Custodial Symmetry Research Articles

Loop induced H±W∓Z vertices in the general two Higgs doublet model with CP violation

Understanding symmetry of extended Higgs models helps to construct the theory beyond the standard model and gives important insights for testing these models by experiments. The custodial symmetry plays an important role as a leading principle to construct non-minimal Higgs sectors. On the other hand, new sources of CP violation in the Higgs sector are essentially important to explain the baryon asymmetry of the Universe. In the two Higgs doublet model, it has been known that the CP violation and the custodial symmetry are not compatible, and thus the CP violating Higgs potential in general violates the custodial symmetry in the Higgs sector. In this paper, we discuss the loop-induced H±W∓Z vertices in the two Higgs doublet model, which are induced due to the custodial symmetry violation of the sector of the particles in the loop, and study how the vertices are affected by the CP violation. We calculate the H±W∓Z vertices in the most general two Higgs doublet model with the CP violation at the one-loop level and evaluate the decay modes H± → W±Z caused by these vertices. We obtain new contributions to the H±W∓Z vertices from the CP violating part of the model, in addition to the known contributions from the CP conserving part. Moreover, we find that an asymmetry between the decays H+ → W+Z and H− → W−Z is caused by the CP violating phases in the model, some of which are important for the electroweak baryogenesis. We also briefly mention testability of these vertices at future collider experiments.

Representation-theoretical characterization of canonical custodial symmetry in NHDM potentials

By considering the basis-covariant constituents of N-Higgs-doublet potentials, we derive necessary and sufficient conditions for canonical SO(4)C Custodial Symmetry (CS) of potentials with N>2 doublets, based on representation-theoretical and geometrical relations. In essence, our characterization relates the presence of canonical CS to basis-covariant vectors corresponding to particular bases of the defining representation of the orthogonal Lie algebras. For N=3,4 and 5, the conditions demand little computational effort to be evaluated, and we provide practical algorithms that may be efficiently implemented in a computer program, for deciding whether or not a potential is custodial-symmetric.

Custodial symmetry breaking and Higgs boson signatures at the LHC

We discuss the simplest model that predicts a tree-level modification of the ρ parameter from a shift in the W-mass without changing the prediction for the Z mass. This model predicts a new neutral Higgs and two charged Higgses, with very similar masses and suppressed couplings to the Standard Model fermions. We discuss the decay properties of these new scalar bosons, and the main signatures at the Large Hadron Collider. Comparing these signatures for the first time to the latest measurements, we show that while masses around 200 GeV are excluded for some scenarios, over a wide range of model parameter space the new bosons can have a mass close to the electroweak scale without violating existing limits from experimental searches or destroying the agreement with measurements. We investigate the scenario where the new neutral Higgs is fermiophobic and has a large branching ratio into W gauge bosons and/or photons, and show that this could lead to a signal in the diphoton mass spectrum at low Higgs masses. We discuss the different signatures that can motivate new measurements and searches at the Large Hadron Collider. Published by the American Physical Society 2024

Pseudo-Nambu-Goldstone dark matter in SU(7) grand unification

We propose a grand unified theory (GUT) pseudo-Nambu-Goldstone boson (pNGB) dark matter (DM) model based on SU(7) gauge symmetry. In the GUT model, the Standard Model (SM) gauge symmetry GSM≔SU(3)C×SU(2)L×U(1)Y and the “dark” gauge symmetry SU(2)D are unified, where the SU(2)D symmetry plays an important role in the stability of DM. The unification of SM fermions and dark sector fermions is partially realized. The gauge symmetry SU(7) is spontaneously broken to SU(5)×SU(2)×U(1) gauge symmetry at the GUT scale by the nonvanishing vacuum expectation values of an SU(7) adjoint scalar field, where the SU(5) gauge symmetry is not usual SU(5) GUT gauge symmetry. The symmetry is further broken to GSM×SU(2)D at an intermediate scale. Furthermore, the SU(2)D symmetry is broken by the SU(2)D doublet and triplet scalar fields at the TeV scale. In the pNGB DM model based on GSM×SU(2)D, the residual global U(1)V dark custodial symmetry guarantees DM stability. On the other hand, in the SU(7) pNGB DM model, this global symmetry is explicitly broken by the Yukawa interaction and the effective Majorana mass terms. In the scalar sector, the cubic coupling constants of the SU(2)D doublet and triplet scalar fields are the order parameters of the U(1)V symmetry breaking. To maintain U(1)V symmetry and thus the DM stability, we need to tune Yukawa coupling constants and cubic scalar couplings at high accuracy. We find that the allowed DM mass region is quite restricted as the gauge coupling constant of SU(2)D is determined by the condition of the gauge coupling unification. To satisfy gauge coupling unification and the current experimental constraint on proton lifetime, we find that three generations of SU(3)C adjoint fermions and another three generations of SU(2)L adjoint fermions with the intermediate mass scale are required. We also find that there is no other solution to satisfy simultaneously the gauge coupling unification and the proton decay constraint if one assumes the other symmetry breaking schemes. Published by the American Physical Society 2024

Constraining the Higgs trilinear coupling from an SU(2) quadruplet with bounded-from-below conditions

Integrating out a heavy scalar can cause the Higgs trilinear coupling to deviate from its Standard Model value: a good example is provided by an SU(2) quadruplet. Constraints on the full theory, however, can limit the size of the deviation. We show that the bounded-from-below conditions for the Standard Model extended by an SU(2) quadruplet strongly constrain the ℤ2-breaking Higgs portal and can bound the Higgs trilinear coupling close to its Standard Model value. For TeV-scale quadruplet masses in models with custodial symmetry violation, these constraints can be a few times stronger than constraints from electroweak precision measurements. For the custodial quadruplet, these are the strongest theoretical constraints available.

Custodial symmetry violation in scalar extensions of the standard model* *X.W. is supported by the Fundamental Research Funds for the Central Universities of China (GK202003018). J.-H.Y. is supported by the National Science Foundation of China (12022514, 11875003, and 12047503), the National Key Research and Development Program of China (2020YFC2201501, 2021YFA0718304), the CAS Project for Young Scientists in Basic Research (YSBR-006),

The new measurement of the W boson mass from the CDF collaboration shows a significant tension with the standard model prediction, which evidences violation of custodial symmetry in the scalar sector. We study the scalar extensions of the standard model, which can be categorized into two classes, the scalar sector with custodial symmetry (Georgi-Machacek model and its generalizations) and the scalar sector without custodial symmetry, and explore how these extensions fit to electroweak precision data and the new CDF . The favored oblique parameters originate from either the large mass splitting in the multiplet via the loop contribution or the large vacuum expectation value, which breaks custodial symmetry at the tree level. In particular, we find that GeV new particles are allowed in the scalar extension scenarios.

Complete NNLO operator bases in Higgs effective field theory

For the first time, we list the complete and independent set of operators at the next-to-next-to-leading order (NNLO) in the Higgs effective field theory (HEFT). The Young tensor technique utilized in this work guarantees the completeness and independence of the on-shell amplitude basis while the Adler zero condition imposes non-linear symmetry on the Nambu-Goldstone bosons that play the central role in the chiral Lagrangian. The spurion fields are incorporated into the gauge structure of operators following the Littlewood-Richardson rule to accommodate custodial symmetry breaking. We construct 11506 (1927574) NNLO operators for one (three) flavor of fermions for the electroweak chiral Lagrangian with the light Higgs, and enumerate 8065(1179181) operators for one (three) flavor of fermions when the right-handed neutrino is absent by Hilbert series technique. Below the electroweak symmetry breaking scale, the dimension-8 standard model effective field theory (SMEFT) operators could be matched to these HEFT operators at both the NLO and NNLO orders.

Explaining the CDF-II W-boson mass anomaly in the Georgi–Machacek extension models

Original Georgi–Machacek model can preserve the custodial symmetry at tree level after the electroweak symmetry breaking. Unless additional SU(2)_c custodial symmetry breaking effects are significant, the new physics contributions to Delta m_W are always very small. Our numerical results show that ordinary GM model can contribute to Delta m_W a maximal amount 0.0012 GeV, which can not explain the new CDF-II anomaly on W boson mass. We propose firstly to introduce small misalignment among the triplet VEVs to increase Delta m_W, which can give large new physics contributions to Delta m_W. Such slightly misaligned triplet VEVs from custodial symmetry preserving scalar potential can still be allowed. Our numerical results indicate that the resulting Delta m_W can easily reach the 1sigma range of CDF-II m_W data and the splitting among the triplet VEVs Delta v(equiv v_xi -v_chi ) can be as small as 0.8 GeV for v_chi lesssim 15 GeV. We also propose to introduce an additional custodial symmetry breaking source by extending the GM model with a low scale RH neutrino sector, which can adopt the leptogenesis mechanism and allow moderately large coupling strength h_{ij} even for triplet VEVs of order GeV. With low scale RH neutrino mass scale of order 10^2–10^4 TeV, the new physics contribution to Delta m_W can reach 0.03 GeV and is much larger than that of ordinary GM model. Combining both custodial SU(2)_c symmetry breaking effects, the small misalignment among the triplet VEVs and the moderately large h_{ij} couplings allowed with RH neutrino sector, the value of Delta m_W can still easily reach the 1sigma range of CDF-II m_W data, with a minimum splitting (among the triplet VEVs) approximately 0.7 GeV for v_chi lesssim 15 GeV.

Hilbert series, the Higgs mechanism, and HEFT

We expand Hilbert series technologies in effective field theory for the inclusion of massive particles, enabling, among other things, the enumeration of operator bases for non-linearly realized gauge theories. We find that the Higgs mechanism is manifest at the level of the Hilbert series, as expected for the partition function of an S-matrix that is subject to the Goldstone equivalence theorem. In addition to massive vectors, we detail how other massive, spinning particles can be studied with Hilbert series; in particular, we spell out the ingredients for massive gravity in general spacetime dimensions. Further methodology is introduced to enable Hilbert series to capture the effect of spurion fields acquiring vevs. We apply the techniques to the Higgs Effective Field Theory (HEFT), providing a systematic enumeration of its operator basis. This is achieved both from a direct and a custodial symmetry spurion-based approach; we compare and contrast the two approaches, and our results to those appearing in previous literature.

Pseudo-Nambu-Goldstone dark matter from non-Abelian gauge symmetry

We propose a pseudo-Nambu-Goldstone boson (pNGB) dark matter (DM) model based on an additional non-Abelian gauge symmetry $SU(2{)}_{D}$. The gauge symmetry $SU(2{)}_{D}$ is spontaneously broken to a global custodial symmetry $U(1{)}_{V}$ via the nonvanishing vacuum expectation values of $SU(2{)}_{D}$ doublet and triplet scalar fields. Due to the exact global symmetry $U(1{)}_{V}$, the lightest $U(1{)}_{V}$ charged particle becomes stable. We assume that the lightest charged particle in the model is the charged complex pNGB, which we regard as DM. It avoids the strong constraints from current DM direct detection experiments due to the property of NGB. We find that the measured energy density of DM can be reproduced when the DM mass is larger than the half of the Higgs mass, where the lower limit generally comes from the constraint of DM invisible decay and the upper limit from DM direct detection experiments depends on the model parameters.

Lepton flavor violation within the simplest little Higgs model

We carry out an exhaustive analysis of lepton flavor violating processes within the Simplest Little Higgs model. Its discovery could be expected from either $\mu\to e$ conversion in nuclei, $\mu\to e\gamma$ or $\mu\to3e$ decays. Then, the tau sector could help discriminating this model not only via $\tau\to\ell\gamma$ ($\ell=\mu,e$) and $\tau\to3\ell$ decays, but also by means of $\ell\to\tau$ conversion in nuclei, which is promising in this respect. Although the model violates slightly custodial symmetry, acommodating the recent CDF $M_W$ measurement is in tension with electroweak precision data.

CDF W mass anomaly with a Stueckelberg-Higgs portal

We propose an explanation to the new W mass measurement recently reported by the CDF collaboration, which is larger than the standard model expectation by about 7 standard deviations. To alleviate the tensions that are imposed on the electroweak sector by the new W mass measurement, we carry out an analysis in the Stueckelberg extended standard model where a new neutral gauge boson appears which mixes with the two neutral gauge bosons in the electroweak sector both via the Stueckelberg mass terms and via the gauge invariant Stueckelberg-Higgs portal interaction and spoils the custodial symmetry at the tree level so that the simple relation between the W boson mass and the Z boson mass does not hold. We find that such an extension increases the W boson mass if the new gauge boson mass is larger than the Z boson mass. We further show that there exists a significant part of the parameter space in the extended model which includes the CDF mass anomaly and is consistent with the various observables at the Z pole and consistent with the ATLAS dilepton limits. The Stueckelberg ZSt′ boson, which resolves the CDF W mass anomaly, should be searchable in future LHC experiments.

Kinetic mixing, custodial symmetry, and a lower bound on the mass of a dark gauge boson

Abstract We consider the extension of the standard model by dark fields with an Abelian spontaneously broken gauge symmetry in a hidden dark matter scenario. The dimension-four gauge-invariant terms include a kinetic mixing term and a Higgs mixing term, and we show that, after spontaneous symmetry breaking, the tree-level relation $M^{2}_{W}=M^{2}_{\tilde{Z}} \cos ^{2} \tilde{\theta }_{w}$ holds and permits us to write the mixing angle induced by the kinetic mixing in the neutral massive gauge boson sector, θζ, in terms of the values of MZ, the weak mixing angle, and of the mass of the physical dark gauge boson ZD. At the loop level, a similar relation is obtained in the $\overline{MS}$ scheme. Using the result extracted from the global fit to electroweak precision data for the ratio $\rho _{0}=M^{2}_{W}/\hat{c}^{2}_{Z} M^{2}_{Z}\hat{\rho }$, we obtain the lower bound $M_{Z_{D}}\gt M_{Z}$ for the dark gauge boson mass at the $94\%$ confidence level. We argue that this lower bound holds in the general case of theories for physics beyond the standard model with an extra U(1) gauge factor subgroup, whenever the extended Higgs potential respects custodial symmetry.

Composite two-Higgs doublet model from dilaton effective field theory

We construct a composite two-Higgs-doublet model (2HDM) within the context of dilaton effective field theory. This EFT describes the particle spectrum observed in lattice simulations of a near-conformal SU(3) gauge field theory. A second Higgs doublet is naturally accommodated within the EFT, used previously to construct a one-Higgs-doublet model. We explore the interplay of phenomenological and EFT requirements, supplemented by information from numerical lattice studies of the SU(3) gauge theory with eight fundamental (Dirac) fermions. We build a viable model with moderate parameter tuning. In the relevant portion of parameter space, we match the model to a conventional general description of 2HDMs, and comment on the role of custodial symmetry (breaking) in the scalar potential. Contributions to the precision electroweak parameter T provide a possible explanation for the recent measurement of the W-boson mass by CDF II.

Violation of custodial symmetry from W -boson mass measurements

The new measurement of the $W$-boson mass from the CDF collaboration shows a significant tension with the Standard Model prediction. We quantify this discrepancy within a state-of-the-art analysis of electroweak precision data and scrutinize the leading deformations of the Standard Model Effective Field Theory arising at dimension six. We find evidence for a non-zero value of the $T$ parameter, i.e. for a novel source of violation of custodial symmetry, pointing to physics beyond the Standard Model at the 4.5$\sigma$ level. We contextualize the implications of our findings in light of other present anomalies in Particle Physics.

W -boson mass in the triplet seesaw model

The CDF collaboration has recently published a precision measurement of the W-boson mass that differs from the Standard Model prediction by seven standard deviations. This result can be explained with additional electroweak multiplets that either break the custodial symmetry or contribute to oblique parameters at loop level. Here, we study one of the best-motivated scenarios involving new multiplets: the type-II seesaw model, which involves a scalar triplet that generates Majorana neutrino masses and can furthermore resolve the W-boson mass discrepancy. This favors a doubly-charged scalar with mass between 100 and 200 GeV as well as other scalars with a fixed mass splitting. The entire preferred parameter space is testable at the LHC.

Custodial symmetry, the Georgi-Machacek model, and other scalar extensions

In an SU(2) gauge theory, if the gauge bosons turn out to be degenerate after spontaneous symmetry breaking, obviously these mass terms are invariant under a global SU(2) symmetry that is unbroken. The pure gauge terms are also invariant under this symmetry. This symmetry is called the custodial symmetry (CS). In $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ gauge theories, CS implies a mass relation between the $W$ and the $Z$ bosons. The Standard Model, as well as various extensions of it in the scalar sector, possesses such a symmetry. In this paper, we critically examine the notion of CS and show that there may be three different classes of CS, depending on the gauge couplings and self-couplings of the scalars. Among old models that preserve CS, we discuss the two-Higgs-doublet model and the one-doublet plus two-triplet model by Georgi and Machacek. We show that for two-triplet extensions, the Georgi-Machacek model is not the most general possibility with CS. Rather, we find, as the most general extension, a new model with more parameters and hence a richer phenomenology. Some of the consequences of this new model have also been discussed.

Gauge invariant operators in the SU(2) Higgs model: Ward identities and renormalization

The renormalization properties of two local gauge invariant composite operators $(O,{R}_{\ensuremath{\mu}}^{a})$ corresponding, respectively, to the gauge invariant description of the Higgs particle and of the massive gauge vector boson, are analyzed to all orders in perturbation theory by means of the algebraic renormalization in the $SU(2)$ Higgs model, with a single scalar in the fundamental representation, when quantized in the Landau gauge in Euclidean space-time. The present analysis generalizes earlier results presented in the case of the $U(1)$ Higgs model. A powerful global Ward identity, related to an exact custodial symmetry, is derived for the first time, with deep consequences at the quantum level. In particular, the gauge invariant vector operators ${R}_{\ensuremath{\mu}}^{a}$ turn out to be the conserved Noether currents of the above-mentioned custodial symmetry. As such, these composite operators do not renormalize, as expressed by the fact that the renormalization $Z$-factors of the corresponding external sources, needed to define the operators ${R}_{\ensuremath{\mu}}^{a}$ at the quantum level, do not receive any quantum corrections. Using this Ward identity, one can also prove that the longitudinal component of the two-point correlation function $⟨{R}_{\ensuremath{\mu}}^{a}(p){R}_{\ensuremath{\nu}}^{b}(\ensuremath{-}p)⟩$ exhibits only a tree level nonvanishing contribution which, moreover, is momentum independent, thus it is not associated to any physical propagating mode. Finally, we point out that the renowned nonrenormalization theorem for the ghost-antighost-vector boson vertex in Landau gauge remains true to all orders, also in the presence of the Higgs field.

Custodial Chiral Symmetry in a Su-Schrieffer-Heeger Electrical Circuit with Memory.

Custodial symmetries are common in the standard model of particle physics. They arise when quantum corrections to a parameter are proportional to the parameter itself. Here, we show that a custodial symmetry of the chiral type is also present in a classical Su-Schrieffer-Heeger (SSH) electrical circuit with memory. In the absence of memory, the SSH circuit supports a symmetry-protected topological edge state. Memory induces nonlinearities that break chiral symmetry explicitly and spread the state across the circuit. However, the resulting state is still protected against perturbations by the ensuing custodial chiral symmetry. These predictions can be verified experimentally and demonstrate the interplay between symmetry and memory.

Status of negative coupling modifiers for extended Higgs sectors

In this work, we study the status of negative coupling modifiers in extended Higgs sectors, focusing on the ratio of coupling modifiers that probes custodial symmetry violation $\lambda_{WZ} = \kappa_{W}/\kappa_{Z}$. Higgs sectors with multiplets larger than doublets are the only weakly coupled models that give tree-level modifications to $\lambda_{WZ}$, and we explore all such models allowed by the constraint from the $\rho$ parameter and perturbative unitarity. This class of models has a custodial symmetry violating potential, while the vacuum configuration preserves the symmetry. We apply precision measurements from ATLAS and CMS and show that each data set can exclude a vast set of models with $\lambda_{WZ} < 0$ at greater than 95\% confidence level. We give evidence that $\lambda_{WZ}<0$ is excluded in all weakly coupled models.