In this paper we consider the effects of top quark compositeness on the electroweak parameters T-hat and S-hat and the Zbb coupling. We do so by using an effective field theory analysis to identify several promising patterns of mixing between standard model like and vector fermions, and then analyzing simple extensions of the Standard Model that realize those patterns. These models illustrate four ways in which an extended O(4) symmetry, which controls the size of radiative corrections to the observables discussed, may be broken. These models may also be viewed as highly-deconstructed versions of five-dimensional gauge theories dual to various strongly-interacting composite Higgs theories. We comment on how our results relate to extra-dimensional models previously considered, and we demonstrate that one pattern of O(4) breaking is phenomenologically favored.
We study precision electroweak constraints within a RS1 model with gauge fields and fermions in the bulk. The electroweak gauge symmetry is enhanced to SU(2)_L \times SU(2)_R \times U(1)_{B-L}, thereby providing a custodial isospin symmetry sufficient to suppress excessive contributions to the T parameter. We then construct complete models, complying with all electroweak constraints, for solving the hierarchy problem, without supersymmetry or large hierarchies in the fundamental couplings. Using the AdS/CFT correspondence our models can be interpreted as dual to a strongly coupled conformal Hi…
We discuss the effects of isospin breaking which appear in the vertex corrections for $Zb \bar{b}$, $Zτ^+τ^-$ and $Wντ$ in a one-family technicolor model without exact custodial symmetry. By means of the effective lagrangian approach we compute the vertex corrections for $Zb\bar{b}$, $Zττ$ and $Wτν$ taking account of the contributions from technivectormesons. If the isospin symmetry in technilepton sector is not exact, technivectormesons contribute to the vertex correction for $Zττ$ but such contributions to the correction for $Wτν$ are absent. If the difference is measured, it is the evidence…
We analyze the tension between naturalness and isospin violation in the Lee-Wick Standard Model (LW SM), by computing tree-level and fermionic one-loop contributions to the post-LEP electroweak parameters and the Zbb coupling. The model is most natural when the LW partners of the gauge bosons and fermions are light, but small partner masses can lead to large isospin violation. The post-LEP parameters yield a simple picture in the LW SM: the gauge sector contributes to Y and W only, with leading contributions arising at tree-level, while the fermion sector contributes to S-hat and T-hat only, w…
We assess current experimental constraints on the bi-doublet + singlet model of top compositeness previously proposed in the literature. This model extends the standard model's spectrum by adding a custodially-embedded vector-like electroweak bi-doublet of quarks and a vector-like electroweak singlet quark. While either of those states alone would produce a model in tension with constraints from precision electroweak data, in combination they can produce a viable model. We show that current precision electroweak data, in the wake of the Higgs discovery, accommodate the model and we explore the…
Composite Higgs models based on SU(5)/SO(5) are characterised by the presence of custodial triplets, like the Georgi-Machacek model. We classify all the operators giving rise to the top mass and Higgs potential in presence of fermion partial compositeness, with top partners in two-index representations of SU(5). A detailed study of each operator allows us to find correlations in the couplings of Higgs and non-Higgs pseudo-Nambu-Goldstone bosons, which depend only on one, two or three independent parameters. We also analyse the Higgs potential, finding that a misalignment along the custodial in…
We show that the recently reported excess in resonant diboson production can be explained in the context of non-custodial composite Higgs models. Dibosons are generated via the s-channel exchange of massive vector bosons present in these models. We discuss the compatibility of the signal excess with other diboson experimental searches. We also discuss the tension between diboson production and other experimental tests of the model that include electroweak precision data, dilepton, dijet and top pair production and show that there is a region of parameter space in which they are all compatible …
We study bounds on Higgs boson masses from perturbative unitarity in the Georgi-Machacek model, whose Higgs sector is composed of a scalar isospin doublet, a real and a complex isospin triplet fields. This model can be compatible with the electroweak precision data without fine tuning because of the imposed global SU(2)_R symmetry in the Higgs potential, by which the electroweak rho parameter is unity at the tree level. All possible two-body elastic-scattering channels are taken into account to evaluate the S-wave amplitude matrix, and then the condition of perturbative unitarity is imposed on…
We study the effects of custodial symmetry violation in the Georgi-Machacek (GM) model. The GM model adds isospin-triplet scalars to the Standard Model in a way that preserves custodial symmetry at tree level; however, this custodial symmetry has long been known to be violated at the one-loop level by hypercharge interactions. We consider the custodial-symmetric GM model to arise at some high scale as a result of an unspecified ultraviolet completion, and quantify the custodial symmetry violation induced as the model is run down to the weak scale. The measured value of the eletroweak rho param…
I study, in the leptonic sector, the role of the $SU(2)_V$ custodial symmetry $\ti{\cal G}$ which was shown in ref. [1] to control the quantization of the electric charge in the $J=0$ mesonic sector. The electroweak theory is considered, according to ref. [2], as a purely vectorial model which interacts with a ``hidden'' sector of composite scalars. $\ti{\cal G}$ can only be a symmetry of the former if the neutrino is a Majorana particle; the latter provides a dynamical modification of the leptonic weak couplings, reconstructing those of the Standard Model with a massless Majorana neutrino.
We present a class of composite Higgs models arising from a warped extra dimension that can satisfy all the electroweak precision tests in a significant portion of their parameter space. A custodial symmetry plays a crucial role in keeping the largest corrections to the electroweak observables below their experimental limits. In these models the heaviness of the top quark is not only essential to trigger the electroweak symmetry breaking, but it also implies that the lowest top resonance and its custodial partners, the custodians, are significantly lighter than the other resonances. These cust…
The composite Higgs models based on the top seesaw mechanism commonly possess an enhanced approximate chiral symmetry, which is spontaneously broken to produce the Higgs field as the pseudo-Nambu-Goldstone bosons. The minimal model with only one extra vector-like singlet quark that mixes with the top quark can naturally give rise to a 126 GeV Higgs boson. However, without having a custodial symmetry it suffers from the weak-isospin violation constraint, which pushes the chiral symmetry breaking scale above a few TeV, causing a substantial fine-tuning for the weak scale. We consider an extensio…
We analyze the bounds on the spectrum of composite Higgs models (CHM) that come from flavor observables, by means of simple two-site effective Lagrangians, which incorporate a custodial symmetry and a Left-Right parity and which could also be adopted in further phenomenological studies on CHM. We derive, in particular, an important constraint on the masses of the $(t_L, b_L)$ partners, which does not depend on the flavor structure of the sector beyond the SM. This bound is obtained from the "infrared" contribution to $b \to sγ$ induced by the flavor-conserving effective vertex $Wt_Rb_R$. We fi…
Twin Higgs models are economical extensions of the Standard Model that stabilize the electroweak scale. In these theories the Higgs field is a pseudo Nambu-Goldstone boson that is protected against radiative corrections up to scales of order 5 TeV by a discrete parity symmetry. We construct, for the first time, a class of composite twin Higgs models based on confining QCD-like dynamics. These theories naturally incoporate a custodial isospin symmetry and predict a rich spectrum of particles with masses of order a TeV that will be accessible at the LHC.
We investigate whether the $2.6σ$ deviation from lepton flavour universality in $B^+\to K^+\ell^+\ell^-$ decays recently observed at the LHCb experiment can be explained in minimal composite Higgs models. We show that a visible departure from universality is indeed possible if left-handed muons have a sizable degree of compositeness. Constraints from $Z$-pole observables are avoided by a custodial protection of the muon coupling. The deficit in the invisible $Z$ width at LEP is explained in the same region of parameters.
Dark mesons are bosonic composites of a new, strongly-coupled sector beyond the Standard Model. We consider several dark sectors with fermions that transform under the electroweak group, as arise from a variety of models including strongly-coupled theories of dark matter (e.g., stealth dark matter), bosonic technicolor (strongly-coupled indcued electroweak symmetry breaking), vector-like confinement, etc. We consider theories with two and four flavors under an $SU(N)$ strong group that acquire variously chiral, vector-like, and mixed contributions to their masses. We construct the non-linear s…
We study the effects of extended fermion sectors, respecting custodial symmetry, on Higgs production and decay. The resulting protection for the Z->b_L b_L and Z->τ_Rτ_R decays allows for potentially interesting signals in Higgs physics, while maintaining the good agreement of the Standard Model with precision tests, without significant fine-tuning. Although being viable setups on their own, the models we study can particularly be motivated as the low energy effective theories of the composite Higgs models MCHM_5 and MCHM_10 or the corresponding gauge-Higgs unification models. The spectra can …
In models of electroweak symmetry breaking in which the SM fermions get their masses by mixing with composite states, it is natural to expect the top quark to show properties of compositeness. We study the phenomenological viability of having a mostly composite top. The strongest constraints are shown to mainly come from one-loop contributions to the T-parameter. Nevertheless, the presence of light custodial partners weakens these bounds, allowing in certain cases for a high degree of top compositeness. We find regions in the parameter space in which the T-parameter receives moderate positive …
We study analytically and numerically the bounds imposed by the electroweak precision tests on a minimal composite Higgs model. The model is based on spontaneous SO(5)/SO(4) breaking, so that an approximate custodial symmetry is preserved. The Higgs arises as a pseudo-Goldstone boson at a scale below the electroweak symmetry breaking scale. We show that one can satisfy the electroweak precision constraints without much fine-tuning. This is the case if the left-handed top quark is fully composite, which gives a mass spectrum within the reach of the LHC. However a composite top quark is strongly…
We explore the collider relevance of a charge-radius coupling among light mesons in composite Higgs models. In particular, we focus of a coupling of the photon to the composite Higgs and a composite singlet, arising from isospin violation in the underlying theory. This coupling offers a deep probe of the composite nature of the Higgs mechanism, being sensitive to the electromagnetic and weak isospin structure of its constituents. The main collider effect consists in the production of the Higgs boson in association with a light composite pseudo-scalar. We present an exploratory cut-and-count an…
We consider a realisation of composite Higgs models in the context of $SU(6)/ SO(6)$ symmetry, which features a custodial bi-triplet, two Higgs doublets and dark matter candidates. This model can arise from an underlying gauge-fermion theory. The general vacuum structure is explored using the top partial compositeness to generate a special vacuum characterised by a single angle aligned with the first Higgs doublet. We present the CP and Dark Matter $\mathbb{Z}_2$ parity in two different pNGB bases and analyse the spectra in the absence of tadpoles and tachyons. For the phenomenology, we discus…
We consider composite Higgs models where the Higgs is a pseudo-Nambu Goldstone boson arising from the spontaneous breaking of an approximate global symmetry by some underlying strong dynamics. We focus on the SO(5) -> SO(4) symmetry breaking pattern, assuming the partial compositeness paradigm. We study the consequences on Higgs physics of the fermionic representations produced by the strong dynamics, that mix with the Standard Model (SM) degrees of freedom. We consider models based on the lowest-dimensional representations of SO(5) that allow for the custodial protection of the Z -> b b coupl…
We propose a composite grand unified theory to study the anomalies in the semileptonic $B$ decays. We show a simple group containing the custodial and Standard Model gauge symmetries, that can deliver a set of composite pseudo Nambu-Goldstone bosons: the Higgs, a colorless SU(2)$_L$-fourplet and three leptoquarks: a triplet and two doublets. We give a description in terms of an effective theory of resonances. By assuming anarchic partial compositeness of the Standard Model fermions, we find representations for the composite fermions that allow to obtain the Higgs Yukawa couplings, as well as l…
The coupling of a composite Higgs to the standard model fields can deviate substantially from the standard model values. In this case perturbative unitarity might break down before the scale of compositeness is reached, which would suggest that additional composites should lie well below this scale. In this paper we account for the presence of an additional spin 1 custodial triplet of rhos. We examine the implications of requiring perturbative unitarity up to the compositeness scale and find that one has to be close to saturating certain unitarity sum rules involving the Higgs and the rho coup…
We classify the four-dimensional purely fermionic gauge theories that give a UV completion of composite Higgs models. Our analysis is at the group theoretical level, addressing the necessary (but not sufficient) conditions for the viability of these models, such as the existence of top partners and custodial symmetry. The minimal cosets arising are those of type SU(5)/SO(5) and SU(4)/Sp(4). We list all the possible "hyper-color" groups allowed and point out the simplest and most promising ones.
In this talk, we describe the effects of extended fermion sectors, respecting custodial symmetry, on Higgs production and decay. The resulting protection for the Z->b_L b_L and Z->τ_R τ_R decays allows for potentially interesting signals in Higgs physics, while maintaining the good agreement of the Standard Model with precision tests. The setups can be motivated as the low energy effective theories of the composite Higgs models MCHM_5 and MCHM_10, where the spectra can be identified with the light custodians present in these theories. We will show that these describe the relevant physics in th…
One of the most challenging hurdles to the construction of realistic composite Higgs models is the generation of Yukawa couplings for the Standard Model fermions. This problem can be successfully addressed in approximate conformal theories that admit a marginally relevant mixing between composite fermionic operators and the SM fermions. I argue that SU(3) gauge theories with light Dirac flavors in the fundamental representation feature all the ingredients under theoretical control, including a strongly-coupled IR fixed point, composite partners for all Standard Model fermions, absence of Landa…
We construct renormalizable Standard Model extensions, valid up to the Planck scale, that give a composite Higgs from a new fundamental strong force acting on fermions and scalars. Yukawa interactions of these particles with Standard Model fermions realize the partial compositeness scenario. Under certain assumptions on the dynamics of the scalars, successful models exist because gauge quantum numbers of Standard Model fermions admit a minimal enough 'square root'. Furthermore, right-handed SM fermions have an SU(2)$_R$-like structure, yielding a custodially-protected composite Higgs. Baryon a…
We study the scattering of longitudinally polarized W bosons in extensions of the Standard Model where anomalous Higgs couplings to gauge sector and higher order O(p^4) operators are considered. These new couplings with respect to the Standard Model should be thought as the low energy remnants of some new dynamics involving the electroweak symmetry breaking sector. By imposing unitarity and causality constraints on the WW scattering amplitudes we find relevant restrictions on the possible values of the new couplings and the presence of new resonances above 300 GeV. We investigate the propertie…
We explore the up flavor structure of composite pseudo Nambu-Goldstone-boson Higgs models, where we focus on the flavor anarchic minimal $SO(5)$ case. We identify the different sources of flavor violation in this framework and emphasise the differences from the anarchic Randall-Sundrum scenario. In particular, the fact that the flavor symmetry does not commute with the symmetries that stabilize the Higgs potential may constrain the flavor structure of the theory. In addition, we consider the interplay between the fine tuning of the model and flavor violation. We find that generically the tunin…
A composite model of quarks and bosons is proposed in which a spin $1/2$ isospin doublet $ψ$ is the basic building block of quarks and bosons in the standard model. The $ψ$ has two components $v$ and $w$ with charges $Q=\frac{1}{3}e$ and $Q=0$, respectively, that combine to form the three generations of colored quark flavors. A strong force described by a triplet of massless gluons bind the constituents called geminis. The confining constituent non-Abelian $SU(2)_C$ field theory is called constituent dynamics with a confining energy scale $Λ_{CD}$. The constituent dynamics condensate $\langle\…
We consider how best to search for top partners in generic composite Higgs models. We begin by classifying the possible group representations carried by top partners in models with and without a custodial $SU(2)\times SU(2) \rtimes \mathbb{Z}_2$ symmetry protecting the rate for $Z \rightarrow b\overline{b}$ decays. We identify a number of minimal models whose top partners only have electric charges of $\frac{1}{3}, \frac{2}{3},$ or $\frac{4}{3}$ and thus decay to top or bottom quarks via a single Higgs or electroweak gauge boson. We develop an inclusive search for these based on a top veto, wh…
We consider a Composite Higgs Model (CHM) with two isospin doublet Higgs fields arising as pseudo Nambu-Goldstone bosons from a ${\rm SO}(6)\to {\rm SO}(4)\times {\rm SO}(2)$ breaking. The main focus of this work is to explicitly compute the properties of these Higgses in terms of the fundamental parameters of the composite sector such as masses, Yukawa and gauge couplings of the new spin-1 and spin-1/2 resonances. Concretely, we calculate the Higgs potential at one-loop level through the Coleman-Weinberg mechanism from the explicit breaking of the ${\rm SO(6)}$ global symmetry by the partial …
A composite model of quarks and bosons is proposed in which a spin $1/2$ isospin doublet $ψ$ is the basic building block of quarks, $W^\pm$, $Z^0$ and Higgs boson $H^0$ in the standard model. The $ψ$ has two components $α$ and $β$ with charges $Q_α=\frac{2}{3}e$ and $Q_β=-\frac{1}{3}e$, respectively. The two constituent flavors $α$ and $β$ combine to form the three generations of colored standard model quark flavors. A triplet of massless hypergluons binds the constituents called geminis. The strong confining non-Abelian $SU(2)_C$ constituent color dynamics has a confining energy scale $Λ_{CD}…
We study the minimal model of conformal technicolor, an SU(2) gauge theory near a strongly coupled conformal fixed point, with conformal symmetry softly broken by technifermion mass terms. Conformal symmetry breaking triggers chiral symmetry breaking in the pattern SU(4) -> Sp(4), which gives rise to a pseudo-Nambu-Goldstone boson that can act as a composite Higgs boson. The top quark is elementary, and the top and electroweak gauge loop contributions to the Higgs mass are cut off entirely by Higgs compositeness. In particular, the model requires no top partners and no "little Higgs" mechanism…
We initiate a phenomenological study of `universal relations' in composite Higgs models, which are dictated by nonlinear shift symmetries acting on the 125 GeV Higgs boson. These are relations among one Higgs couplings with two electroweak gauge bosons (HVV), two Higgses couplings with two electroweak gauge bosons (HHVV), one Higgs couplings with three electroweak gauge bosons (HVVV), as well as triple gauge boson couplings (TGC), which are all controlled by a single input parameter: the decay constant $f$ of the pseudo-Nambu-Goldstone Higgs boson. Assuming custodial invariance in strong secto…
These lectures start with a detailed pedagogical introduction to electroweak symmetry breaking in the Standard Model, including gauge boson and fermion mass generation and the resulting predictions for Higgs boson interactions. I then survey Higgs boson decays and production mechanisms at hadron and $e^+e^-$ colliders. I finish with two case studies of Higgs physics beyond the Standard Model: two-Higgs-doublet models, which I use to illustrate the concept of minimal flavor violation, and models with isospin-triplet scalar(s), which I use to illustrate the concept of custodial symmetry.
Composite states of electrically charged and QCD-colored hyperquarks (HQs) in a confining SU(N_HC) hypercolor gauge sector are a plausible extension of the standard model at the TeV scale, and have been widely considered as an explanation for the tentative LHC diphoton excess. Additional new physics is required to avoid a stable charged hyperbaryon in such theories. We classify renormalizable models allowing the decay of this unwanted relic directly into standard model states, showing that they are significantly restricted if the new scalar states needed for UV completion are at the TeV scale.…
In gauge theories, observable quantities have to be gauge-invariant. In general, this requires composite operators, which usually have substantially different properties, e.g. masses, than the elementary particles. Theories with a Higgs field, in which the Brout-Englert-Higgs effect is active, provide an interesting exception to this rule. Due to an intricate mechanism, the Fröhlich-Morchio-Strocchi mechanism, the masses of the composite operators with the same $J^P$ quantum numbers, but modified internal quantum numbers, have the same masses. This mechanism is supported using lattice gauge th…
Exotic decays of top partners in new bosons are the norm in realistic models of a composite Higgs. We focus on the custodial charge-$5/3$ partner, which normally decays exclusively into $tW^+$. The new channels include a colour-sextet, $X_{5/3} \to \bar{b} π_6$, as well as singly and doubly charged scalars, $X_{5/3} \to t ϕ^+$, $b ϕ^{++}$. We use existing same-sign lepton searches to show that the new final states are constrained at the same level as the standard one. At the same time, exotic final states also offer opportunities for improvement: examples include a hard photon in $X_{5/3}\righ…
Within composite Higgs models based on the top seesaw mechanism, we show that the Higgs field can arise as the pseudo Nambu-Goldstone boson of the broken U(3)_L chiral symmetry associated with a vector-like quark and the t-b doublet. As a result, the lightest CP-even neutral state of the composite scalar sector is lighter than the top quark, and can be identified as the newly discovered Higgs boson. Constraints on weak isospin violation push the chiral symmetry breaking scale above a few TeV, implying that other composite scalars are probably too heavy to be probed at the LHC, but may be withi…
The Standard Model with a light Higgs provides a very accurate description of the electroweak precision observables. The largest deviation between the Standard Model predictions and the experimental measurements, the forward-backward asymmetry of the bottom quark A{FB}^b, can be interpreted as an indication of new physics at the TeV scale. The strong agreement between theory and experiment in the branching fraction of the Z into b-quarks puts strong constraints for new physics aiming to solve the A{FB}^b puzzle. We study a class of natural composite Higgs models that can solve the A{FB}^b anom…
We consider a class of composite Higgs models based on asymptotically free $SO(d)$ gauge theories with $d$ odd, with fermions in two irreducible representations, and in which the Higgs field arises as a pseudo Nambu-Goldstone boson and the top quark is partially composite. The Nambu-Goldstone coset containing the Higgs field, or Higgs coset, is either $SU(4)/Sp(4)$ or $SU(5)/SO(5)$, whereas the top partners live in two-index representations of the relevant flavor group ($SU(4)$ or $SU(5)$). In both cases, there is a large number of terms in the most general four-fermion lagrangian describing t…
We provide a pedagogical introduction to extensions of the Standard Model in which the Higgs is composite. These extensions are known as models of dynamical electroweak symmetry breaking or, in brief, Technicolor. Material covered includes: motivations for Technicolor, the construction of underlying gauge theories leading to minimal models of Technicolor, the comparison with electroweak precision data, the low energy effective theory, the spectrum of the states common to most of the Technicolor models, the decays of the composite particles and the experimental signals at the Large Hadron Colli…
The appearance of a light composite $0^+$ scalar resonance in nearly conformal gauge-fermion theories motivates further study of the low energy structure of these theories. To this end, we present a nonperturbative lattice calculation of s-wave scattering of Goldstone bosons in the maximal-isospin channel in SU(3) gauge theory with $N_f=8$ light, degenerate flavors. The scattering phase shift is measured both for different values of the underlying fermion mass and for different values of the scattering momentum. We examine the effect of a light flavor-singlet scalar (reported in earlier studie…
In the framework of an approach to bosonization based on the use of fermionic composites as fundamental variables, a quadratic action in even Grassmann variables with the quantum numbers of the pions has been constructed. It includes the sigma-field in order to be invariant under SU(2)_L times SU(2)]_R transformations over the quarks. This action exhibits the Goldstone phenomenon reducing its symmetry to the O(3) isospin invariance. A squared mass for the pions is generated according to PCAC. The model has been investigated in the Stratonovitch-Hubbard representation, in which form it is remin…
We make a theoretical study of the $η(1405) \to π^{0} f_0(980)$ and $η(1405) \to π^{0} a_0(980)$ reactions with an aim to determine the isospin violation and the mixing of the $f_0(980)$ and $a_0(980)$ resonances. We make use of the chiral unitary approach where these two resonances appear as composite states of two mesons, dynamically generated by the meson-meson interaction provided by chiral Lagrangians. We obtain a very narrow shape for the $f_0(980)$ production in agreement with a BES experiment. As to the amount of isospin violation, or $f_0(980)$ and $a_0(980)$ mixing, assuming constant…
Gauge-invariant perturbation theory is an extension of ordinary perturbation theory which describes strictly gauge-invariant states in theories with a Brout-Englert-Higgs effect. Such gauge-invariant states are composite operators which have necessarily only global quantum numbers. As a consequence, flavor is exchanged for custodial quantum numbers in the standard model, recreating the fermion spectrum in the process. Here, we study the implications of such a description, possibly also for the generation structure of the standard model. In particular, this implies that scattering processes are…
