Search

We present the results of the "Fast Calorimeter Simulation Challenge 2022" - the CaloChallenge. We study state-of-the-art generative models on four calorimeter shower datasets of increasing dimensionality, ranging from a few hundred voxels to a few tens of thousand voxels. The 31 individual submissions span a wide range of current popular generative architectures, including Variational AutoEncoders (VAEs), Generative Adversarial Networks (GANs), Normalizing Flows, Diffusion models, and models based on Conditional Flow Matching. We compare all submissions in terms of quality of generated calorimeter showers, as well as shower generation time and model size. To assess the quality we use a broad range of different metrics including differences in 1-dimensional histograms of observables, KPD/FPD scores, AUCs of binary classifiers, and the log-posterior of a multiclass classifier. The results of the CaloChallenge provide the most complete and comprehensive survey of cutting-edge approaches to calorimeter fast simulation to date. In addition, our work provides a uniquely detailed perspective on the important problem of how to evaluate generative models. As such, the results presented here should be applicable for other domains that use generative AI and require fast and faithful generation of samples in a large phase space., Comment: 204 pages, 100+ figures, 30+ tables

In the quest to build generative surrogate models as computationally efficient alternatives to rule-based simulations, the quality of the generated samples remains a crucial frontier. So far, normalizing flows have been among the models with the best fidelity. However, as the latent space in such models is required to have the same dimensionality as the data space, scaling up normalizing flows to high dimensional datasets is not straightforward. The prior L2LFlows approach successfully used a series of separate normalizing flows and sequence of conditioning steps to circumvent this problem. In this work, we extend L2LFlows to simulate showers with a 9-times larger profile in the lateral direction. To achieve this, we introduce convolutional layers and U-Net-type connections, move from masked autoregressive flows to coupling layers, and demonstrate the successful modelling of showers in the ILD Electromagnetic Calorimeter as well as Dataset 3 from the public CaloChallenge dataset.

There have been many applications of deep neural networks to detector calibrations and a growing number of studies that propose deep generative models as automated fast detector simulators. We show that these two tasks can be unified by using maximum likelihood estimation (MLE) from conditional generative models for energy regression. Unlike direct regression techniques, the MLE approach is prior-independent and non-Gaussian resolutions can be determined from the shape of the likelihood near the maximum. Using an ATLAS-like calorimeter simulation, we demonstrate this concept in the context of calorimeter energy calibration., Comment: 12 pages, 7 figures

Recently, several normalizing flow-based deep generative models have been proposed to accelerate the simulation of calorimeter showers. Using CaloFlow as an example, we show that these models can simultaneously perform unsupervised anomaly detection with no additional training cost. As a demonstration, we consider electromagnetic showers initiated by one (background) or multiple (signal) photons. The CaloFlow model is designed to generate single photon showers, but it also provides access to the shower likelihood. We use this likelihood as an anomaly score and study the showers tagged as being unlikely. As expected, the tagger struggles when the signal photons are nearly collinear, but is otherwise effective. This approach is complementary to a supervised classifier trained on only specific signal models using the same low-level calorimeter inputs. While the supervised classifier is also highly effective at unseen signal models, the unsupervised method is more sensitive in certain regions and thus we expect that the ultimate performance will require a combination of these approaches., Comment: 14 pages, 8 figures

In modern collider experiments, the quest to explore fundamental interactions between elementary particles has reached unparalleled levels of precision. Signatures from particle physics detectors are low-level objects (such as energy depositions or tracks) encoding the physics of collisions (the final state particles of hard scattering interactions). The complete simulation of them in a detector is a computational and storage-intensive task. To address this computational bottleneck in particle physics, alternative approaches have been developed, introducing additional assumptions and trade off accuracy for speed.The field has seen a surge in interest in surrogate modeling the detector simulation, fueled by the advancements in deep generative models. These models aim to generate responses that are statistically identical to the observed data. In this paper, we conduct a comprehensive and exhaustive taxonomic review of the existing literature on the simulation of detector signatures from both methodological and application-wise perspectives. Initially, we formulate the problem of detector signature simulation and discuss its different variations that can be unified. Next, we classify the state-of-the-art methods into five distinct categories based on their underlying model architectures, summarizing their respective generation strategies. Finally, we shed light on the challenges and opportunities that lie ahead in detector signature simulation, setting the stage for future research and development., Comment: v2: Accepted in Reviews in Physics

Whenever invertible generative networks are needed for LHC physics, normalizing flows show excellent performance. In this work, we investigate their performance for fast calorimeter shower simulations with increasing phase space dimension. We use fast and expressive coupling spline transformations applied to the CaloChallenge datasets. In addition to the base flow architecture we also employ a VAE to compress the dimensionality and train a generative network in the latent space. We evaluate our networks on several metrics, including high-level features, classifiers, and generation timing. Our findings demonstrate that invertible neural networks have competitive performance when compared to autoregressive flows, while being substantially faster during generation., Comment: 34 pages, 14 figures, 7 tables, journal version

In many well-motivated models of the electroweak scale, cascade decays of new particles can result in highly boosted hadronic resonances (e.g. $Z/W/h$). This can make these models rich and promising targets for recently developed resonant anomaly detection methods powered by modern machine learning. We demonstrate this using the state-of-the-art CATHODE method applied to supersymmetry scenarios with gluino pair production. We show that CATHODE, despite being model-agnostic, is nevertheless competitive with dedicated cut-based searches, while simultaneously covering a much wider region of parameter space. The gluino events also populate the tails of the missing energy and $H_T$ distributions, making this a novel combination of resonant and tail-based anomaly detection., Comment: 13 pages, 15 figures

Machine learning--based anomaly detection (AD) methods are promising tools for extending the coverage of searches for physics beyond the Standard Model (BSM). One class of AD methods that has received significant attention is resonant anomaly detection, where the BSM is assumed to be localized in at least one known variable. While there have been many methods proposed to identify such a BSM signal that make use of simulated or detected data in different ways, there has not yet been a study of the methods' complementarity. To this end, we address two questions. First, in the absence of any signal, do different methods pick the same events as signal-like? If not, then we can significantly reduce the false-positive rate by comparing different methods on the same dataset. Second, if there is a signal, are different methods fully correlated? Even if their maximum performance is the same, since we do not know how much signal is present, it may be beneficial to combine approaches. Using the Large Hadron Collider (LHC) Olympics dataset, we provide quantitative answers to these questions. We find that there are significant gains possible by combining multiple methods, which will strengthen the search program at the LHC and beyond., Comment: 27 pages, 21 figures. Updated with revisions for journal acceptance

Well-trained classifiers and their complete weight distributions provide us with a well-motivated and practicable method to test generative networks in particle physics. We illustrate their benefits for distribution-shifted jets, calorimeter showers, and reconstruction-level events. In all cases, the classifier weights make for a powerful test of the generative network, identify potential problems in the density estimation, relate them to the underlying physics, and tie in with a comprehensive precision and uncertainty treatment for generative networks., Comment: 32 pages, 19 figures

Simulating particle detector response is the single most expensive step in the Large Hadron Collider computational pipeline. Recently it was shown that normalizing flows can accelerate this process while achieving unprecedented levels of accuracy, but scaling this approach up to higher resolutions relevant for future detector upgrades leads to prohibitive memory constraints. To overcome this problem, we introduce Inductive CaloFlow (iCaloFlow), a framework for fast detector simulation based on an inductive series of normalizing flows trained on the pattern of energy depositions in pairs of consecutive calorimeter layers. We further use a teacher-student distillation to increase sampling speed without loss of expressivity. As we demonstrate with Datasets 2 and 3 of the CaloChallenge2022, iCaloFlow can realize the potential of normalizing flows in performing fast, high-fidelity simulation on detector geometries that are ~ 10 - 100 times higher granularity than previously considered., Comment: 19 pages, 15 figures; v2: title changed, matches published version

We explore the use of normalizing flows to emulate Monte Carlo detector simulations of photon showers in a high-granularity electromagnetic calorimeter prototype for the International Large Detector (ILD). Our proposed method -- which we refer to as "Layer-to-Layer-Flows" (L$2$LFlows) -- is an evolution of the CaloFlow architecture adapted to a higher-dimensional setting (30 layers of $10\times 10$ voxels each). The main innovation of L$2$LFlows consists of introducing $30$ separate normalizing flows, one for each layer of the calorimeter, where each flow is conditioned on the previous five layers in order to learn the layer-to-layer correlations. We compare our results to the BIB-AE, a state-of-the-art generative network trained on the same dataset and find our model has a significantly improved fidelity., Comment: v2: 28 pages, 13 figures; matches version accepted for publication in JINST. Neither SISSA Medialab Srl nor IOP Publishing Ltd is responsible for any errors or omissions in this version of the manuscript or any version derived from it. Published version available via DOI

Theory predictions for the LHC require precise numerical phase-space integration and generation of unweighted events. We combine machine-learned multi-channel weights with a normalizing flow for importance sampling, to improve classical methods for numerical integration. We develop an efficient bi-directional setup based on an invertible network, combining online and buffered training for potentially expensive integrands. We illustrate our method for the Drell-Yan process with an additional narrow resonance., Comment: 33 pages, 15 figures, minor fixes to v1

Modern machine learning is transforming particle physics fast, bullying its way into our numerical tool box. For young researchers it is crucial to stay on top of this development, which means applying cutting-edge methods and tools to the full range of LHC physics problems. These lecture notes lead students with basic knowledge of particle physics and significant enthusiasm for machine learning to relevant applications. They start with an LHC-specific motivation and a non-standard introduction to neural networks and then cover classification, unsupervised classification, generative networks, and inverse problems. Two themes defining much of the discussion are well-defined loss functions and uncertainty-aware networks. As part of the applications, the notes include some aspects of theoretical LHC physics. All examples are chosen from particle physics publications of the last few years., Comment: Further expanded v3, we very much appreciate feedback

CaloFlow is a new and promising approach to fast calorimeter simulation based on normalizing flows. Applying CaloFlow to the photon and charged pion Geant4 showers of Dataset 1 of the Fast Calorimeter Simulation Challenge 2022, we show how it can produce high-fidelity samples with a sampling time that is several orders of magnitude faster than Geant4. We demonstrate the fidelity of the samples using calorimeter shower images, histograms of high-level features, and aggregate metrics such as a classifier trained to distinguish CaloFlow from Geant4 samples., Comment: 36 pages, 21 figures, v3: match published version

The full optimization of the design and operation of instruments whose functioning relies on the interaction of radiation with matter is a super-human task, given the large dimensionality of the space of possible choices for geometry, detection technology, materials, data-acquisition, and information-extraction techniques, and the interdependence of the related parameters. On the other hand, massive potential gains in performance over standard, "experience-driven" layouts are in principle within our reach if an objective function fully aligned with the final goals of the instrument is maximized by means of a systematic search of the configuration space. The stochastic nature of the involved quantum processes make the modeling of these systems an intractable problem from a classical statistics point of view, yet the construction of a fully differentiable pipeline and the use of deep learning techniques may allow the simultaneous optimization of all design parameters. In this document we lay down our plans for the design of a modular and versatile modeling tool for the end-to-end optimization of complex instruments for particle physics experiments as well as industrial and medical applications that share the detection of radiation as their basic ingredient. We consider a selected set of use cases to highlight the specific needs of different applications., Comment: 109 pages, 32 figures. To be submitted to Reviews in Physics

The computational cost for high energy physics detector simulation in future experimental facilities is going to exceed the current available resources. To overcome this challenge, new ideas on surrogate models using machine learning methods are being explored to replace computationally expensive components. Additionally, differentiable programming has been proposed as a complementary approach, providing controllable and scalable simulation routines. In this document, new and ongoing efforts for surrogate models and differential programming applied to detector simulation are discussed in the context of the 2021 Particle Physics Community Planning Exercise (`Snowmass')., Comment: contribution to Snowmass 2021

First-principle simulations are at the heart of the high-energy physics research program. They link the vast data output of multi-purpose detectors with fundamental theory predictions and interpretation. This review illustrates a wide range of applications of modern machine learning to event generation and simulation-based inference, including conceptional developments driven by the specific requirements of particle physics. New ideas and tools developed at the interface of particle physics and machine learning will improve the speed and precision of forward simulations, handle the complexity of collision data, and enhance inference as an inverse simulation problem., Comment: Review article based on a Snowmass 2021 contribution

Recently, we introduced CaloFlow, a high-fidelity generative model for GEANT4 calorimeter shower emulation based on normalizing flows. Here, we present CaloFlow v2, an improvement on our original framework that speeds up shower generation by a further factor of 500 relative to the original. The improvement is based on a technique called Probability Density Distillation, originally developed for speech synthesis in the ML literature, and which we develop further by introducing a set of powerful new loss terms. We demonstrate that CaloFlow v2 preserves the same high fidelity of the original using qualitative (average images, histograms of high level features) and quantitative (classifier metric between GEANT4 and generated samples) measures. The result is a generative model for calorimeter showers that matches the state-of-the-art in speed (a factor of $10^4$ faster than GEANT4) and greatly surpasses the previous state-of-the-art in fidelity., Comment: 24 pages, 15 figures, 4 tables; v2: matches accepted version

We propose a new model-agnostic search strategy for physics beyond the standard model (BSM) at the LHC, based on a novel application of neural density estimation to anomaly detection. Our approach, which we call Classifying Anomalies THrough Outer Density Estimation (CATHODE), assumes the BSM signal is localized in a signal region (defined e.g. using invariant mass). By training a conditional density estimator on a collection of additional features outside the signal region, interpolating it into the signal region, and sampling from it, we produce a collection of events that follow the background model. We can then train a classifier to distinguish the data from the events sampled from the background model, thereby approaching the optimal anomaly detector. Using the LHC Olympics R&D dataset, we demonstrate that CATHODE nearly saturates the best possible performance, and significantly outperforms other approaches that aim to enhance the bump hunt (CWoLa Hunting and ANODE). Finally, we demonstrate that CATHODE is very robust against correlations between the features and maintains nearly-optimal performance even in this more challenging setting., Comment: 17 pages, 12 figures; v2: minor updates; v3 (published version): added study of background sculpting and minor fixes

We introduce CaloFlow, a fast detector simulation framework based on normalizing flows. For the first time, we demonstrate that normalizing flows can reproduce many-channel calorimeter showers with extremely high fidelity, providing a fresh alternative to computationally expensive GEANT4 simulations, as well as other state-of-the-art fast simulation frameworks based on GANs and VAEs. Besides the usual histograms of physical features and images of calorimeter showers, we introduce a new metric for judging the quality of generative modeling: the performance of a classifier trained to differentiate real from generated images. We show that GAN-generated images can be identified by the classifier with nearly 100% accuracy, while images generated from CaloFlow are better able to fool the classifier. More broadly, normalizing flows offer several advantages compared to other state-of-the-art approaches (GANs and VAEs), including: tractable likelihoods; stable and convergent training; and principled model selection. Normalizing flows also provide a bijective mapping between data and the latent space, which could have other applications beyond simulation, for example, to detector unfolding., Comment: 33 pages, 19 figures, 5 tables; v2: improved handling of datasets, conclusions unchanged; v3: matches accepted version

Electroweak symmetry non-restoration up to high temperatures well above the electroweak scale offers new alternatives for baryogenesis. We propose a new approach for electroweak symmetry non-restoration via an inert Higgs sector that couples to the Standard Model Higgs as well as an extended scalar singlet sector. We implement renormalization group improvements and thermal resummation, necessary to evaluate the effective potential spanning over a broad range of energy scales and temperatures. We present examples of benchmark scenarios that allow for electroweak symmetry non-restoration all the way up to hundreds of TeV temperatures, and also feature suppressed sphaleron washout factors down to the electroweak scale. Our method for transmitting the Standard Model broken electroweak symmetry to an inert Higgs sector has several intriguing implications for (electroweak) baryogenesis, early universe thermal histories, and can be scrutinized through Higgs physics phenomenology and electroweak precision measurements at the HL-LHC., Comment: 32 pages, 6 figures, 4 tables; v2: matches published version

We present a novel integrator based on normalizing flows which can be used to improve the unweighting efficiency of Monte-Carlo event generators for collider physics simulations. In contrast to machine learning approaches based on surrogate models, our method generates the correct result even if the underlying neural networks are not optimally trained. We exemplify the new strategy using the example of Drell-Yan type processes at the LHC, both at leading and partially at next-to-leading order QCD., Comment: 9 pages, 2 figures, 2 tables; v2: matches Journal version

In many fields of science, high-dimensional integration is required. Numerical methods have been developed to evaluate these complex integrals. We introduce the code i-flow, a python package that performs high-dimensional numerical integration utilizing normalizing flows. Normalizing flows are machine-learned, bijective mappings between two distributions. i-flow can also be used to sample random points according to complicated distributions in high dimensions. We compare i-flow to other algorithms for high-dimensional numerical integration and show that i-flow outperforms them for high dimensional correlated integrals. The i-flow code is publicly available on gitlab at https://gitlab.com/i-flow/i-flow., Comment: 21 pages, 5 figures, 4 tables; v2: improved presentation and discussion, matches published version. Mach. Learn.: Sci. Technol (2020)

Due to the mass gap between the Standard Model and possible New Physics states, electroweak effective approaches are appropriate. Although a linear realization of the electroweak symmetry breaking with the Higgs forming a doublet together with the Goldstone bosons of the EWSB is a first possibility (SMEFT), we adopt the more general non-linear realization, where the Higgs is a singlet with independent couplings (EWET, HEFT or EWChL). We present the effective Lagrangian at low energies (the EWET, with only the SM fields) and at high energies (the resonance theory, with also a set of resonances). Taking into account the high scale of these resonances, their experimental searches seem to be more accessible by considering their imprints at low-energies, i.e., their imprints in the Low Energy Constants (LECs) of the EWET at energies lower than the resonance masses. We give some examples of these phenomenological connections., Comment: 6 pages, 3 figures. Talk given at European Physical Society Conference on High Energy Physics (EPS-HEP2019), 10-17 July (2019), Ghent (Belgium)

The electroweak sector of the Standard Model can be formulated in a way similar to Chiral Perturbation Theory (ChPT), but extended by a singlet scalar. The resulting effective field theory (EFT) is called Higgs-Electroweak Chiral Lagrangian (EWCh$\mathcal{L}$) and is the most general approach to new physics in the Higgs sector. It solely assumes the pattern of symmetry breaking leading to the three electroweak Goldstone bosons (i.e. massive $W$ and $Z$) and the existence of a Higgs-like scalar particle. The power counting of the EWCh$\mathcal{L}$ is given by a generalization of the momentum expansion of ChPT. It is connected to a loop expansion, making the theory renormalizable order by order in the EFT. I will briefly review the construction of the EWCh$\mathcal{L}$ and its power counting. Then, I will discuss the complete one-loop renormalization of the EWCh$\mathcal{L}$ employing the background-field method and the super-heat-kernel expansion. This computation confirms the power counting assumptions, is consistent with the completeness of the operator basis, and reproduces known results of subsectors in the appropriate limits., Comment: 8 pages; Proceedings of The 9th International workshop on Chiral Dynamics, 17-21 September 2018, Durham, NC, USA

Using background-field method and super-heat-kernel expansion, we derive a master formula for the one-loop UV divergences of the bosonic dimension-6 operators in Standard Model Effective Field Theory (SMEFT). This approach reduces the calculation of all the UV divergences to algebraic manipulations. Using this formula we corroborate results in the literature for the one-loop anomalous dimension matrix of SMEFT obtained via diagrammatic methods, considering contributions from the operators $X^3, \phi^6, \phi^4 D^2, X^2 \phi^2$ of the Warsaw basis. The formula is derived in a general way and can be applied to other quantum field theories as well., Comment: 42 pages, no figures

Taking into account the negative results of direct searches for beyond the Standard Model fields and the consequent mass gap between Standard Model and possible unknown states, the use of electroweak effective theories is justified. Whereas at low energies we consider a non-linear realization of the electroweak symmetry breaking with a singlet Higgs and a strongly-coupled ultraviolet completion, at higher energies the known particles are assumed to be coupled to heavy states: bosonic fields with $J^P=0^\pm$ and $J^P=1^\pm$ (in electroweak triplets or singlets and in QCD octets or singlets) and fermionic states with $J=\frac{1}{2}$ (in electroweak doublets and in QCD triplets or singlets). By integrating out these heavy resonances, the pattern of next-to-leading order low-energy constants among the light fields can be studied. A phenomenological study trying to estimate the scale of these resonances is also shown., Comment: 4 pages. Talk given at The 39th International Conference on High Energy Physics (ICHEP2018), 4-11 July (2018), Seoul (Korea)

We analyze heavy states from generic ultraviolet completions of the Standard Model in a model-independent way and investigate their implications on the low-energy couplings of the electroweak effective theory. We build a general effective Lagrangian, implementing the electroweak symmetry breaking $SU(2)_L\otimes SU(2)_R\to SU(2)_{L+R}$ with a non-linear Nambu-Goldstone realization, which couples the known particles to the heavy states. We generalize the formalism developed in previous works~[1,2] to include colored resonances, both of bosonic and fermionic type. We study bosonic heavy states with $J^P=0^\pm$ and $J^P=1^\pm$, in singlet or triplet $SU(2)_{L+R}$ representations and in singlet or octet representations of $SU(3)_C$, and fermionic resonances with $J=\frac{1}{2}$ that are electroweak doublets and QCD triplets or singlets. Integrating out the heavy scales, we determine the complete pattern of low-energy couplings at the lowest non-trivial order. Some specific types of (strongly- and weakly-coupled) ultraviolet completions are discussed to illustrate the generality of our approach and to make contact with current experimental searches., Comment: 51 pages, 2 figures, 12 tables; v2: matches Journal version

We perform a Bayesian statistical analysis of the constraints on the nonlinear Effective Theory given by the Higgs electroweak chiral Lagrangian. We obtain bounds on the effective coefficients entering in Higgs observables at the leading order, using all available Higgs-boson signal strengths from the LHC runs 1 and 2. Using a prior dependence study of the solutions, we discuss the results within the context of natural-sized Wilson coefficients. We further study the expected sensitivities to the different Wilson coefficients at various possible future colliders. Finally, we interpret our results in terms of some minimal composite Higgs models., Comment: 45 pages, 9 figures, 8 tables; v2: updated references, experimental input now includes data of Moriond 2018, extended discussion of projection to future colliders; v3: added Appendix, matches Journal version

If the electroweak phase transition (EWPT) is of strongly first order due to higher dimensional operators, the scale of new physics generating them is at the TeV scale or below. In this case the effective-field theory (EFT) neglecting operators of dimension higher than six may overlook terms that are relevant for the EWPT analysis. In this article we study the EWPT in the EFT to dimension eight. We estimate the reach of the future gravitational wave observatory LISA for probing the region in which the EWPT is strongly first order and compare it with the capabilities of the Higgs measurements via double-Higgs production at current and future colliders. We also match different UV models to the previously mentioned dimension-eight EFT and demonstrate that, from the top-down point of view, the double-Higgs production is not the best signal to explore these scenarios., Comment: 31 pages, 17 figures. V2: text shortened and reshaped, small mistakes fixed, conclusions unchanged. Matches published version

Taking into account the negative searches for New Physics at the LHC, electroweak effective theories are appropriate to deal with current energies. Tracks of new, higher scales can be studied through next-to leading order corrections of the electroweak effective theory. We assume a generic non-linear realization of the electroweak symmetry breaking with a singlet Higgs and a strongly-coupled UV-completion. We further consider a high-energy Lagrangian that incorporates explicitly a general set of new heavy fields. After integrating out these heavy resonances, we study the pattern of low-energy constants among the light fields, which are generated by the massive states., Comment: 7 pages, 1 figure. Talk given at The European Physical Society Conference on High Energy Physics, 5-12 July (2017), Venice (Italy)

We discuss effective field theories (EFTs) for the Higgs particle, which is not necessarily the Higgs of the Standard Model. We distinguish two different consistent expansions: EFTs that describe decoupling new-physics effects and EFTs that describe non-decoupling new-physics effects. We briefly discuss the first case, the SM-EFT. The focus of this thesis is on the non-decoupling EFTs. We argue that the loop expansion is the consistent expansion in the second case. We introduce the concept of chiral dimensions, equivalent to the loop expansion. Using the chiral dimensions, we expand the electroweak chiral Lagrangian up to next-to-leading order, $\mathcal{O}(f^{2}/\Lambda^{2})=\mathcal{O}(1/16\pi^{2})$. We then compare the decoupling and the non-decoupling EFT. We also consider scenarios in which the new-physics sector is non-decoupling at a scale $f$, far above the electroweak-scale $v$. We discuss the relevance of the resulting double expansion in $\xi=v^{2}/f^{2}$ and $f^{2}/\Lambda^{2}$ for the data analysis at the LHC. In the second part, we discuss the applications of the EFTs, especially of the electroweak chiral Lagrangian. First, we connect the EFT with explicit models of new physics. We show how different regions of the parameter space of the same model generate a decoupling and a non-decoupling EFT. Second, we use the expansion at leading order to describe the current LHC Higgs data. We show how the current parametrization of the Higgs data, the $\kappa$-framework, can be justified quantum field theoretically by the EFT. We fit the data of Run-1 (2010--2013). The effective Lagrangian describing this data can be reduced to six free parameters. The result of this fit is consistent with the SM, but it has statistical uncertainties of about ten percent., Comment: 181 pages, 26 figures. Ph.D. Thesis, based on arXiv:1307.5017, arXiv:1312.5624, arXiv:1412.6356, arXiv:1504.01707, arXiv:1511.00988, arXiv:1603.03062, and arXiv:1608.03564. Abstract shortened to match arXiv requirements. The library version of this thesis can be found at https://edoc.ub.uni-muenchen.de/19873/

We consider the electroweak chiral Lagrangian, including a light scalar boson, in the limit of small $\xi=v^2/f^2$. Here $v$ is the electroweak scale and $f$ is the corresponding scale of the new strong dynamics. We show how the conventional SILH Lagrangian, defined as the effective theory of a strongly-interacting light Higgs (SILH) to first order in $\xi$, can be obtained as a limiting case of the complete electroweak chiral Lagrangian. The approach presented here ensures the completeness of the operator basis at the considered order, it clarifies the systematics of the effective Lagrangian, guarantees a consistent and unambiguous power counting, and it shows how the generalization of the effective field theory to higher orders in $\xi$ has to be performed. We point out that terms of order $\xi^2$, which are usually not included in the SILH Lagrangian, are parametrically larger than terms of order $\xi/16\pi^2$ that are retained, as long as $\xi > 1/16\pi^2$. Conceptual issues such as custodial symmetry and its breaking are also discussed. For illustration, the minimal composite Higgs model based on the coset $SO(5)/SO(4)$ is considered at next-to-leading order in the chiral expansion. It is shown how the effective Lagrangian for this model is contained as a special case in the electroweak chiral Lagrangian based on $SU(2)_L\otimes SU(2)_R/SU(2)_V$., Comment: 22 pages, 1 figure; improved presentation of the results in section 4

We discuss the systematics of power counting in general effective field theories, focussing on those that are nonrenormalizable at leading order. As an illuminating example we consider chiral perturbation theory gauged under the electromagnetic $U(1)$ symmetry. This theory describes the low-energy interactions of the octet of pseudo-Goldstone bosons in QCD with photons and has been discussed extensively in the literature. Peculiarities of the standard approach are pointed out and it is shown how these are resolved within our scheme. The presentation follows closely our recent discussion of power counting for the electroweak chiral Lagrangian. The systematics of the latter is reviewed and shown to be consistent with the concept of chiral dimensions. The results imply that naive dimensional analysis (NDA) is incomplete in general effective field theories, while still reproducing the correct counting in special cases., Comment: 14 pages, no figures

We consider the Standard Model, including a light scalar boson $h$, as an effective theory at the weak scale $v=246\,{\rm GeV}$ of some unknown dynamics of electroweak symmetry breaking. This dynamics may be strong, with $h$ emerging as a pseudo-Goldstone boson. The symmetry breaking scale $\Lambda$ is taken to be at $4\pi v$ or above. We review the leading-order Lagrangian within this framework, which is nonrenormalizable in general. A chiral Lagrangian can then be constructed based on a loop expansion. A systematic power counting is derived and used to identify the classes of counterterms that appear at one loop order. With this result the complete Lagrangian is constructed at next-to-leading order, ${\cal O}(v^2/\Lambda^2)$. This Lagrangian is the most general effective description of the Standard Model containing a light scalar boson, in general with strong dynamics of electroweak symmetry breaking. Scenarios such as the SILH ansatz or the dimension-6 Lagrangian of a linearly realized Higgs sector can be recovered as special cases., Comment: 25 pages, no figures; v2: references added, matches published version; v3: corrected matching of Higgs-portal example to chiral Lagrangian in sec. 7.2