Standard routes to a light composite Higgs either rely on tuning a single channel near criticality or protect a pseudo-Nambu–Goldstone coordinate of a coset. We introduce a third mechanism class in which the protected object is an eigenvalue of the renormalized multi-operator scalar kernel of the strong sector. Two basis-invariant diagnostics, a sector participation number and a sector gap ratio , identify collective lightness, but they cannot distinguish an accidental small determinant from a protected zero mode; the missing discriminator is the microscopic sensitivity Δ_g=|∂ln|m_H^2|/∂ln g|. A protected collective Higgs is defined by ≫1, >1, and Δ_g=𝒪(1). We prove that this class is nonempty. A rank-one TC–DTC locking invariant forbids tree-level aligned curvature, while universal vectorlike DQCD bridge fermions, massless in the microscopic Lagrangian but acquiring a common DQCD constituent mass, obey ∂_h^2∑_Aℳ_A^2|_0=0. The aligned scalar is therefore lifted only at joint two-spurion order, m^2_br=-(d_X/2π^2)g_T^2g_D^2(f_B^4/f_H^2)L_X, giving Δ_g_T=Δ_g_D=2 at leading logarithmic order. The same DTC topology admits a collective top completion and a vector-decoupling route to reducing the positive technicolor contribution to S. The mechanism is falsifiable by sector-restricted lattice spectroscopy and coupling-response scans.

We demonstrate that QCD-like gauge dynamics can be consistently embedded within the Dark Technicolor paradigm by invoking the extended Most Attractive Channel hypothesis, thereby revitalizing conventional technicolor scenarios. In this framework, the Higgs mass is generated dynamically while remaining consistent with electroweak precision tests, including constraints from the S parameter. The flavor problem is resolved by incorporating the Standard Hierarchical VEVs Model, whereas a simple Froggatt–Nielsen construction is shown to be incompatible. Couplings of techni-hadrons such as ρ _TC and η _TC^' to Standard Model fermions are highly suppressed, leading to negligible direct fermionic signatures. Nevertheless, DTC mesons remain testable at the HL-LHC, HE-LHC, and future 100 TeV collider, with promising discovery channels including b̅b , τ ^+τ ^- , tt̅ , and γγ .

We investigate higher-order perturbative corrections to hadronic $τ$ decays by applying nonlinear sequence-transformation techniques to the QCD correction $δ^{(0)}$. In particular, we employ the Shanks transformation and several of its generalisations constructed through Wynn's $\varepsilon$-algorithm, which are known to accelerate the convergence of slowly convergent or divergent series. These methods are used to extract higher-order information from the fixed-order perturbative expansion of $δ^{(0)}$. Within this framework, we estimate the perturbative coefficients $c_{5,1}$-$c_{12,1}$. In particular, we obtain $c_{5,1}=294^{+41}_{-21}$, $c_{6,1}=3415^{+450}_{-467}$, and $c_{7,1}=2.23^{+0.75}_{-0.49}\times 10^4$, where the quoted uncertainties reflect the spread among the different sequence transformations employed. Our analysis demonstrates that Shanks-type sequence transformations based on Wynn's $\varepsilon$-algorithm provide an efficient and systematic tool for probing higher-order perturbative effects in hadronic $τ$ decays in the absence of explicit multi-loop calculations.

We investigate the collider phenomenology of the standard Hierarchical VEVs Model by proposing a new version, which avoids large flavor-changing neutral current interactions, thus rendering the scale of new physics as low as the electroweak scale. The resulting collider signatures are distinctive and testable at the High-Luminosity LHC, the High-Energy LHC, and future 100 TeV hadron colliders. Remarkably, one of the pseudoscalars in the model can account for the 95.4 GeV di-photon excess observed by ATLAS and CMS. In addition, the model naturally accommodates a new class of neutrino-philic dark matter candidate, neutrinic dark matter, that interacts exclusively with neutrino pairs.

We employ Levin-type sequence transformations to accelerate the convergence of the perturbative fixed-order expansion of the QCD correction δ(0) in terms of the strong coupling αs. The method efficiently resums the series, yielding a stable and self-consistent determination of higher-order QCD corrections to hadronic τ decays, consistent with existing results. We find δLevin-FOPT(0)=0.2089±0.0040±0.0060αs, and predict c5,1=278−19+27,c6,1=3375−209+489,c7,1=(2.03−0.25+0.41)×104. Our results demonstrate that Levin-type transformations provide an efficient framework for analyzing asymptotic perturbative series, and studying the higher order perturbative behaviour of the hadronic τ decays.

By employing the extended most attractive channel hypothesis, we show that QCD-like gauge dynamics is possible in the dark-technicolour paradigm. The dark-technicolour paradigm can produce the mass of the Higgs boson, and at the same time, the S-parameter is also satisfied. Moreover, the dark-technicolour paradigm can provide a solution of the flavour problem by accommodating the standard HVM or the Froggatt-Nielsen mechanism within its framework. We present the inclusive collider signatures, such as tt̅ and γγ, of the lowest lying scalars and pseudoscalars of the dark-technicolour paradigm, for the scenarios, when the standard HVM or the Froggatt-Nielsen mechanism are implemented within its framework. The collider signatures are investigated at the high-luminosity Large Hadron Collider, the high-energy Large Hadron Collider, and a 100 TeV future hadron collider. Several signatures are within the reach of the high-luminosity Large Hadron Collider. Additionally, one of the dark-Higgs of the model can address 95.4 GeV excess reported by the ATLAS and CMS experiments. Furthermore, one of the dark-Higgs can act like a new class of chirality-dependent dark-matter particles.

We review the problem of flavour tracing back to the days when the standard model was just coming together. We focus on the recently discussed new solutions of this problem, namely models based on the discrete $\mathcal{Z}_{\rm N} \times \mathcal{Z}_{\rm M}$ flavour symmetry and the VEVs hierarchy, and discuss their various phenomenological consequences. In particular, there appears a novel feature that the solution of the flavour problem based on the discrete flavour symmetry can provide the so-called flavonic dark matter. This predicts a specific relation between the mass and the symmetry-breaking scale, which can be contrasted with the standard QCD axion.

The 𝒵_N ×𝒵_M symmetry is a novel flavour symmetry, that can provide an explanation to the flavour structure of the Standard Model through the Froggatt-Nielsen mechanism. We have investigated the flavour bounds on the minimal 𝒵_2 ×𝒵_5 , and a non-minimal 𝒵_2 ×𝒵_9 version of this symmetry using the current quark and lepton flavour physics data as well as the future projected sensitivities of the quark and lepton flavour effects. It turns out that the future high-luminosity phase-I and II of the LHCb are going to play a crucial role in constraining the parameters of the minimal and the non-minimal 𝒵_2 ×𝒵_N flavour symmetries.

We conduct a comprehensive investigation into the flavor phenomenology and collider signatures of flavon of ZN x ZM flavor symmetries for the soft symmetry-breaking scenario and a new symmetry- conserving mechanism at the high-luminosity LHC, high-energy LHC, and a 100 TeV hadron collider. The flavor physics of quark and leptonic observables places different bounds on the parameter space of flavons of ZN x ZM flavor symmetries. On the collider side, the decay t -> ca can be probed by the high-luminosity LHC, high-energy LHC, and a 100 TeV hadron collider for the Z8 x Z 22 flavor symmetry. The inclusive production signatures can be used to probe the flavon of all the ZN x ZM flavor symmetries for the soft symmetry-breaking scenario for a heavy flavon at a 100 TeV collider. Flavons of all the ZN x ZM flavor symmetries can be probed at high-energy LHC and a 100 TeV collider for a low mass in the case of soft symmetry breaking. The diflavon production is within reach of the high-luminosity LHC, high-energy LHC, and a 100 TeV collider only for a light flavon. The 14 TeV high-luminosity LHC can probe only the Z2 x Z5 and Z8 x Z 22 flavor symmetries for a few specific inclusive signatures. The symmetry-conserving scenario remains beyond the detection capabilities of any collider.

We investigate the renormalization-group scale and scheme dependence of the H → gg decay rate at the order N ^4 LO in the renormalization-group summed perturbative theory, which employs the summation of all renormalization-group accessible logarithms including the leading and subsequent four sub-leading logarithmic contributions to the full perturbative series expansion. Moreover, we study the higher-order behaviour of the H → gg decay width using the asymptotic Padé approximant method in four different renormalization schemes. Furthermore, the higher-order behaviour is independently investigated in the framework of the asymptotic Padé–Borel approximant method where generalized Borel-transform is used as an analytic continuation of the original perturbative expansion. The predictions of the asymptotic Padé–Borel approximant method are found to be in agreement with that of the asymptotic Padé approximant method. Finally, we provide the H → gg decay rate at the order N ^5 LO in the fixed-order Γ _N^5LO = Γ _0 (1.8375 ± 0.047 _α _s(M_Z),1%± 0.0004_M_t± 0.0066_M_H± 0.0036_P± 0.007_s± 0.0005_sc ), and Γ _RGSN^5LO = Γ _0 (1.841 ± 0.047 _α _s(M_Z),1%± 0.0005_M_t± 0.0066_M_H± 0.0002_μ± 0.0027_P± 0.001_sc ) in the renormalization-group summed perturbative theories.

We discuss models of the flavour problem and dark matter based on the discrete 𝒵_ N×𝒵_ M×𝒵_ P flavour symmetry. A new class of dark-matter emerges out of these models, which is defined as the flavonic dark matter. An ultra-violet completion of these models based on the dark-technicolour paradigm is also presented.

We investigate flavour bounds on the Z 2 × Z 5 and Z 2 × Z 9 flavour symmetries. These flavour symmetries are a minimal and a non-minimal forms of the Z 2 × Z N flavour symmetry, that can provide a simple set-up for the Froggatt–Nielsen mechanism. The Z 2 × Z 5 and Z 2 × Z 9 flavour symmetries are capable of explaining the fermionic masses and mixing pattern of the standard model including that of the neutrinos. The bounds on the parameter space of theflavonfieldofthe Z 2 × Z 5 and Z 2 × Z 9 flavoursymmetries are derived using the current quark and lepton flavour physics data and future projected sensitivities of quark and lepton flavoureffects.Thestrongestboundsontheflavonofthe Z 2 × Z 5 symmetry come from the D 0 − ¯ D 0 mixing. The bounds on the Z 2 × Z 9 flavour symmetry are stronger than that of the minimal Z 2 × Z 5 symmetry. The ratio R μμ provides rather robust bounds on the flavon parameters in the future phase-I and phase-II of the LHCb by leaving only a very small region in the allowed parameter space of the models.

We first time show that a common solution to dark matter and the flavor problem of the standard model can be obtained in the framework of the $\mathcal{Z}_{\rm N} \times \mathcal{Z}_{\rm M}$ flavor symmetry where the flavonic Goldstone boson of this flavor symmetry acts as a good dark matter candidate through the misalignment mechanism. Hierarchical mass pattern of quarks and charged leptons naturally follows from the discrete symmetry. For light active neutrinos, we construct the Dirac-type mass matrix which is preferred to fit the observed neutrino oscillation data with normal hierarchy. Our model predicts the axion-like photon coupling characteristically different from the standard QCD axion, and could be probed by the future X-ray or radio observations.

We discuss a low-scale realization of the dark-technicolour paradigm, where the dark-technicolour scale is close to the electroweak scale. This scenario provides an ultraviolet completion of the standard HVM, and predicts a dark-Higgs with mass m_ DH = 95.4 GeV. Moreover, the grand-unification scale in this framework can be as low as 1.18 × 10^8 GeV.

We discuss the standard hierarchical VEVs model, which predicts the leptonic mixing angles in terms of the Cabibbo angle, and masses of strange and charm quarks as sinθ_12^ℓ≥ 1 - 2 sinθ_12, sinθ_23^ℓ≥ 1 - sinθ_12, and sinθ_13^ℓ≥sinθ_12 - m_s/m_c for the normal mass ordering. This results in very precise predictions of the leptonic mixing angles given by sinθ_12^ℓ = 0.55 ± 0.00134, sinθ_23^ℓ = 0.775 ± 0.00067, and sinθ_13^ℓ = 0.1413 - 0.1509. Furthermore, we predict neutrinos to be the Dirac kind. The flavour bounds on the model are discussed, and the impact of the standard hierarchical VEVs model on the flavonic dark matter is highlighted by showing its natural emergence.

We investigate flavour bounds on the 𝒵_2 ×𝒵_5 and 𝒵_2 ×𝒵_9 flavour symmetries. These flavour symmetries are a minimal and a non-minimal forms of the 𝒵_2 ×𝒵_N flavour symmetry, that can provide a simple set-up for the Froggatt–Nielsen mechanism. The 𝒵_2 ×𝒵_5 and 𝒵_2 ×𝒵_9 flavour symmetries are capable of explaining the fermionic masses and mixing pattern of the standard model including that of the neutrinos. The bounds on the parameter space of the flavon field of the 𝒵_2 ×𝒵_5 and 𝒵_2 ×𝒵_9 flavour symmetries are derived using the current quark and lepton flavour physics data and future projected sensitivities of quark and lepton flavour effects. The strongest bounds on the flavon of the 𝒵_2 ×𝒵_5 symmetry come from the D^0 - D̅^0 mixing. The bounds on the 𝒵_2 ×𝒵_9 flavour symmetry are stronger than that of the minimal 𝒵_2 ×𝒵_5 symmetry. The ratio R_μμ provides rather robust bounds on the flavon parameters in the future phase-I and phase-II of the LHCb by leaving only a very small region in the allowed parameter space of the models.

We investigate flavour bounds on the Z(2) Z(5) and Z(2) Z(9) flavour symmetries. These flavour symmetries are a minimal and a non-minimal forms of the Z2 ZN flavour symmetry, that can provide a simple set-up for the Froggatt-Nielsen mechanism. The Z(2) Z(5) and Z(2) Z(9) flavour symmetries are capable of explaining the fermionic masses and mixing pattern of the standard model including that of the neutrinos. The bounds on the parameter space of the flavon field of the Z(2) Z(5) and Z(2) Z(9) flavour symmetries are derived using the current quark and lepton flavour physics data and future projected sensitivities of quark and lepton flavour effects. The strongest bounds on the flavon of the Z(2) Z(5) symmetry come from the D-0 - (D) over bar (0) mixing. The bounds on the Z(2)Z(9) flavour symmetry are stronger than that of the minimal Z(2) Z(5) symmetry. The ratio R-mu mu provides rather robust bounds on the flavon parameters in the future phase-I and phase-II of the LHCb by leaving only a very small region in the allowed parameter space of the models.

We present an origin of the vacuum expectation values hierarchy in a nonminimal technicolor (TC) framework which is capable of explaining the flavor spectrum of the Standard Model (SM) along with neutrino masses and mixing, and simultaneously satisfying crucial experimental bounds. The TC scale in this framework can be lower such that a SM-like Higgs boson emerges from within the model. We also derive lower bound on the mass of the vector TC state using the latest experimental bound on the S-parameter.