We construct a weakly coupled, renormalizable ultraviolet completion of the Littlest Higgs model with T-parity (LHT), based on an SU(5)xSU(2)xU(1) gauge theory with a discrete Z_2 symmetry. Our model reproduces the complete structure of the LHT below the 10 TeV scale, including the collective symmetry breaking mechanism which solves the little hierarchy problem. The model is manifestly free of anomalies, including both gauge/gravitational anomalies and anomalies involving T-parity. At the TeV scale, the model contains additional states not present in the LHT. We estimate the impact of these states on precision electroweak observables, and show that the model is realistic. We also discuss how our model can be embedded into a supersymmetric theory or a five-dimensional setup with a warped extra dimension, stabilyzing the hierarchy between the 10 TeV and the Planck scale.
Flavor changing neutral current processes are studied in the littlest Higgs model with T-parity. It is found that the logarithmic divergence reported earlier in Z boson flavor changing processes is exactly canceled by contributions from additional interaction terms of heavy fermions and the Z boson. Phenomenological impact on the K --> pi nu nu_bar processes is discussed.
We compute the relevant parameters of the combined Higgs and ϕscalar effective potential in the Littlest Higgs (LH) model. These parameters are obtained as the sum of two kind of contributions. The first one is the one-loop radiative corrections coming from fermions and gauge bosons. The second one is obtained at the tree level from the higher order effective operators needed for the ultraviolet completion of the model. Finally we analyze the restrictions that the requirement of reproducing the standard electroweak symmetry breaking of the SM set on the LH model parameters.
We recalculate the leading one-loop contributions to mu > e gamma and mu -> eee in the Littlest Higgs model with T-parity, recovering previous results for the former. When all the Goldstone interactions are taken into account, the latter is also ultraviolet finite. The present experimental limits on these processes require a somewhat heavy effective scale ~2.5 TeV, or the flavour alignment of the Yukawa couplings of light and heavy leptons at the ~10% level, or the splitting of heavy lepton masses to a similar precision. Present limits on tau decays set no bounds on the corresponding parameter…
We have examined in detail the nonrestoration of symmetry at high temperature in a finite-temperature Littlest Higgs model, without and with T-parity, by evaluating the one-loop-order finite-temperature integrals of the effective potential numerically, without the high-temperature approximation, We observe that in the model without T-parity it is not possible to find a transition temperature within the allowed temperature range of the model (0<T<4f) if the UV completion factors are those which give the Standard Model electroweak minimum, as the effective potential always increases with tempera…
We calculate the corrections to the partial widths of the light Higgs boson in the Littlest Higgs model due to effects of the TeV-scale physics. We focus on the loop-induced Higgs coupling to photon pairs, which is especially sensitive to the effects of new particles running in the loop. This coupling can be probed with high precision at a photon collider in the process gamma gamma -> H -> b bbar for a light Higgs boson with mass 115 GeV <= MH <~ 140 GeV. Using future LHC measurements of the parameters of the Littlest Higgs model, one can calculate a prediction for this process, which will ser…
Large extra dimensions are one of the constructions addressing the hierarchy problem of the Standard Model. Their main theoretical and phenomenological challenge is that already predicting LHC effects requires an ultraviolet completion of TeV-scale gravity. In these lecture notes we first give a basic introduction into TeV-scale gravity models and their collider effects and then discuss possible ultraviolet completions, like string theory and fixed-point gravity.
Effective Flavour Models do not address questions related to the nature of the fundamental renormalisable theory at high energies. We study the ultraviolet completion of Flavour Models, which in general has the advantage of improving the predictivity of the effective models. In order to illustrate the important features we provide minimal completions for two known A4 models. We discuss the phenomenological implications of the explicit completions, such as lepton flavour violating contributions that arise through the exchange of messenger fields.
This letter constructs, making use of the on-shell spinor-helicity formalism, a possible ultraviolet completion of gravity following a "bottom-up" approach. The assumptions of locality, unitarity and causality i) require an infinite tower of resonances with increasing spin and quantized mass, ii) introduce a duality relation among crossed scattering channels, and iii) dress all gravitational amplitudes in the Standard Model with a form factor that closely resembles either the Veneziano or the Virasoro-Shapiro amplitude in string theory. As a consequence of unitarity, the theory predicts leadin…
We highlight how the existence of an ultraviolet completion for interacting Standard-Model type matter puts constraints on the viable microscopic dynamics of asymptotically safe quantum gravity within truncated Renormalization Group flows. A first constraint -- the weak-gravity bound -- is rooted in the destruction of quantum scale-invariance in the matter system by strong quantum-gravity fluctuations. A second constraint arises by linking Planck-scale dynamics to the dynamics at the electroweak scale. Specifically, we delineate how to extract a prediction of the top quark mass from asymptotic…
The static electromagnetic properties of the $W$ boson, $Δκ$ and $ΔQ$, are calculated in the T-parity and non T-parity littlest Higgs model (LHM) including terms up to the order of $(v/f)^4$, with $v$ the standard model (SM) vacuum expectation value and $f$ the scale of the global symmetry breaking. There are contributions from the virtual effects of the new heavy particles and also from the new corrections to the SM vertices, which depend on the mixing parameter $c$ and decrease quickly as $f$ increases. Depending on the value of $c$, the partial contributions to $Δκ$ and $ΔQ$ can add constru…
We investigate the Lagrangian terms of scalar-gauge interactions in a weakly-coupled gauge theory beyond the ultraviolet momentum cutoff of models where the collective symmetry breaking mechanism protects the Higgs mass squared against having quadratic divergence. We also propose a five-dimensional model in which the gauge symmetry breaking is achieved by boundary conditions consistent with local gauge transformations. It is shown that the scalar-gauge operators in the two models differ in the absence of quadratic divergence.
We propose an ultraviolet completion model for pseudo-Nambu-Goldstone dark matter with a hidden $\mathrm{U}(1)$ gauge symmetry. Compared to previous studies, this setup is simpler, introducing less interactions. Dark matter scattering off nucleons is highly suppressed by the ultraviolet scale and direct detection constraints can be easily evaded. The kinetic mixing between the hidden $\mathrm{U}(1)$ and the $\mathrm{U}(1)_\mathrm{Y}$ gauge fields would lead to dark matter decays. We find that the current bound on the dark matter lifetime implies that the ultraviolet scale should be higher than…
We calculate the partial widths of the light Higgs boson in the Littlest Higgs model. The loop-induced Higgs coupling to photon pairs, which is especially sensitive to effects of the new TeV-scale particles running in the loop, can be probed with high precision at a photon collider in the process gamma gamma --> H --> b bbar. Using the parameters of the Littlest Higgs model measured at the LHC one can calculate a prediction for this process, which will serve as a test of the model and as a probe for a strongly-coupled UV completion at the 10 TeV scale.
In a gauge theory with SU(5) broken to SU(2)$\times$U(1), the Higgs mass squared receives only logarithmic divergence from all scalar-gauge interactions at one loop. The same pattern of gauge symmetry breaking can be achieved without any scalar fields in a five-dimensional model.
The Scotogenic model is an economical scenario that generates neutrino masses at the 1-loop level and includes a dark matter candidate. This is achieved by means of an ad-hoc $\mathbb{Z}_2$ symmetry, which forbids the tree-level generation of neutrino masses and stabilizes the lightest $\mathbb{Z}_2$-odd state. Neutrino masses are also suppressed by a quartic coupling, usually denoted by $λ_5$. While the smallness of this parameter is natural, it is not explained in the context of the Scotogenic model. We construct an ultraviolet completion of the Scotogenic model that provides a natural expla…
Composite asymmetric dark matter scenarios naturally explain why the dark matter mass density is comparable with the visible matter mass density. Such scenarios generically require some entropy transfer mechanism below the composite scale; otherwise, their late-time cosmology is incompatible with observations. A tiny kinetic mixing between a dark photon and the visible photon is a promising example of the low-energy portal. In this paper, we demonstrate that grand unifications in the dark and the visible sectors explain the origin of the tiny kinetic mixing. We particularly consider an ultravi…
In the presence of the T-parity violating Wess-Zumino-Witten (WZW) anomaly term, the otherwise stable heavy photon A_H in the Littlest Higgs model with T-parity (LHT) decays to either Standard Model (SM) gauge boson pairs, or to SM fermions via loop diagrams. We make a detailed study of the collider signatures where the A_H can be reconstructed from invariant mass peaks in the opposite sign same flavor dilepton or the four-lepton channels. This enables us to obtain information about the fundamental symmetry breaking scale f in the LHT and thereby the low-lying mass spectrum of the theory. In a…
Neutrino masses and mixings can be generated in many different ways, with some of these scenarios featuring new physics at energy scales relevant for Large Hadron Collider searches. A systematic approach to constructing a large class of models for Majorana neutrinos may be founded upon a list of gauge-invariant effective operators -- formed from quarks, leptons and the Higgs doublet -- that violate lepton-number conservation by two units. By opening up these operators in all possible ways consistent with some minimality assumptions, a complete catalogue of a class of minimal radiative neutrino…
We calculate the most interesting rare and CP-violating K and B decays in the Littlest Higgs model with T-parity. We give a collection of Feynman rules including v^2/f^2 contributions that are presented here for the first time and could turn out to be useful also for applications outside flavour physics. We adopt a model-independent parameterization of rare decays in terms of the gauge independent functions X_i,Y_i,Z_i (i=K,d,s), which is in particular useful for the study of the breaking of the universality between K, B_d and B_s systems through non-MFV interactions. Performing the calculatio…
We study the Gauss-Bonnet (GB) term as the leading higher-curvature correction to pure Einstein gravity. Assuming a tree-level ultraviolet completion free of ghosts or tachyons, we prove that the GB term has a nonnegative coefficient in dimensions greater than four. Our result follows from unitarity of the spectral representation for a general ultraviolet completion of the GB term.
Institute for Advanced Study（高等研究院）；Sorbonne Université, CNRS（索邦大学，法国国家科学研究中心）；Scuola Normale Superiore and INFN（比萨高等师范学院和意大利国家核物理研究所）；Academy of Athens, Division of Natural Sciences（雅典科学院自然科学部）；Texas A&M University（德克萨斯农工大学）
We compute all the tree-level contributions to the Wilson coefficients of the dimension-six Standard-Model effective theory in ultraviolet completions with general scalar, spinor and vector field content and arbitrary interactions. No assumption about the renormalizability of the high-energy theory is made. This provides a complete ultraviolet/infrared dictionary at the classical level, which can be used to study the low-energy implications of any model of interest, and also to look for explicit completions consistent with low-energy data.
We explore a simple parameterization of new physics that results in an ultraviolet complete gauge-quark sector of the Standard Model. Specifically, we add an antiscreening contribution to the beta functions of the gauge couplings and a flavor-independent, antiscreening contribution to the beta functions of the Yukawa couplings. These two free parameters give rise to an intricate web of Renormalization Group fixed points. Their predictive power extends to the flavor structure and mixing patterns, which we investigate to demonstrate that some of the free parameters of the Standard Model could be…
The ultraviolet completion of flavour models can strongly improve the predictivity of the respective effective models. We consider $A_4$ models, existing minimal UV completions and construct several next-to-minimal UV complete models. We compare the results of these possibilities to the experimental data including $θ_{13}$. Through the predictive power of the UV completions, we are able to either rule out or constrain several minimal and next-to-minimal alternatives.
The CDF and D0 collaborations have observed a forward-backward asymmetry in t-tbar production at large invariant mass in excess of the standard model prediction. One explanation involves a heavy color octet particle with axial vector couplings to quarks (an axigluon). We describe and contrast various aspects of axigluons obtained from the breaking of a chiral SU(3)_L x SU(3)_R gauge theory both from the standpoint of a string-inspired field theory and from a quiver analysis of a local type IIa intersecting brane construction. Special attention is paid to the additional constraints and issues t…
It is shown that, $(a Λ^2 + b |H|^2)R$ in a spacetime of curvature $R$ is a natural ultraviolet $(U\!V)$ completion of $(a Λ^4 + b Λ^2 |H|^2)$ in the flat-spacetime Standard Model $(S\!M)$ with Higgs field $H$, $U\!V$ scale $Λ$ and loop factors $a$, $b$. This curvature completion rests on the fact that a $Λ$-mass gauge theory in flat spacetime turns, on the cut-view $R = 4 Λ^2$, into a massless gauge theory in curved spacetime. It provides a symmetry reason for curved spacetime, wherein gravity and matter are both low-energy effective phenomena. Gravity arises correctly if new physics exists w…
We introduce a scalar-less anomaly free chiral gauge theory that serves as natural ultraviolet completion of models of fundamental composite (Goldstone) Higgs dynamics. The new theory is able to generate the top mass and furthermore features a built-in protection mechanism that naturally suppresses the bottom mass. At low energies the theory predicts new fractionally charged fermions, and a number of four-fermion operators that, besides being relevant for the generation of the top mass, also lead to an intriguing phenomenology for the new states predicted by the theory.
Motivated by the observation that the Higgs quartic coupling runs to zero at an intermediate scale, we propose a new framework for models of split supersymmetry, in which gauginos acquire intermediate scale Dirac masses of $\sim 10^{8-11}$ GeV. Scalar masses arise from one-loop finite contributions as well as direct gravity-mediated contributions. Like split supersymmetry, one Higgs doublet is fine-tuned to be light. The scale at which the Dirac gauginos are introduced to make the Higgs quartic zero is the same as is necessary for gauge coupling unification. Thus, gauge coupling unification pe…
Because of ultraviolet/infrared (UV/IR) mixing, the low energy physics of noncommutative gauge theories in the Moyal-Weyl approach seems to depend crucially on the details of the ultraviolet completion. However, motivated by recent string theory analyses, we argue that their phenomenology with a very general class of UV completions can be accurately modelled by a cutoff close to the Planck scale. In the infrared the theory tends continuously to the commutative gauge theory. If the photon contains contributions from a trace-U(1), we would observe vacuum birefringence, i.e. a polarisation depend…
A review of the Higgs portal-vector dark matter interpretation of the spin-independent dark-matter nucleon elastic scattering cross section is presented, where the invisible Higgs decay width measured at the LHC is used. Effective Field Theory and ultraviolet complete models are discussed. LHC interpretations show only the scalar and Majorana dark-matter scenarios; we propose to include interpretation for vector dark matter in the EFT and UV completions theoretical framework. In addition, our studies suggest an extension of the LHC dark matter interpretations to the sub-GeV regime.
We parametrise the space of all possible flavour non-universal $\mathfrak{u}(1)_X$ extensions of the Standard Model that embed inside anomaly-free semi-simple gauge theories, including up to three right-handed neutrinos. More generally, we parametrise all abelian extensions (i.e.) by any number of $\mathfrak{u}(1)$'s) of the SM with such semi-simple completions. The resulting space of abelian extensions is a collection of planes of dimensions $\leq 6$. Numerically, we find that roughly $2.5\%$ of anomaly-free $\mathfrak{u}(1)_X$ extensions of the SM with a maximum charge ratio of $\pm 10$ can …
The minimal $SO(5)/SO(4)$ sigma model is used as a template for the ultraviolet completion of scenarios in which the Higgs particle is a low-energy remnant of some high-energy dynamics, enjoying a (pseudo) Nambu-Goldstone boson ancestry. Varying the $σ$ mass allows to sweep from the perturbative regime to the customary non-linear implementations. The low-energy benchmark effective non-linear Lagrangian for bosons and fermions is obtained, determining as well the operator coefficients including linear corrections. At first order in the latter, three effective bosonic operators emerge which are …
Electroweak precision tests put stringent constraints on the parameter space of little Higgs models. Tree-level exchange of TeV scale particles in a generic little Higgs model produce higher dimensional operators that make contributions to electroweak observables that are typically too large. To avoid this problem a discrete symmetry dubbed T-parity can be introduced to forbid the dangerous couplings. However, it was realized that in simple group models such as the littlest Higgs model, the implementation of T-parity in a UV completion could present some challenges. The situation is analogous …
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 previously presented the minimal neutral naturalness model, in which neutral naturalness is realized with the minimal coset and the minimal field content in the color-neutral sector. To fully eliminating the cutoff dependence in the Higgs potential, its ultraviolet completion was laid out. In this work, we investigate in details the holographic realization in warped five-dimensional framework. Using the holographic method, we obtain the effective action on the UV brane and derive the finite Higgs potential, with the Higgs being a pseudo Nambu-Goldstone boson. The Higgs potential is fully ra…
We study the predictions of generic ultraviolet completions of two-Higgs doublet models. We assume that at the matching scale between the two-Higgs doublet model and a ultraviolet complete theory -- which can be anywhere between the TeV and the Planck scale -- arbitrary but perturbative values for the quartic couplings are present. We evaluate the couplings down from the matching scale to the weak scale and study the predictions for the scalar mass spectrum. In particular, we show the importance of radiative corrections which are essential for both an accurate Higgs mass calculation as well as…
The ``Little Higgs'' opens up a new avenue for natural electroweak symmetry breaking in which the standard model Higgs particle is realized as a pseudo-Goldstone boson and thus is generically light. The symmetry breaking structure of the Little Higgs models predicts a large multiplet of (pseudo-)Goldstone bosons and their low energy interactions below the ultraviolet (UV) completion scale $Λ\sim 4πf \sim O(10)$ TeV, where $f$ is the Goldstone decay constant. We study unitarity of the Little Higgs models by systematically analyzing the high energy scatterings of these (pseudo-)Goldstone bosons.…
We explore whether quantum gravity effects within the asymptotic safety paradigm can provide a predictive ultraviolet completion for Abelian gauge theories. We evaluate the effect of quantum gravity fluctuations on the running couplings in the gauge sector and discover an asymptotically safe fixed point of the Renormalization Group. In particular, if the strength of gravitational interactions remains below a critical strength, the minimal gauge coupling becomes asymptotically free. Further, we point out that a completely asymptotically free dynamics for the gauge field is impossible to achieve…
Quark and lepton mixing in the asymptotically safe Standard Model
Heidelberg University（海德堡大学）；Institut de Physique Théorique Philippe Meyer, Laboratoire de Physique de l’École normale supérieure (ENS), Université PSL, CNRS, Sorbonne Université, Université Paris Cité（菲利普·迈耶理论物理研究所，巴黎高等师范学院（ENS）物理实验室，PSL大学，法国国家科学研究中心，索邦大学，巴黎西岱大学）
In the popular littlest Higgs model, T-parity can be broken by Wess-Zumino-Witten (WZW) terms induced by a strongly coupled UV completion. On the other hand, certain models with multiple scalar multiplets (called moose models) permit the implementation of an exchange symmetry (X-parity) such that it is not broken by the WZW terms. Here we present a concrete and realistic construction of such a model. The little Higgs model with X-Parity is a concrete and realistic implementation of this idea. In this contribution, the properties of the model are reviewed and the collider phenomenology is discu…
This article reviews the Little Higgs models of electroweak symmetry breaking and their phenomenology. Little Higgs models incorporate a light composite Higgs boson and remain perturbative until a scale of order 10 TeV, as required by precision electroweak data. The collective symmetry breaking mechanism, which forms the basis of Little Higgs models, is introduced. An explicit, fully realistic implementation of this mechanism, the Littlest Higgs model, is then discussed in some detail. Several other implementations, including simple group models and models with T parity, are also reviewed. Pre…
When computing radiative corrections in models with compactified extra dimen- sions, one has to sum over the entire tower of Kaluza-Klein excitations inside the loops. The loop corrections generate a difference between the coupling strength of a zero-mode gauge boson and the coupling strength of its Kaluza-Klein excitation, although both originate from the same higher-dimensional gauge interaction. Furthermore, this dis- crepancy will in general depend on the cutoff scale and assumptions about the UV completion of the extra-dimensional theory. In this article, these effects are studied in deta…
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 w…
The Twin Higgs model provides a natural theory for the electroweak symmetry breaking without the need of new particles carrying the standard model gauge charges below a few TeV. In the low energy theory, the only probe comes from the mixing of the Higgs fields in the standard model and twin sectors. However, an ultraviolet completion is required below ~ 10 TeV to remove residual logarithmic divergences. In non-supersymmetric completions, new exotic fermions charged under both the standard model and twin gauge symmetries have to be present to accompany the top quark, thus providing a high energ…
Institut de Física Corpuscular (IFIC), Consejo Superior de Investigaciones Científicas (CSIC) and Universitat de València (UV)（粒子物理研究所（IFIC），西班牙国家科学研究委员会（CSIC）和瓦伦西亚大学（UV））；University of Basel（巴塞尔大学）
We discuss aspects of the hierarchy problem in effective theories with light scalars and a large, physical ultraviolet (UV) cutoff. We make two main points: (1) The (naive) fine-tuning observed in an effective theory does not automatically imply that the UV completion is fine tuned. Instead, it gives a type of upper bound on the severity of the actual tuning in the UV completion; the actual tuning can be less severe than the naive tuning or even non-existent. (2) Within an effective theory, there appear to be two types of parameter relations that can alleviate the sensitivity of the scalar mas…
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…
Ultraviolet completions for axion-like particles (ALPs) lighter than the neutral pion generically induce ALP-neutral pion mixing, and are therefore sensitive to direct constraints on the mixing angle. For ALPs below the pion mass, we demonstrate that strong and novel bounds on the ALP-pion mixing angle can be extracted from existing rare pion decay data, measured by the PIENU and PIBETA experiments.
We introduce the Hyperbolic Higgs, a novel solution to the little hierarchy problem that features Standard Model neutral scalar top partners. At one-loop order, the protection from ultraviolet sensitivity is due to an accidental non-compact symmetry of the Higgs potential that emerges in the infrared. Once the general features of the effective description are detailed, a completion that relies on a five dimensional supersymmetric framework is provided. Novel phenomenology is compared and contrasted with the Twin Higgs scenario.
We present an ultraviolet extension of the Twin Higgs in which the radial mode of twin symmetry breaking is itself a pseudo-goldstone boson. This "turtle" structure raises the scale of new colored particles in exchange for additional states in the Higgs sector, making multiple Higgs-like scalars the definitive signature of naturalness in this context. We explore the parametrics and phenomenology of a concrete Twin Turtle model and demonstrate its robustness in two different supersymmetric completions. Along the way, we also introduce a new mechanism for inducing hard twin symmetry-breaking qua…
Using a $SU(4)\rightarrow Sp(4)$ pattern of chiral symmetry breaking, we investigate the pseudo-Goldstone nature of the Higgs boson in an elementary realisation that at the same time provides an ultraviolet completion. The renormalizability of the model together with the perturbative corrections determine dynamically the direction of vacuum of the theory and the corresponding Higgs chiral symmetry breaking scale $ f \sim 14~$TeV. The Higgs boson is radiatively generated and the scalar mass spectrum, together with a second massive Higgs boson, lie in the multi-TeV range.
We present a simple microscopic realization of a pseudo-Nambu-Goldstone (pNGB) boson Higgs scenario arising from the breaking of $SO(5) \rightarrow SO(4)$. The Higgs constituents are explicitly identified as well as the interactions responsible for forming the bound state and breaking the electroweak symmetry. This outcome follows from the presence of four-fermion interactions with a super-critical coupling, and uses the Nambu-Jona-Lasinio mechanism to break the global $SO(5)$ symmetry. The Higgs potential is found to be insensitive to high energy scales due to the existence of an approximate …
We calculate the tree-level electroweak precision constraints on a wide class of little Higgs models including: variations of the Littlest Higgs SU(5)/SO(5), SU(6)/Sp(6), and SU(4)^4/SU(3)^4. By performing a global fit to the precision data we find that for generic regions of the parameter space the bound on the symmetry breaking scale f is several TeV, where we have kept the normalization of f constant in the different models. For example, the ``minimal'' implementation of SU(6)/Sp(6) is bounded by f>3.0 TeV throughout most of the parameter space, and SU(4)^4/SU(3)^4 is bounded by f^2 = f_1^2…
