Custodial Symmetry Research Articles

Does unitarity imply finiteness of electroweak oblique corrections at one loop? Constraining extra neutral Higgs bosons

Introducing an arbitrary number of neutral Higgs bosons in the electroweak symmetry breaking sector, we derive a set of conditions among Higgs couplings which need to be satisfied to maintain the unitarity of the high energy scattering amplitudes of weak gauge bosons at the tree level (unitarity sum rules). It is shown that the unitarity sum rules require the tree level $\ensuremath{\rho}$ parameter to be 1, without explicitly invoking the custodial symmetry arguments. The one loop finiteness of the electroweak oblique corrections is automatically guaranteed once these unitarity sum rules are imposed among Higgs couplings. Severe constraints on the lightest Higgs coupling (125 GeV Higgs coupling) and the mass of the second lightest Higgs boson are obtained from the unitarity and the results of the electroweak precision tests (oblique parameter measurements). These results are compared with the effective theory of the light Higgs boson, and we find simple relationships between the mass of the second lightest Higgs boson in our framework and the ultraviolet cutoff in the effective theory framework.

Supersymmetric custodial Higgs triplets and the breaking of universality

Higgs triplet models are known to have difficulties obtaining agreement with electroweak precision data and in particular constraints on the $\rho$ parameter. Either a global $SU(2)_L \otimes SU(2)_R$ symmetry has to be imposed on the scalar potential at the electroweak scale, as done in the well-known Georgi-Machacek (GM) model, or the triplet vacuum expectation values must be very small. We construct a supersymmetric model that can satisfy constraints on the $\rho$ parameter, even if these two conditions are not fulfilled. We supersymmetrize the GM model by imposing the $SU(2)_L \otimes SU(2)_R$ symmetry at a scale $\mathcal M$, which we argue should be at or above the messenger scale, where supersymmetry breaking is transmitted to the observable sector. We show that scales $\mathcal M$ well above 100 TeV and sizable contributions from the triplets to electroweak symmetry breaking can be comfortably accommodated. We discuss the main phenomenological properties of the model and demonstrate that the departure from custodial symmetry at the electroweak scale, due to radiative breaking, can show up at the LHC as a deviation in the `universal' relation for the Higgs couplings to $WW$ and $ZZ$. As a by-product of supersymmetry, we also show that one can easily obtain both large tree-level \emph{and} one loop corrections to the Higgs mass. This allows for stops that can be significantly lighter and with smaller mixing than those needed in the MSSM.

Non-standard higgs decays in U(1) extensions of the MSSM

In U(1) extensions of the Minimal Supersymmetric extension of the Standard Model there is a simple mechanism that leads to a heavy Z' boson with a mass which is substantially larger than the supersymmetry breaking scale. This mechanism may also result in a pseudoscalar state that is light enough for decays of the 125 GeV Standard Model-like Higgs boson into a pair of such pseudoscalars to be kinematically allowed. We study these decays within E6 inspired supersymmetric models with an exact custodial symmetry that forbids tree-level flavor-changing transitions and the most dangerous baryon and lepton number violating operators. We argue that the branching ratio of the lightest Higgs boson decays into a pair of the light pseudoscalar states may not be negligibly small.

An ultra-weak sector, the strong CP problem and the pseudo-Goldstone dilaton

In the context of a Coleman–Weinberg mechanism for the Higgs boson mass, we address the strong CP problem. We show that a DFSZ-like invisible axion model with a gauge-singlet complex scalar field S, whose couplings to the Standard Model are naturally ultra-weak, can solve the strong CP problem and simultaneously generate acceptable electroweak symmetry breaking. The ultra-weak couplings of the singlet S are associated with underlying approximate shift symmetries that act as custodial symmetries and maintain technical naturalness. The model also contains a very light pseudo-Goldstone dilaton that is consistent with cosmological Polonyi bounds, and the axion can be the dark matter of the universe. We further outline how a SUSY version of this model, which may be required in the context of Grand Unification, can avoid introducing a hierarchy problem.

Sigma decomposition

In composite Higgs models the Higgs is a pseudo-Goldstone boson of a high-energy strong dynamics. We have constructed the effective chiral Lagrangian for a generic symmetric coset, restricting to CP-even bosonic operators up to four momenta which turn out to depend on seven parameters, aside from kinetic terms. Once the same sources of custodial symmetry breaking as in the Standard Model are considered, the total number of operators in the basis increases up to ten, again aside from kinetic terms. Under these assumptions, we have then particularised the discussion to three distinct frameworks: the original $SU(5)/SO(5)$ Georgi-Kaplan model, the minimal custodial-preserving $SO(5)/SO(4)$ model and the minimal $SU(3)/(SU(2)\times U(1))$ model, which intrinsically breaks custodial symmetry. The projection of the high-energy electroweak effective theory into the bosonic sector of the Standard Model is shown to match the low-energy chiral effective Lagrangian for a dynamical Higgs, and it uncovers strong relations between the operator coefficients. Finally, the relation with the bosonic basis of operators describing linear realisations of electroweak symmetry breaking is clarified.

On the flavor structure of natural composite Higgs models & top flavor violation

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 tuning of this class of models is worsen in the anarchic case due to the contributions from the additional fermion resonances. We show that, even in the presence of custodial symmetry, large top flavor violating rate are naturally expected. In particular, $t\to cZ$ branching ratio of order of $10^{-5}$ is generic for this class of models. Thus, this framework can be tested in the next run of the LHC as well as in other future colliders. We also find that the top flavor violation is weakly correlated with the increase amount of fine tuning. Finally, other related flavor violation effects, such as $t \to ch$ and in the $D$ system, are found to be too small to be observed by the current and near future colliders.

Maximally symmetric two Higgs doublet model with natural standard model alignment

We study the Higgs mass spectrum as predicted by a Maximally Symmetric Two Higgs Doublet Model (MS-2HDM) potential based on the SO(5) group, softly broken by bilinear Higgs mass terms. We show that the lightest Higgs sector resulting from this MS-2HDM becomes naturally aligned with that of the Standard Model (SM), independently of the charged Higgs boson mass and $\tan \beta$. In the context of Type-II 2HDM, SO(5) is the simplest of the three possible symmetry realizations of the scalar potential that can naturally lead to the SM alignment. Nevertheless, renormalization group effects due to the hypercharge gauge coupling $g'$ and third-generation Yukawa couplings may break sizeably this SM alignment, along with the custodial symmetry inherited by the SO(5) group. Using the current Higgs signal strength data from the LHC, which disfavour large deviations from the SM alignment limit, we derive lower mass bounds on the heavy Higgs sector as a function of $\tan\beta$, which can be stronger than the existing limits for a wide range of parameters. In particular, we propose a new collider signal based on the observation of four top quarks to directly probe the heavy Higgs sector of the MS-2HDM during the run-II phase of the LHC.

Top seesaw with a custodial symmetry, and the 126 GeV Higgs

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 extension to the minimal model to incorporate the custodial symmetry by adding a vector-like electroweak doublet of quarks with hypercharge +7/6. Such a setup also protects the $$ Zb\overline{b} $$ coupling which is another challenge for many composite Higgs models. With this addition, the chiral symmetry breaking scale can be lowered to around 1 TeV, making the theory much less fine-tuned. The Higgs is a pseudo-Nambu-Goldstone boson of the broken O(5) symmetry. For the Higgs mass to be 126 GeV, the hypercharge +7/6 quarks should be around or below the chiral symmetry breaking scale, and are likely to be the lightest new states. The 14 TeV LHC will significantly extend the search reach of these quarks. To probe the rest of the spectrum, on the other hand, would require a higher-energy future collider.

Ultra-weak sector, Higgs boson mass, and the dilaton

The Higgs boson mass may arise from a portal coupling to a singlet field σ which has a very large VEV f≫mHiggs. This requires a sector of “ultra-weak” couplings ζi, where ζi≲mHiggs2/f2. Ultra-weak couplings are technically naturally small due to a custodial shift symmetry of σ in the ζi→0 limit. The singlet field σ has properties similar to a pseudo-dilaton. We engineer explicit breaking of scale invariance in the ultra-weak sector via a Coleman–Weinberg potential, which requires hierarchies amongst the ultra-weak couplings.

Symmetries, sum rules and constraints on effective field theories

Using unitarity, analyticity and crossing symmetry, we derive universal sum rules for scattering amplitudes in theories invariant under an arbitrary symmetry group. The sum rules relate the coefficients of the energy expansion of the scattering amplitudes in the IR to total cross sections integrated all the way up to the UV. Exploiting the group structure of the symmetry, we systematically determine all the independent sum rules and positivity conditions on the expansion coefficients. For effective field theories the amplitudes in the IR are calculable and hence the sum rules set constraints on the parameters of the effective Lagrangian. We clarify the impact of gauging on the sum rules for Goldstone bosons in spontaneously broken gauge theories. We discuss explicit examples that are relevant for WW-scattering, composite Higgs models, and chiral perturbation theory. Certain sum rules based on custodial symmetry and its extensions provide constraints on the Higgs boson coupling to the electroweak gauge bosons.

6D gauge-Higgs unification on T 2 /Z N with custodial symmetry

We investigate the gauge-Higgs unification models compactified on T 2 /Z N that have the custodial symmetry. We select possible gauge groups, orbifolds and representations of the matter fermions that are consistent with the custodial symmetry, by means of the group theoretical analysis. The best candidate we found is 6D SU(3) C × U(4) gauge theory on T 2 /Z 3 and the third generation quarks are embedded into bulk fermions that are the symmetric traceless rank-2 tensor of SO(6).

Unitarity and causality constraints in composite Higgs models

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 $\mathcal{O}({p}^{4})$ operators are considered. These new couplings with respect to the Standard Model should be thought of 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 dynamical resonances above 300 GeV. We investigate the properties of these new resonances and their experimental detectability. Custodial symmetry is assumed to be exact throughout, and the calculation avoids using the equivalence theorem as much as possible.

The unobservable axial-pion mixing in QCD and near-conformal dynamics

We study a chiral quark model of π, σ, ρ, and a 1 mesons in the large-N c limit. We show that the quadratic a 1 − π mixing can be set to zero at zero momentum, thus protecting the g σππ and g ρππ couplings from being contaminated by the corresponding vertices with one or both pions replaced by the axial-vector meson. We further require the chiral-quark Lagrangian to feature an approximate classical scale invariance, the latter being only broken by dimension-two mass terms, and that the longitudinal vector meson scattering amplitudes grow at most like s, where √s is the center-of-mass energy. This allows us to accurately predict the correct value of g σππ , g ρππ , and the a 1 decay constant $$f_{a_1 }$$ . We further show that strongly coupled theories with near-conformal dynamics are expected to feature an approximate custodial chiral symmetry, with interesting phenomenological consequences.

UV completions of partial compositeness: the case for a SU(4) gauge group

We present a model of partial compositeness arising as the IR limit of a SU(4) gauge theory with only fermionic matter. This group is one of the most promising ones among a handful of possible choices allowing a symmetry breaking pattern incorporating custodial symmetry and a top partner candidate, while retaining asymptotic freedom. It is favored for not giving rise to lepto-quarks or Landau poles in the SM gauge couplings. The minimal UV theory consists of five hyperfermions in the anti-symmetric representation and three in the fundamental and anti-fundamental. The IR theory is centered around the coset SU(5)/SO(5), with top partners in the fundamental of SO(5), giving rise to one composite fermion of electric charge 5/3, three of charge 2/3 and one of charge −1/3. Electro-Weak symmetry breaking occurs via top-quark-driven vacuum misalignment. The top quark mass is generated via the mechanism of partial compositeness, while the remaining fermions acquire a mass via a standard quadratic coupling to the Higgs. We compute the top and bottom quark mass matrix and the Electro-Weak currents of the composite fermions. The model does not give rise to unacceptably large deviations from the SM Z → $$ b\overline{b} $$ decay width.

Erratum: Unitarity bounds in the Higgs model including triplet fields with custodial symmetry [Phys. Rev. D77, 095009 (2008)

Erratum: Unitarity bounds in the Higgs model including triplet fields with custodial symmetry [Phys. Rev. D<b>77</b>, 095009 (2008)

Fermionic UV completions of composite Higgs models

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.

The Higgs mass coincidence problem: why is the Higgs mass $$m_H^2=m_Z m_t$$ m H 2 = m Z m t ?

In the light of the recent LHC boson discovery, we present a phenomenological evaluation of the ratio \(\rho _t=m_Z m_t/m_H^2\), from the LHC combined \(m_H\) value, we get (\((1\sigma )\)) $$\begin{aligned} \rho _t^{({\mathrm {exp}})}= 0.9956 \pm 0.0081. \end{aligned}$$ This value is close to 1 with a precision of the order \({\sim }1~\%\). Similarly we evaluate the ratio \(\rho _{Wt}=(m_W + m_t)/(2 m_H)\). From the up-to-date mass values we get \(\rho _{Wt}^{({\mathrm {exp}})}= 1.0066\pm 0.0035 (1\sigma )\). The Higgs mass is numerically close (at the \(1~\%\) level) to the \(m_H\sim (m_W+m_t)/2\). From these relations we can write any two mass ratios as a function of, exclusively, the Weinberg angle (with a precision of the order of \(1~\%\) or better): $$\begin{aligned} \frac{m_i}{m_j}&\simeq f_{ij}(\theta _W),\quad i,j=W,Z,H,t. \end{aligned}$$ (1) For example: \(m_H/m_Z \simeq 1+\sqrt{2} s_{\theta _W/2}^2\), \(m_H/m_t c_{\theta _W} \simeq 1-\sqrt{2}s_{ \theta _W/2}^2\). In the limit \(\cos \theta _W\rightarrow 1\) all the masses would become equal \(m_Z=m_W=m_t=m_H\). We review the theoretical situation of this ratio in the SM and beyond. In the SM these relations are rather stable under RGE pointing out to some underlying UV symmetry. In the SM such a ratio hints for a non-casual relation of the type \(\lambda \simeq \kappa (g^2+{g'}^2 )\) with \(\kappa \simeq 1+o(g/g_t)\). Moreover the existence of relations \(m_i/m_j \simeq f_{ij}(\theta _W)\) could be interpreted as a hint for a role of the \(SU(2)_c\) custodial symmetry, together with other unknown mechanism. Without a symmetry at hand to explain then in the SM, there arises a Higgs mass coincidence problem, why the ratios \(\rho _t,\rho _{Wt}\) are so close to one, can we find a mechanism that naturally gives \(m_H^2=m_Z m_t\), \(2m_H= m_W+m_t\)?

Exotic Higgs decays in the [formula omitted] inspired SUSY models

We study the decays of the SM-like Higgs state within the E6 inspired supersymmetric (SUSY) models with exact custodial symmetry that forbids tree-level flavor-changing transitions and the most dangerous baryon and lepton number violating operators. In these models there are two states which are absolutely stable and can contribute to the dark matter density. One of them is the lightest SUSY particle (LSP) which is expected to be lighter than 1 eV forming hot dark matter in the Universe. The presence of another stable neutral state allows to account for the observed cold dark matter density. In the considered SUSY models next-to-lightest SUSY particle (NLSP) also tend to be light. We argue that the NLSP with GeV scale mass can result in the substantial branching ratio of the nonstandard decays of the lightest Higgs boson.

Supersymmetric custodial triplets

We analyze the extension of the minimal supersymmetric standard model which includes extra $Y=(0,\ifmmode\pm\else\textpm\fi{}1)$ supersymmetric triplets with a global $SU(2{)}_{L}\ensuremath{\bigotimes}SU(2{)}_{R}$ symmetry spontaneously broken to the custodial $SU(2{)}_{V}$ by the vacuum expectation value of the neutral scalar components of doublets and triplets. The model is the supersymmetrization of the nonsupersymmetric model introduced long ago by Georgi and Machacek where the $\ensuremath{\rho}$ parameter is kept to unity at the tree level by the custodial symmetry. Accordingly, the scalar sector is classified into degenerate $SU(2{)}_{V}$ multiplets: singlets, triplets (including the one containing the Godstone bosons) and fiveplets. The singly and doubly charged chiral superfields play a key role in the unitarization of the theory. The couplings of the standard model-like Higgs to vector bosons (including $\ensuremath{\gamma}\ensuremath{\gamma}$) and fermions, and the corresponding Higgs signal strengths, are in agreement with LHC experimental data for a large region of the parameter space. Breaking of custodial invariance by radiative corrections suggests a low-scale mechanism of supersymmetry breaking.

Stückelberg mechanism in the presence of physical scalar resonances

We show that it is possible to accommodate physical scalar resonances within a minimal nonlinearly realized electroweak theory in a way compatible with a natural Hopf algebra selection criterion (Weak Power Counting) and the relevant functional identities of the model (Local Functional Equation, Slavnov-Taylor identity, ghost equations, b-equations). The Beyond-the-Standard-Model (BSM) sector of the theory is studied by BRST techniques. The presence of a mass generation mechanism \`a la St\"uckelberg allows for two mass invariants in the gauge boson sector. The corresponding 't Hooft gauge-fixing is constructed by respecting all the symmetries of the theory. The model interpolates between the Higgs and a purely St\"uckelberg scenario. Despite the presence of physical scalar resonances, we show that tree-level violation of unitarity in the scattering of longitudinally polarized charged gauge bosons occurs at sufficiently high energies, if a fraction of the mass is generated by the St\"uckelberg mechanism. The formal properties of the physically favoured limit after LHC7-8 data, where BSM effects are small and custodial symmetry in the gauge boson sector is respected, are studied.