On the Power Counting in Effective Field Theories

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 figure

Complete Electroweak Chiral Lagrangian with a Light Higgs at NLO

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.

A Systematic Approach to the SILH Lagrangian

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

Current and future constraints on Higgs couplings in the nonlinear Effective Theory

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 versio

CaloFlow II: Even Faster and Still Accurate Generation of Calorimeter Showers with Normalizing Flows

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: 28 pages, 15 figures, 4 table

Higgs effective field theories

Am 4. Juli 2012 wurde am großen Hadronenbeschleuniger LHC am europäischen Kernforschungszentrum CERN bei Genf die Entdeckung eines neuen Teilchens bekannt gegeben. Die Eigenschaften des Teilchens stimmen, im Rahmen der noch relativ großen experimentellen Unsicherheiten, mit denen des lang gesuchten Higgsbosons überein. Teilchenphysiker in aller Welt stellen sich nun die Frage: "Ist es das Standardmodell Higgs-Teilchen, das wir beobachten; oder ist es ein anderes Teilchen mit ähnlichen Eigenschaften?" Effektive Feldtheorien (EFTs) ermöglichen eine allgemeine, modellunabhängige Beschreibung des Teilchens. Dabei benutzen wir wenige minimale Annahmen - nur Standardmodell Teilchen als Freiheitsgrade und eine Skalenseparation zur neuen Physik - welche durch aktuelle experimentelle Ergebnisse gestützt werden. Per Konstruktion beschreiben effektive Theorien daher ein physikalisches System nur bei einer bestimmten Energieskala, in unserem Fall der elektroschwachen Skala $v$. Effekte von neuer Physik bei höheren Energien, $\Lambda$, werden durch modifizierte Wechselwirkungen der leichten Teilchen parametrisiert. In dieser Dissertation, "Effektive Feldtheorien für das Higgs - Systematik und Anwendung", diskutieren wir effektive Feldtheorien für das Higgs Teilchen, welches nicht notwendigerweise das Higgs-Teilchen des Standardmodells ist. Besonderes Augenmerk richten wir auf eine systematische und konsistente Entwicklung der EFT. Diese Systematik ist abhängig von der Dynamik der neuen Physik. Wir unterscheiden zwei verschiedene konsistente Entwicklungen. Zum einen effektive Theorien von Modellen neuer Physik, die bei niedrigen Energien entkoppeln und zum anderen effektive Beschreibungen von nicht entkoppelnden Modellen. Wir diskutieren den ersten Fall, die Standardmodell EFT, kurz, da der Fokus dieser Arbeit auf nicht entkoppelnden effektiven Theorien liegt. Wir erläutern, dass die konsistente Entwicklung im zweiten Fall in Quantenschleifen erfolgen muss und führen das dazu äquivalente Konzept der chiralen Dimensionen ein. Mithilfe der chiralen Dimensionen entwickeln wir die elektroschwache chirale Lagrangedichte bis einschließlich nächstführender Ordnung, $\mathcal{O}(f^{2}/\Lambda^{2})=\mathcal{O}(1/16\pi^{2})$. Wir diskutieren auch den Einfluss verschiedener Annahmen über die schützende (custodial) Symmetrie im Higgssektor auf die Liste der Operatoren. Wir beenden die Diskussion über die Systematik mit einem Vergleich der entkoppelnden und nicht entkoppelnden EFT. Wir betrachten dabei auch den Fall, dass die neue Physik einen nicht entkoppelnden Sektor bei einer Energieskala $f$ besitzt, welcher deutlich über der elektroschwachen Skala $v$ liegt. Wir diskutieren die Relevanz der daraus resultierenden Doppelentwicklung in $\xi=v^{2}/f^{2}$ und $f^{2}/\Lambda^{2}$ für die Datenanalyse am LHC. Im zweiten Teil dieser Dissertation diskutieren wir Anwendungen der effektiven Theorien, insbesondere der elektroschwachen chiralen Lagrangedichte. Als Erstes verbinden wir die EFT mit expliziten Modellen für neue Physik. Dies illustriert, wie die Vorhersagen des Entwicklungsschemas in einem konkreten Fall realisiert werden. Wir zeigen auch an einem Beispiel, wie verschiedene Parameterbereiche derselben Theorie sowohl eine entkoppelnde als auch eine nicht entkoppelnde EFT generieren. Als Zweites nutzen wir die effektive Entwicklung in führender Ordnung um die aktuellen Higgsdaten des LHCs zu beschreiben. Wir zeigen, dass die aktuelle Parametrisierung der Higgsdaten, welche von den Experimentatoren am CERN verwendet wird (der $\kappa$-Formalismus), sich durch diese Entwicklung quantenfeld-theoretisch begründen lässt. Das Ergebnis eines Fits zeigt daher nicht nur, ob das beobachtete Teilchen das Standardmodell Higgs-Teilchen ist, sondern auch, sofern sich Abweichungen manifestieren, welche Art von neuer Physik bevorzugt wird. In unserem konkreten Fall nutzen wir die Daten von 2010-2013. Die effektive Lagrangedichte, die diese Daten beschreibt, lässt sich auf sechs freie Parameter reduzieren. Das Ergebnis ist konsistent mit dem Standardmodell, weist aber noch statistische Unsicherheiten von etwa $10\%$ auf.Researchers of the Large Hadron Collider (LHC) at the European Organization for Nuclear Research (CERN) announced on July 4th, 2012, the observation of a new particle. The properties of the particle agree, within the relatively large experimental uncertainties, with the properties of the long-sought Higgs boson. Particle physicists around the globe are now wondering, "Is it the Standard Model Higgs that we observe; or is it another particle with similar properties?" We employ effective field theories (EFTs) for a general, model-independent description of the particle. We use a few, minimal assumptions - Standard Model (SM) particle content and a separation of scales to the new physics - which are supported by current experimental results. By construction, effective field theories describe a physical system only at a certain energy scale, in our case at the electroweak-scale $v$. Effects of new physics from a higher energy-scale, $\Lambda$, are described by modified interactions of the light particles. In this thesis, "Higgs Effective Field Theories - Systematics and Applications", we discuss effective field theories for the Higgs particle, which is not necessarily the Higgs of the Standard Model. In particular, we focus on a systematic and consistent expansion of the EFT. The systematics depends on the dynamics of the new physics. 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, however, 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, $\mathcal{O}(f^{2}/\Lambda^{2})=\mathcal{O}(1/16\pi^{2})$. Further, we discuss how different assumptions on the custodial symmetry in the Higgs sector influences the list of operators in the basis. Finally, we 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 of this thesis, we discuss the applications of the EFTs, especially of the electroweak chiral Lagrangian. First, we connect the EFT with explicit models of new physics. This illustrates how the power counting works in a specific example. 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, which is used by the experimentalists at CERN (the $\kappa$-framework), can be justified quantum field theoretically by the EFT. The result of a fit does therefore not only indicate whether we observe the SM-Higgs, but also, in case there are deviations, what kind of new physics is preferred. In this thesis, 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. It has, however, statistical uncertainties of about ten percent

Colorful Imprints of Heavy States in the Electroweak Effective Theory

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 versio