Higgs Effective Field Theory can be used to study vector-boson elastic scattering at the high energies relevant for the LHC. For most of the parameter space, the scattering is strongly interacting, with the minimal Standard Model being a remarkable exception. From its one-loop treatment complemented with dispersion relations and the Equivalence Theorem, we derive two different unitarization methods which produce analytical amplitudes corresponding to different approximate solutions to the dispersion relations: the Inverse Amplitude method (IAM) and the N/D method. The partial waves obtained can show poles in the second Riemann sheet whose natural interpretation is that of dynamical resonances with masses and widths a function of the starting HEFT parameters. We compare the different unitarizations and we find that they are qualitatively, and in many cases quantitatively, very similar. However, for different reason it is more interesting to use one of the two methods depending on the particular channel for WW, ZZ, WZ, Zh, Wh or hh scattering. In this note we briefly describe the possible I and J channels for these reactions and give the unitarization method of choice in each case. The amplitudes obtained provide realistic resonant and nonresonant cross sections to be compared with and to be used for a proper interpretation of the LHC data.
Higgs Effective Field Theory (HEFT) is deployed to study elastic vector-boson scattering at the high LHC energies. The interaction is strong over most of the parameter space, with the minimal Standard Model being a remarkable exception. One-loop HEFT complemented with dispersion relations and the Equivalence Theorem leads to two different unitarization methods which produce analytical amplitudes corresponding to different approximate solutions to the dispersion relations: the Inverse Amplitude method (IAM) and the N/D method. The partial waves obtained can show poles in the second Riemann shee…
We define and calculate helicity partial-wave amplitudes for processes linking the Electroweak Symmetry Breaking Sector (EWSBS) to $γγ$, employing (to NLO) the Higgs-EFT (HEFT) extension of the Standard Model and the Equivalence Theorem, while neglecting all particle masses. The resulting amplitudes can be useful in the energy regime ($500\,{\rm GeV}-3\,{\rm TeV}$). We also deal with their unitarization so that resonances of the EWSBS can simultaneously be described in the $γγ$ initial or final states. Our resulting amplitudes satisfy unitarity, perturbatively in $α$, but for all $s$ values. I…
Due to the gap between the known 100 GeV scale and new physics if any, it is natural to employ an effective one-loop Lagrangian (HEFT) for the particles of the Electroweak Symmetry Breaking Sector (W_L, Z_L and h). To describe any new particles and resonances that may be found at the LHC we employ its unitarized amplitudes, valid even in the presence of new strong interactions. We have assessed the systematics by comparing several such methods, and find that they give qualitatively similar results and succesfully produce unitary amplitudes in the nonperturbative regime. We are thus in a positi…
There are two canonical approaches to treating the Standard Model as an Effective Field Theory (EFT): Standard Model EFT (SMEFT), expressed in the electroweak symmetric phase utilizing the Higgs doublet, and Higgs EFT (HEFT), expressed in the broken phase utilizing the physical Higgs boson and an independent set of Goldstone bosons. HEFT encompasses SMEFT, so understanding whether SMEFT is sufficient motivates identifying UV theories that require HEFT as their low energy limit. This distinction is complicated by field redefinitions that obscure the naive differences between the two EFTs. By re…
We analyze the electroweak interactions in the framework of the Higgs effective field theory using the available Higgs and electroweak diboson production results from LHC run 2 as well as the electroweak precision data. Assuming universality of the weak current, our study considers 25 possible anomalous couplings. To unveil the nature of the Higgs boson, i.e. isosinglet versus part of SU(2)L doublet, we explore the correlation effects between observables that are predicted to exist in the linear realization of the electroweak gauge symmetry but not in its non-linear counterpart. This improves …
We study the existence of skyrmions in the presence of all the electroweak degrees of freedom, including a dynamical Higgs boson, with the electroweak symmetry being non-linearly realized in the scalar sector. For this, we use the formulation of the Higgs Effective Field Theory (HEFT). In contrast with the linear realization, a well-defined winding number exists in HEFT for all scalar field configurations. We classify the effective operators that can potentially stabilize the skyrmions and numerically find the region in parameter spaces that support them. We do so by minimizing the static ener…
We report computations of the production cross-section of longitudinal electroweak and Higgs boson pairs within Effective Field Theory for the Electroweak sector (including the Higgs). We have recently reported theoretical studies of gauge boson-gauge boson and Higgs-Higgs resonance production with various quantum numbers. We are now focusing on gauge boson-Higgs boson two-body axial-vector resonances and show a typical cross section. At last, we point out that photon-photon production has also been studied (in $e^-e^+$ as well as pp machines), as this is a very clean process allowing access t…
The complete effective chiral Lagrangian for a dynamical Higgs is presented and constrained by means of a global analysis including electroweak precision data together with Higgs and triple gauge boson coupling data from the LHC Run~I. The operators' basis up to next-to-leading order in the expansion consists of 148 (188 considering right-handed neutrinos) flavour universal terms and it is presented here making explicit the custodial nature of the operators. This effective Lagrangian provides the most general description of the physical Higgs couplings once the electroweak symmetry is assumed,…
We present a unified analysis of the two main production processes of vector boson pairs at the LHC, VV-fusion and qqbar annihilation, in a minimal strongly interacting electroweak symmetry breaking sector. Using a unitarized electroweak chiral Lagrangian formalism and modeling the final V_L V_L strong rescattering effects by a form factor, we describe qqbar annihilation processes in terms of the two chiral parameters that govern elastic V_L V_L scattering. Depending on the values of these two chiral parameters, the unitarized amplitudes may present resonant enhancements in different angular m…
The LHC is now exploring the 1-3 TeV scale where resonances of the Electroweak Symmetry Breaking Sector might exist. If so, Unitarized Effective Theory can be used to describe the data with all the constraints of unitarity, causality and global-symmetry breaking, and to find the resonance positions in the complex s-plane. From any resonances found, one can infer the parameters of the universal Effective Lagrangian, and those may be used to inform higher-energy theories (UV completions) that can be matched to it. We exemplify with two-body resonances in the coupled channels hh and W_L W_L- Z_L …
We present a study of the production of vector resonances at the LHC via $W^+Z$ vector boson scattering and explore the sensitivities to these resonances for expected LHC luminosities. We work in the framework of the electroweak chiral Lagrangian, where these resonances can be generated dynamically by unitarizing the scattering amplitudes. We implement all these features into a model adapted for MonteCarlo, the IAM-MC, that allows us to give predictions for the sensitivity to these resonances and to the relevant parameters involved for $pp \to W^+Zjj$, $pp\to \ell_1^+\ell_1^-\ell_2^+νjj$, and …
Effective Field Theories such as HEFT, organized as momentum expansions, are a controllable approximation to strong dynamics only near threshold, as they miss exact elastic unitarity, reducing their predictive power at a higher scale if small separations from the Standard Model are found at the LHC or elsewhere. Unitarized chiral perturbation theory extends their reach to the saturation of unitarity but, generally with unknown systematics. Here I present a brief "handbook" on how to put to use the Inverse Amplitude Method to predict a resonance (associated with a new physics scale in Beyond th…
Electroweak interactions assign a central role to the gauge group $SU(2)_L \times U(1)_Y$, which is either realized linearly (SMEFT) or nonlinearly (e.g., HEFT) in the effective theory obtained when new physics above the electroweak scale is integrated out. Although the discovery of the Higgs boson has made SMEFT the default assumption, nonlinear realization remains possible. The two can be distinguished through their predictions for the size of certain low-energy dimension-6 four-fermion operators: for these, HEFT predicts $O(1)$ couplings, while in SMEFT they are suppressed by a factor $v^2/…
Recently, a new boson h has been discovered at the LHC which, so far, is compatible with the properties of the SM Higgs. However, the SM is not the most general low-energy dynamics for the minimal electroweak symmetry breaking sector with three Goldstone bosons and one light scalar. By using non-linear effective Lagrangian for these four particles we study different processes at one-loop precision, identifying the counterterms needed to cancel the divergences. Then we apply the IAM unitarization method on the partial waves, both to make more realistic predictions which could be tested at the L…
The existence of a mass gap between the Standard Model (SM) and possible new states encourages us to use effective field theories. Here we follow the non-linear realization of the electroweak symmetry breaking: the electroweak effective theory (EWET), also known as Higgs effective field theory (HEFT) or electroweak chiral Lagrangian (EWChL). At short distances an effective resonance Lagrangian which couples the SM states to bosonic and fermionic resonances is considered. After integrating out the resonances and assuming a well-behaved high-energy behavior, we estimate or bound purely bosonic l…
In the light of the mass gap between Standard Model (SM) states and possible new particles, effective field theories are a suitable approach. We take on the non-linear realization of the electroweak symmetry breaking: the electroweak effective theory (EWET), also known as Higgs effective field theory (HEFT) or electroweak chiral Lagrangian (EWChL). At higher scales we consider a resonance electroweak Lagrangian, coupling SM fields to resonances. Integrating out these resonances and assuming a well-behaved high-energy behavior, some of the bosonic low-energy constants are determined or constrai…
We explore the use of the Inverse Amplitude Method for unitarization of scattering amplitudes to derive the existence and properties of possible new heavy states associated with perturbative extensions of the electroweak breaking sector of the Standard Model starting from the low energy effective theory. We use a toy effective theory generated by integrating out a heavy singlet scalar and compare the pole mass and width of the unitarized amplitudes with those of the original model. Our results show that the Inverse Amplitude Method reproduces correctly the singlet mass up to factors of O(1-3),…
Vector boson scatterings are fundamental processes to shed light on the nature of the electroweak symmetry breaking mechanism. Deviations from the Standard Model predictions on the corresponding observables can be interpreted in terms of effective field theories, that however undergo consistency conditions. In this paper, the same-sign W W scattering is considered within the HEFT context and the correct usage of the effective field theory approach is discussed. Regions of the parameters space are identified where a signal of new physics could be measured at HL-LHC with a significance of more t…
We calculate fermion-loop corrections to high energy $W^+W^-$ scattering in the context of a Strongly Interacting Electroweak Symmetry Breaking Sector (EWSBS) using Higgs Effective Field Theory(HEFT). We test the assumption that these corrections are negligible when compared to the boson-loop ones, as it is commonly taken for granted in the literature. While this is correct inmost cases, we find that, for some particular regions of the parameter space, fermion-loops can be important: deviations in the couplings of the HEFT from their Standard Model values may lead to fermion-loop corrections a…
In the present work we study the production of vector resonances at the LHC by means of the vector boson scattering $WZ \to WZ$ and explore the sensitivities to these resonances for the expected future LHC luminosities. We are assuming that these vector resonances are generated dynamically from the self interactions of the longitudinal gauge bosons, $W_L$ and $Z_L$, and work under the framework of the electroweak chiral Lagrangian to describe in a model independent way the supposedly strong dynamics of these modes. The properties of the vector resonances, mass, width and couplings to the $W$ a…
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)…
We estimate production cross sections for 2-body resonances of the Electroweak Symmetry Breaking sector (in $W_LW_L$ and $Z_LZ_L$ rescattering) from $γγ$ scattering. We employ unitarized Higgs Effective Field Theory amplitudes previously computed coupling the two photon channel to the EWSBS. We work in the Effective Photon Approximation and examine both $e^-e^+$ collisions at energies of order 1-2 TeV (as relevant for future lepton machines) and $pp$ collisions at LHC energies. Dynamically generating a spin-0 resonance around 1.5 TeV (by appropriately choosing the parameters of the effective t…
If new resonances of the electroweak symmetry breaking sector (longitudinal-gauge and Higgs) bosons are found in the 1-3 TeV region, the right tool to assess their properties and confront experimental data in a largely model-independent yet simple manner is Unitarized Effective Theory. Its ingredients are: 1) custodial symmetry and the Equivalence Theorem, that allow to approximate W_L and Z_L by an isospin-triplet of Goldstone bosons omega^a in the 1-TeV region. 2) The effective coupling of a generic, approximately massless scalar-isoscalar h to those Goldstone bosons, and the chiral Lagrangi…
This article presents a number of technical tools and results that may be instrumental to discern the nature of the Higgs particle. In scenarios where an additional strongly interacting sector is present in the electroweak theory resulting in a composite Higgs and longitudinal components of the massive gauge bosons, unitarity, analyticity and related techniques will be crucial to understand the properties of such a sector. The situation today may be reminiscent of the pre-QCD days: a strongly interacting theory governs the short-distances, but we have only access to long-distance experiment in…
In this work we interpret the Einstein-Hilbert (EH) Lagrangian of gravitation as the first term of a low-energy effective theory similar to those considered in the chiral Lagrangian approach to low-energy hadron physics or the electroweak chiral Lagragians describing the symmetry breaking sector of the Standard Model (SM). Starting from the one-loop computation of the elastic graviton-graviton scattering amplitude by Dunbar and Norridge, we unitarize the IR regularized partial waves by using the Inverse Amplitude Method (IAM). This method enlarges the regime of applicability of the perturbativ…
Characterizing LHC-Resonances in extended HEFT: information on the nature of extended scalar sectors
Dipartimento di Scienze Matematiche e Informatiche, Scienze Fisiche e Scienze della Terra, Universita degli Studi di Messina（墨西拿大学数学、信息与计算机科学系，物理科学与地球科学系）；INFN Sezione di Catania（意大利国家核物理研究所卡塔尼亚分部）；Institut für Theoretische Physik, Georg-August-Universität Göttingen（哥廷根大学理论物理研究所）；Departament de Física Quàntica i Astrofísica, Universitat de Barcelona（巴塞罗那大学量子物理与天体物理学系）；Max-Planck-Institut für Kernphysik（马克斯·普朗克核物理研究所）
We formulate a generalization of Higgs effective field theory (HEFT) including arbitrary number of extra neutral and charged Higgs bosons (generalized HEFT, GHEFT) to describe non-minimal electroweak symmetry breaking models. Using the geometrical form of the GHEFT Lagrangian, which can be regarded as a nonlinear sigma model on a scalar manifold, it is shown that the scalar boson scattering amplitudes are described in terms of the Riemann curvature tensor (geometry) of the scalar manifold and the covariant derivatives of the potential. The coefficients of the one-loop divergent terms in the ob…
In view of the annnouncement that in 2012 the LHC will run at 8 TeV, we study the possibility of detecting signals of alternative mechanisms of ElectroWeak Symmetry Breaking, described phenomenologically by unitarized models, at energies lower than 14 TeV. A complete calculation with six fermions in the final state is performed using the PHANTOM event generator. Our results indicate that at 8 TeV some of the scenarios with TeV scale resonances are likely to be identified while models with no resonances or with very heavy ones will be inaccessible, unless the available luminosity will be much h…
Tantalizing LHC hints suggest that resonances of the Electroweak Symmetry Breaking Sector might exist at the TeV scale. We recall a few key meson-meson resonances in the GeV region that could have high-energy analogues which we compare, as well as the corresponding unitarized effective theories describing them. While detailed dynamics may be different, the constraints of unitarity, causality and global-symmetry breaking, incorporated in the Inverse Amplitude Method, allow to carry some intuition over to the largely unmeasured higher energy domain. If the 2 TeV ATLAS excess advances one such ne…
We study the one-loop corrections to Vector Boson Scattering (in particular $W^+W^-$ elastic scattering) within the framework of effective theories. Re-scattering via intermediate electroweak would-be-Goldstone bosons dominate at high energies, as the corresponding loop diagrams with these intermediate bosons scale like $\mathcal{O}(s^2/v^4)$ in the chiral effective counting. In the present article, we focus our attention on fermion-loop corrections which scale like $\mathcal{O}(M_{\rm Fer}^2 s/v^4)$ in the Higgs Effective Field Theory (HEFT). Although this dependency is formally suppressed fo…
In this work we study the phenomenology of the process $pp\to W^+W^-jj$ at the LHC, in the scenario of the resonant vector boson scattering subprocess $W^+W^-\to W^+W^-$ which we describe within the effective field theory framework of the Electroweak Chiral Lagrangian. We assume a strongly interacting electroweak symmetry breaking sector in which dynamically generated resonances with masses in the TeV scale appear as poles in the Electroweak Chiral Lagrangian amplitudes unitarized with the Inverse Amplitude Method. The relevant resonance here, $V_0$, is the neutral component of the triplet of …
Weak vector boson scattering at high energies will be one of the key measurements in current and upcoming LHC runs. It is most sensitive to any new physics associated with electroweak symmetry breaking. However, a conventional EFT analysis will fail at high energies. To address this problem, we present a parameter-free prescription valid for arbitrary perturbative and non-perturbative models: the T-matrix unitarization. We describe its implementation as an asymptotically consistent reference model matched to the low-energy effective theory. We show examples of typical observables of vector-bos…
We study the production of vector resonances at the LHC via $WZ$ scattering processes and explore the sensitivities to these resonances for the expected future LHC luminosities. The electroweak chiral Lagrangian and the Inverse Amplitude Method (IAM) are used for analyzing a dynamically generated vector resonance, whose origin would be the (hypothetically strong) self interactions of the longitudinal gauge bosons, $W_L$ and $Z_L$. We implement the unitarized scattering amplitudes into a single model, the IAM-MC, that has been adapted to MadGraph~5. It is written in terms of the electroweak chi…
Effective Field Theories (EFTs) constructed as derivative expansions in powers of momentum, in the spirit of Chiral Perturbation Theory (ChPT), are a controllable approximation to strong dynamics as long as the energy of the interacting particles remains small, as they do not respect exact elastic unitarity. This limits their predictive power towards new physics at a higher scale if small separations from the Standard Model are found at the LHC or elsewhere. Unitarized chiral perturbation theory techniques have been devised to extend the reach of the EFT to regimes where partial waves are satu…
We show that a five dimensional Universal Extra Dimension model, compactified on a line segment, is consistently formulated even when the gauge symmetry is broken solely by non-zero Dirichlet boundary conditions on a bulk Higgs field, without any quartic interaction. We find that the longitudinal W+W- elastic scattering amplitude, under the absence of the Higgs zero-mode, is unitarized by exchange of infinite towers of KK Higgs bosons. Resultant amplitude scales linearly with the scattering energy, exhibiting five dimensional nature. A tree-level partial-wave unitarity condition is satisfied u…
