution we will introduce recent developments in the standard model effective field theory, showing how the whole movement from renormalization to predictions plays from Veltman to SMEFT.

In this work we address three questions: can we successfully describe (observed) deviations from the standard model in the SMEFT language? Can we learn something about the underlying, beyond the standard model, physics using the SMEFT language? If no deviation is observed, how to proceed? Given the myriad of viable BSM options with extended scalar sectors, we suggest a widespread use of SMEFT not just as a global fitting tool (that could miss out on deviations from extended scalar sectors) but also as a bookkeeping framework in which the results from SMEFT fits to individual observables are provided, reported, and archived in a consistent way. The compatibility of such individual results can then be assessed in the light of BSM models with extended scalar sectors.

The increasing interest in the phenomenology of the Standard Model Effective Field Theory (SMEFT), has led to the development of a wide spectrum of public codes which implement automatically different aspects of the SMEFT for phenomenological applications. In order to discuss the present and future of such efforts, the "SMEFT-Tools 2019" Workshop was held at the IPPP Durham on the 12th-14th June 2019. Here we collect and summarize the contents of this workshop.

There is increasing need to assess the impact and the interpretation of dim = 6 and dim = 8 operators within the context of the Standard Model Effective Field Theory (SMEFT). The observational and mathematical consistency of a construct based on dim = 6 and dim = 8 operators is critically examined in the light of known theoretical results. The discussion is based on a general dim = 4 theory X and its effective extension, XEFT; it includes elimination of redundant operators and their higher order compensation, SMEFT in comparison with ultraviolet completions incorporating a proliferation of scalar and mixings, canonical normalization of effective field theories, gauge invariance and gauge fixing, role of tadpoles when constructing XEFT at NLO, heavy-light contributions to the low energy limit of theories containing bosons and fermions, one-loop matching, EFT fits and their interpretation and effective field theory interpretation of derivative-coupled field theories.

The behavior of scattering amplitudes in the vicinity of a physical-region Landau singularity is considered. The impact on LHC processes is discussed.

This report summarises some of the activities of the HiggsTools initial training network working group in the period 2015-2017. The main goal of this working group was to produce a document discussing various aspects of state-of-the-art Higgs physics at the large hadron collider (LHC) in a pedagogic manner The first part of the report is devoted to a description of phenomenological searches for new physics (NP) at the LHC. All of the available studies of the couplings of the new resonance discovered in 2012 by the ATLAS and CMS experiments (Aad et al (ATLAS Collaboration) 2012 Phys. Lett. B 716 1-29; Chatrchyan et al (CMS Collaboration) 2012 Phys. Lett. B 716 30-61) conclude that it is compatible with the Higgs boson of the standard model (SM) within present precision. So far the LHC experiments have given no direct evidence for any physical phenomena that cannot be described by the SM. As the experimental measurements become more and more precise, there is a pressing need for a consistent framework in which deviations from the SM predictions can be computed precisely. Such a framework should be applicable to measurements in all sectors of particle physics, not only LHC Higgs measurements but also electroweak precision data, etc. We critically review the use of the k-framework, fiducial and simplified template cross sections, effective field theories, pseudoobservables and phenomenological Lagrangians. Some of the concepts presented here are well known and were used already at the time of the large electron-positron collider (LEP) experiment. However, after years of theoretical and experimental development, these techniques have been refined, and we describe new tools that have been introduced in order to improve the comparison between theory and experimental data. In the second part of the report, we propose Phi(eta)* as a new and complementary observable for studying Higgs boson production at large transverse momentum in the case where the Higgs boson decays to two photons. The Phi(eta)* variable depends on measurements of the angular directions and rapidities of the two Higgs decay products rather than the energies, and exploits the information provided by the calorimeter in the detector. We show that, even without tracking information, the experimental resolution for Phi(eta)* is better than that of the transverse momentum of the photon pair, particularly at low transverse momentum. We make a detailed study of the phenomenology of the Phi(eta)* variable, contrasting the behaviour with the Higgs transverse momentum distribution using a variety of theoretical tools including event generators and fixed order perturbative computations. We consider the theoretical uncertainties associated with both p TH and Phi(eta)* distributions. Unlike the transverse momentum distribution, the Phi(eta)* distribution is well predicted using the Higgs effective field theory in which the top quark is integrated out-even at large values of Phi(eta)*-thereby making this a better observable for extracting the parameters of the Higgs interaction. In contrast, the potential of the Phi(eta)* distribution as a probe of NP is rather limited, since although the overall rate is affected by the presence of additional heavy fields, the shape of the Phi(eta)* distribution is relatively insensitive to heavy particle thresholds.

This Report summarizes some of the activities of the HiggsTools Initial Training Network working group in the period 2015-2017. The main goal of this working group was to produce a document discussing various aspects of state-of-the-art Higgs physics at the Large Hadron Collider (LHC) in a pedagogic manner. The first part of the Report is devoted to a description of phenomenological searches for New Physics at the LHC. As the experimental measurements become more and more precise, there is a pressing need for a consistent framework in which deviations from the SM predictions can be computed precisely. We critically review the use of the \k{appa}-framework, fiducial and simplified template cross sections, effective field theories, pseudo-observables and phenomenological Lagrangians. In the second part of the Report, we propose $\varphi_{\eta}^*$ as a new and complementary observable for studying Higgs boson production at large transverse momentum in the case where the Higgs boson decays to two photons. We make a detailed study of the phenomenology of the $\varphi_{\eta}^*$ variable, contrasting the behaviour with the Higgs transverse momentum distribution using a variety of theoretical tools including event generators and fixed order perturbative computations.

This report will review the Higgs boson properties: the mass, the total width and the couplings to fermions and bosons. The measurements have been performed with the data collected in 2011 and 2012 at the LHC accelerator at CERN by the ATLAS and CMS experiments. Theoretical frameworks to search for new physics are also introduced and discussed.

Debate topic for effective field theory (EFT) is the choice of a “basis” for \({\rm dim}= 6\) operators. Clearly all bases are equivalent as long as they are a “basis”, containing a minimal set of operators after the use of equations of motion and respecting the \(SU(3)\times SU(2)\times U(1)\) gauge invariance. From a more formal point of view a basis is characterized by its closure with respect to renormalization. Equivalence of bases should always be understood as a statement for the S-matrix and not for the Lagrangian, as dictated by the equivalence theorem. Any phenomenological approach that misses one of these ingredients is still acceptable for a preliminar analysis, as long as it does not pretend to be an EFT. Here we revisit the equivalence theorem and its consequences for EFT when two sets of higher-dimensional operators are connected by a set of non-linear, non-invariant, field reparametrizations.

Multiple elliptic polylogarithms can be written as (multiple) integrals of products of basic hypergeometric functions. The latter are computable, to arbitrary precision, using a q-difference equation and q-contiguous relations.

During Run 1 of LHC Standard Model deviations have been analyzed in the context of the socalled kappa framework.Perspectives on a new framework for improving the experimental strategy are briefly discussed.

We review the status of calculations in the Standard Model Effective Field Theory (SMEFT) beyond leading order (LO). Improving the SMEFT beyond LO allows theoretical errors to be characterized and reduced when considering SMEFT interpretations of the data, which is essential considering the improving experimental precision at LHC. Next to leading order results also allow a more consistent analysis of measurements with different effective scales in the SMEFT. Going beyond LO is clearly important in the event that deviations from the SM are large enough that experimental indications of physics beyond the SM emerge. We discuss a consistent and well defined approach to LO in the SMEFT, so that the improvement to NLO is straightforward. We discuss the basic issues involved in improving calculations to NLO in the SMEFT, and review the advances in this direction that have been achieved to date.

Debate topic for Effective Field Theory (EFT) is the choice of a "basis" for $\mrdim = 6$ operators Clearly all bases are equivalent as long as they are a "basis", containing a minimal set of operators after the use of equations of motion and respecting gauge invariance. From a more formal point of view a basis is characterized by its closure with respect to renormalization. Equivalence of bases should always be understood as a statement for the S-matrix and not for the Lagrangian, as dictated by the equivalence theorem. Any phenomenological approach that misses one of these ingredients is still acceptable for a preliminar analysis, as long as it does not pretend to be an EFT. Here we revisit the equivalence theorem and its consequences for EFT when two sets of higher dimensional operators are connected by a set of non-linear, noninvariant, field reparametrizations.

A set of constructs, definitions, and propositions that present a systematic view of the Standard Model Effective Field Theory (SMEFT), i.e. how the influence of higher energy processes is localizable in a few structural properties which can be captured by a handful of Wilson coefficients.

After the LHC Run 1, the standard model (SM) of particle physics has been completed. Yet, despite its successes, the SM has shortcomings vis-à-vis cosmological and other observations. At the same time, while the LHC restarts for Run 2 at 13 TeV, there is presently a lack of direct evidence for new physics phenomena at the accelerator energy frontier. From this state of affairs arises the need for a consistent theoretical framework in which deviations from the SM predictions can be calculated and compared to precision measurements. Such a framework should be able to comprehensively make use of all measurements in all sectors of particle physics, including LHC Higgs measurements, past electroweak precision data, electric dipole moment, g−2, penguins and flavor physics, neutrino scattering, deep inelastic scattering, low-energy e+e− scattering, mass measurements, and any search for physics beyond the SM. By simultaneously describing all existing measurements, this framework then becomes an intermediate step, pointing us toward the next SM, and hopefully revealing the underlying symmetries. We review the role that the standard model effective field theory (SMEFT) could play in this context, as a consistent, complete, and calculable generalization of the SM in the absence of light new physics. We discuss the relationship of the SMEFT with the existing kappa-framework for Higgs boson couplings characterization and the use of pseudo-observables, that insulate experimental results from refinements due to ever-improving calculations. The LHC context, as well as that of previous and future accelerators and experiments, is also addressed.

The integration of heavy scalar fields is discussed in a class of BSM models, containing more that one representation for scalars and with mixing. The interplay between integrating out heavy scalars and the Standard Model decoupling limit is examined. In general, the latter cannot be obtained in terms of only one large scale and can only be achieved by imposing further assumptions on the couplings. Systematic low-energy expansions are derived in the more general, non-decoupling scenario, including mixed tree-loop and mixed heavy-light generated operators. The number of local operators is larger than the one usually reported in the literature.

This review summarizes the results of the activities which have taken place in 2014 within the Standard Model Working Group of the “What Next” Workshop organized by INFN, Italy. We present a framework, general questions, and some indications of possible answers on the main issue for Standard Model physics in the LHC era and in view of possible future accelerators.

A consistent framework for studying Standard Model deviations is developed. It assumes that New Physics becomes relevant at some scale beyond the present experimental reach and uses the Effective Field Theory approach by adding higher-dimensional operators to the Standard Model Lagrangian and by computing relevant processes at the next-to-leading order, extending the original κ -framework. The generalized κ -framework provides a useful technical tool to decompose amplitudes at NLO accuracy into a sum of well defined gauge-invariant sub components.