The major goal of this work is to find solutions of Einstein-Maxwell field equations for anisotropic, expansion-free, non-static, spherically distributed matter content. The analytical models that highlight the major benefit of simplicity are shown and this makes it possible to use them as a toy model to illustrate how cavities evolve. Furthermore, the transport equations, quasi-homologous constraints and the junction conditions are also evaluated along with their useful implications. Eventually, the consequences of electric force on this system are summed up in the last section.

In this paper, we study the time-dependent Aharonov–Casher effect and its corrections due to spatial noncommutativity. Given that the charge of the infinite line in the Aharonov–Casher effect can adiabatically vary with time, we show that the original Aharonov–Casher phase receives an adiabatic correction, which is characterized by the time-dependent charge density. Based on Seiberg–Witten map, we show that noncommutative corrections to the time-dependent Aharonov–Casher phase contains not only an adiabatic term but also a constant contribution depending on the frequency of the varying electric field.

A novel detection of sub-GeV dark matter is proposed in the paper. The electron cloud is boosted by the dark matter and emits an electron when it is dragged back by the heavy nucleus, namely the coherent scattering of the electron cloud of the atom. The survey in the x-ray diffraction shows that the atomic form factors are much more complex than the naive consideration. The results of the relativistic Hartree–Fock (RHF) method give non-trivial shapes of the atoms. The detailed calculation of the recoil of the electron cloud, the kinetics, the fiducial cross section and the corresponding calculation of detection rate are given analytically. The numerical results show that the limits of the RHF form factors are much more stringent than the recoil of a single electron, almost 4 orders stronger, and also give tight limitations compared to the Migdal effect below about several hundred MeV. The physical picture and the corresponding results are promising and need further exploration.

In this paper, we calculate the scalar a₀(980)-meson leading-twist wave function by using the light-cone harmonic oscillator model (LCHO), where the model parameters are determined by fitting the ξ-moments $\langle {\xi }_{{a}_{0}}^{n}{\rangle }_{\zeta }$ of its light-cone distribution amplitudes. Then, the a₀(980)-meson leading-twist light-cone distribution amplitudes with three different scales ζ = (1.0, 2.0, 5.2)GeV are given. After constructing the relationship between the a₀(980)-meson leading-twist parton distribution functions/valence quark distribution function and its LCHO wave function, we exhibit the ${q}^{{a}_{0}}(x,\zeta )$ and ${{xq}}^{{a}_{0}}(x,\zeta )$ with different scales. Furthermore, we also calculate the Mellin moments of the a₀(980)-meson's valence quark distribution function $\langle {x}^{n}{q}^{{a}_{0}}{\rangle }_{\zeta }$ with n = (1, 2, 3), i.e. $\langle {{xq}}^{{a}_{0}}{\rangle }_{{\zeta }_{5}}=0.027$, $\langle {x}^{2}{q}^{{a}_{0}}{\rangle }_{{\zeta }_{5}}=0.018$ and $\langle {x}^{3}{q}^{{a}_{0}}{\rangle }_{{\zeta }_{5}}=0.013$. Finally, the scale evolution for the ratio of the Mellin moments ${x}_{\,{a}_{0}}^{n}(\zeta ,{\zeta }_{k})$ are presented.

In the framework of the perturbative Quantum Chromodynamics factorization, the production of a hadron includes contributions from fragmentation as well as combination, with the latter being of higher twist. In particular, the heavy meson production can be via the combination of a heavy quark with a light one, and the cross section can be factorized to be the convolution of the combination matrix element, the light quark distribution function, and the hard partonic sub-cross section of the heavy quark production. The partonic distribution and the combination matrix element are functions of a scaling variable, respectively, which is the momentum fraction of the corresponding quark with respect to the heavy meson. We studied the D^*± production in jet via combination in pp collision at the LHC. The total result is comparable with the experimental data. The combination matrix elements can be further studied in various hadron production processes.

In this paper, we discuss some non-trivial relations for ordered exponentials on smooth Riemannian manifolds. As an example of application, we study the dependence of the four-dimensional quantum Yang–Mills effective action on the special gauge transformation with respect to the background field. Also, we formulate some open questions about a structure of divergences for a special type of regularization in the presence of the background field formalism.

We study the lattice QCD Λ_cN phase shifts for the ³S₁-³D₁ coupled channel using both the leading order covariant chiral effective theory and the next-to-leading order non-relativistic chiral effective field theory (ChEFT). We show that although it is possible to describe simultaneously the ³S₁ and ³D₁ phase shifts and the inelasticity η₁, the fitted energy range is quite small, only up to E_c.m. = 5 MeV. This raises concerns regarding the consistency between leading/next-to-leading order ChEFT and the lattice QCD simulations.

In the framework of a non-relativistic quark model, we have calculated the masses of the ${QQ}\bar{Q^{\prime} }\bar{Q^{\prime} }$ (Q = c, b, s and $\bar{Q}^{\prime} =\bar{c},\bar{b},\bar{s}$) tetraquark states. Using the linear and quadratic confinement potentials inside a Cornell-type potential along with all possible spin and color configurations, the Schrödinger equation masses of these tetraquark states have been calculated. Based on our numerical analysis, linear confinement and quadratic confinement produce acceptable results. Models that use linear confinement estimate charmonium-like tetraquark mass close to experimental data, as well as bottomonium-like tetraquark mass below the threshold of their meson-meson channels.

The relativistic quantum motions of the oscillator field (via the Klein–Gordon oscillator equation) under a uniform magnetic field in a topologically non-trivial space-time geometry are analyzed. We solve the Klein–Gordon oscillator equation using the Nikiforov-Uvarov method and obtain the energy profile and the wave function. We discuss the effects of the non-trivial topology and the magnetic field on the energy eigenvalues. We find that the energy eigenvalues depend on the quantum flux field that shows an analogue of the Aharonov–Bohm effect. Furthermore, we obtain the persistent currents, the magnetization, and the magnetic susceptibility at zero temperature in the quantum system defined in a state and show that these magnetic parameters are modified by various factors.

We investigate a cogenesis mechanism within the twin Higgs setup that can naturally explain the nature of dark matter, the cosmic coincidence puzzle, little hierarchy problem, leptogenesis, and the tiny neutrino masses. Three heavy Majorana neutrinos are introduced to the standard model sector and the twin sector respectively, which explain the tiny neutrino masses and generate the lepton asymmetry and the twin lepton asymmetry at the same time. The twin cogenesis mechanism applies to any viable twin Higgs model without an explicit ${{\mathbb{Z}}}_{2}$ breaking in the leptonic sector and evading the ΔN_eff constraint. We illustrate the twin cogenesis mechanism using the neutrino-philic twin two Higgs doublet model, a newly proposed model to lift the twin neutrino masses with spontaneous ${{\mathbb{Z}}}_{2}$ breaking. The dark photon with a Stueckelberg mass ${ \mathcal O }(10)$ MeV ensures the energy in the twin sector as well as the symmetric component of twin sector particles can be depleted. The lightest twin baryons are the dark matter candidates with masses of approximately 5.5 GeV, which explains naturally the amount of dark matter and visible matter in the Universe are of the same order. We also demonstrate twin cogenesis in the fraternal twin Higgs setup, in which the dark matter candidate is the twin bottom bound state ${{\rm{\Omega }}}_{{b}^{{\prime} }{b}^{{\prime}}{b}^{{\prime\prime}}}$.

For loop integrals, reduction is the standard method. Having an efficient way to find reduction coefficients is an important topic in scattering amplitudes. In this paper, we present the generation functions of reduction coefficients for general one-loop integrals with an arbitrary tensor rank in their numerator.

The exclusive η and π⁰ electroproduction is studied in the handbag approach based on the generalized parton distributions (GPDs) factorization. Predictions of π⁰ and η mesons are calculated for future electron-ion collider in China (EicC) energy ranges, using obtained cross sections we extract information on the transversity GPDs contributions to these processes.

In a simplified model including an SU(2) singlet vector-like top quark T (VLT) with charge 2/3, we investigate the associated production of Wb with VLT at a Compact Linear Collider (CLIC). Focusing on the T → Wb decay channel, we discuss two decay models: ${e}^{+}{e}^{-}\to \bar{T}{W}^{+}b\to ({l}^{-}\bar{\nu }\bar{b})({jjb})$ and ${e}^{+}{e}^{-}\to \bar{T}{W}^{+}b\to ({jj}\bar{b})({l}^{+}\nu b)$. In this paper, we consider recent experimental limits including the electroweak precision observables, the width of the top quark, and the direct LHC searches. Through a detailed detector simulation, we show the exclusion and discovery capability on the T at 3 TeV CLIC, where the initial state radiation effect has also been considered. The results show that it is promising to detect the VLT at future high-energy e⁺e⁻ colliders.

In this paper, following the Occam’s razor principle, we have put forward a very simple form of the Dirac neutrino mass matrix ${M}_{{\rm{D}}}^{}$ in the minimal seesaw model with the right-handed neutrino mass matrix being diagonal ${M}_{{\rm{R}}}^{}=\mathrm{diag}({M}_{1}^{},{M}_{2}^{})$; it has one texture zero and only contains three real parameters, whose values can be determined from the neutrino oscillation experimental results. Such a model leads to a neutrino mass matrix ${M}_{\nu }^{}\simeq -{M}_{{\rm{D}}}^{}{M}_{{\rm{R}}}^{-1}{M}_{{\rm{D}}}^{T}$ that obeys the TM1 and μ-τ reflection symmetries simultaneously. In this way all the lepton flavor mixing parameters except for ${\theta }_{13}^{}$ are predicted; the value of ${\theta }_{12}^{}$ is predicted by the TM1 symmetry, while those of ${\theta }_{23}^{}$, δ, ρ and σ by the μ-τ reflection symmetry. And the neutrino masses are predicted to be of the NO case with ${m}_{1}^{}=0$, for which all three light neutrino masses will be pinned down with the help of the experimental results for the neutrino mass squared differences. For these results, the effective Majorana neutrino mass $| {\left({M}_{\nu }^{}\right)}_{{ee}}^{}| $ that controls the rate of the neutrinoless double beta decay is predicted to be 1.6 or 3.8 meV in the case of σ = 0 or π/2. We have also studied the implications of the model for leptogenesis. It turns out that only in the two-flavor leptogenesis regime (which holds in the temperature range 10⁹–10¹² GeV) can leptogenesis have a chance to be successful. And a successful leptogenesis can be achieved at ${M}_{1}^{}\simeq 1.2\times {10}^{11}$ GeV in the case of σ = π/2, but not in the case of σ = 0.

We search for Lorentz symmetry violation effects at low-energy regime by exploring the Dirac equation in (1 + 1)-dimensions and the possibility of dealing with quantum systems with spherical symmetry. We bring a discussion about the influence of the Lorentz symmetry violation effects on the spectrum of molecular vibrations caused by the coupling between a fixed vector field and the derivative of the fermionic field. Further, we discuss the influence of this Lorentz symmetry violation background on the revival time.

We present the first numerical solution that corresponds to a pair of Cho-Maison monopoles and antimonopoles (MAPs) in the SU(2) × U(1) Weinberg-Salam (WS) theory. The monopoles are finitely separated, while each pole carries a magnetic charge ±4π/e. The positive pole is situated in the upper hemisphere, whereas the negative pole is in the lower hemisphere. The Cho-Maison MAP is investigated for a range of Weinberg angles, $0.4675\leqslant \tan {\theta }_{{\rm{W}}}\leqslant 10$, and Higgs self-coupling, 0 ≤ β ≤ 1.7704. The magnetic dipole moment (μ_m) and pole separation (d_z) of the numerical solutions are calculated and analyzed. The total energy of the system, however, is infinite due to point singularities at the locations of monopoles.

In this paper, the proton structure function ${F}_{2}^{p}(x,{Q}^{2})$ at small-x is investigated using an analytical solution to the Balitsky-Kovchegov (BK) equation. In the context of the color dipole description of deep inelastic scattering (DIS), the structure function ${F}_{2}^{p}(x,{Q}^{2})$ is computed by applying the analytical expression for the scattering amplitude N(k, Y) derived from the BK solution. At transverse momentum k and total rapidity Y, the scattering amplitude N(k, Y) represents the propagation of the quark-antiquark dipole in the color dipole description of DIS. Using the BK solution we extracted the integrated gluon density xg(x, Q²) and then compared our theoretical estimation with the LHAPDF global data fits, NNPDF3.1sx and CT18. Finally, we have investigated the behavior of ${F}_{2}^{p}(x,{Q}^{2})$ in the kinematic region of 10⁻⁵ ≤ x ≤ 10⁻² and 2.5 GeV² ≤ Q² ≤ 60 GeV². Our predicted results for ${F}_{2}^{p}(x,{Q}^{2})$ within the specified kinematic region are in good agreement with the recent high-precision data for ${F}_{2}^{p}(x,{Q}^{2})$ from HERA (H1 Collaboration) and the LHAPDF global parametrization group NNPDF3.1sx.

Inspired by the recent observation of the first doubly charmed tetraquark T_cc, we apply the linear Regge relation and mass scaling to study low-lying mass spectra of the doubly heavy tetraquark in a heavy-diquark−light-antidiquark picture. The measured data and other compatible estimates of ground-state masses of doubly heavy baryons are employed to evaluate masses of heavy diquark M_QQ (Q = c, b) and the D/D_s meson masses are used in mass scaling to determine the hyperfine mass splitting. Our mass computation indicates that all low-lying states of doubly heavy tetraquarks are unstable against strong decays to two heavy-light mesons, except for the ground states of nonstrange bb tetraquarks.

We present a lattice quantum chromodynamics (QCD) simulation with 2 + 1 + 1 flavor full QCD ensembles using near-physical quark masses and different spatial sizes L, at a ∼ 0.055 fm. The results show that the scalar and pesudoscalar 2-point correlator with a valence pion mass of approximately 230 MeV become degenerated at L ≤ 1.0 fm, and such an observation suggests that the spontaneous chiral symmetry breaking disappears effectively at this point. At the same time, the mass gap between the nucleon and pion masses remains larger than Λ_QCD in the entire L ∈ [0.2, 0.7] fm range.

One of the simplest ways to account for the observed W-boson mass shift is to introduce the SU(2)_L triplet Higgs boson with zero hypercharge, whose vacuum expectation value is about 3 GeV. If the triplet is heavy enough at ${ \mathcal O }(1)$ TeV, it essentially contributes only to T parameter without any conflict with the observation. The presence of a complex triplet Higgs boson raises the SU(2)_L gauge coupling constant to α₂(M_PL) ≃ 1/44 at the Planck scale. Thanks to this larger gauge coupling constant, we show that the electroweak axion vacuum energy explains the observed cosmological constant provided that the axion field is located near the hill top of the potential at present.

The current study aims to investigate the particular case of two zeros in a Majorana neutrino mass matrix based on A₄ symmetry, where charged lepton mass matrix is diagonal. The texture is ${M}_{\nu }^{{S}_{7}}$ with (μ, μ) and (τ, τ) vanishing element of the neutrino mass matrix. The texture ${M}_{\nu }^{{S}_{7}}$ has magic and μ − τ symmetry, with a tribimaximal form of the mixing matrix, which leads to θ₁₃ = 0 that it is not consistent with experimental data and at first, does not seem to be allowed. Since θ₁₃ a small mixing angle compared to others neutrino mixing angles justifies the use of perturbation theory. We propose that, θ₁₃, and the Dirac phase δ, and two Majorana phases ρ and σ could be generated by using a complex symmetric perturbation mass matrix in the mass basis and find that $\delta {m}^{2}\equiv \,{m}_{2}^{2}-{m}_{1}^{2}\ne 0$ affect to the atmospheric mixing angle. We show that only the predictions of the case I, with Δ < 0 and ${\rm{Re}}(\alpha )\lt 0$, are consistent with the experimental data. Furthermore, the allowed range of our parameter space and complex elements of perturbation mass matrix are found, which led to finding the allowed region of the neutrino masses, the Majorana phases, the effective neutrino mass for the neutrinoless double beta decay, the allowed deviation of θ₂₃ from 45°, and to predict the normal neutrino mass hierarchy. The predicted region of $\langle {m}_{{\nu }_{\beta \beta }}\rangle $ and θ₂₃ are in line with the current experimental data which indicate the accuracy of our model and its results. The results of the case II, with Δ > 0 and ${\rm{Re}}(\alpha )\gt 0$, are ruled out.

A new physics scenario to explain PeV neutrinos observed in the IceCube experiment is introduced, with dark matter and dark energy considered. A slowly decaying very heavy fermion with a PeV mass as the dark matter particle is the origin of the PeV neutrinos. They couple to an extremely light field and this light field constitutes the dark energy.

The internal structure of the charm-strange mesons ${D}_{s0}^{* }(2317)$ and D_s1(2460) are the subject of intensive studies. Their widths are small because they decay dominantly through isospin-breaking hadronic channels ${D}_{s0}^{* }{(2317)}^{+}\to {D}_{s}^{+}{\pi }^{0}$ and ${D}_{s1}{(2460)}^{+}\to {D}_{s}^{* +}{\pi }^{0}$. The D_s1(2460) can also decay into the hadronic final states ${D}_{s}^{+}\pi \pi $, conserving isospin. In that case there is, however, a strong suppression from phase space. We study the transition ${D}_{s1}{(2460)}^{+}\to {D}_{s}^{+}{\pi }^{+}{\pi }^{-}$ in the scenario that the D_s1(2460) is a D^*K hadronic molecule. The ππ final state interaction is taken into account through dispersion relations. We find that the ratio of the partial widths of the ${\rm{\Gamma }}{({D}_{s1}{(2460)}^{+}\to {D}_{s}^{+}{\pi }^{+}{\pi }^{-})/{\rm{\Gamma }}({D}_{s1}(2460)}^{+}\to {D}_{s}^{* +}{\pi }^{0})$ obtained in the molecular picture is consistent with the existing experimental measurement. More interestingly, we demonstrate that the π⁺π⁻ invariant mass distribution shows a double bump structure, which can be used to disentangle the hadronic molecular picture from the compact state picture for the D_s1(2460)⁺. Predictions on the ${B}_{s1}^{0}\to {B}_{s}^{0}{\pi }^{+}{\pi }^{-}$ are also made.

In this paper, we investigate the behavior of charm and bottom distribution functions at low x with respect to the parametrization of the gluon distribution function in the framework of the nonlinear Gribov-Levin-Ryskin-Mueller-Qiu, Zhu-Ruan-Shen (GLR-MQ-ZRS) evolution equation. Also, we solve the GLR-MQ-ZRS equation using the parametrization behavior of the gluon distribution function. The computed results are compared with the NNPDF3.0, CT14 and GRV92 collaborations.

The potential of muon colliders opens up new possibilities for the exploration of new physics beyond the Standard Model. It is worthwhile to investigate whether muon colliders are suitable for studying gluonic quartic gauge couplings (gQGCs), which can be contributed by dimension-8 operators in the framework of the Standard Model effective field theory, and are intensively studied recently. In this paper, we study the sensitivity of the process ${\mu }^{+}{\mu }^{-}\to {jj}\nu \bar{\nu }$ to gQGCs. Our results indicate that the muon colliders with c.m. energies larger than 4 TeV can be more sensitive to gQGCs than the Large Hadron Collider.

This analysis explores the new wormhole (WH) solution in the background of teleparallel gravity with minimal matter coupling. To complete this study, we consider the conformal symmetry with non-zero Killing vectors. The exact shape function is computed by considering the linear equation of state with the phantom regime. The energy conditions are investigated for the calculated shape function with the equation of state parameter. The presence of exotic matter is confirmed due to the violation of the null energy condition. The current study also explores the physical properties of the epicyclic frequencies with quasi-periodic oscillations. In the astrophysical, epicyclic frequencies are extensively employed to explore the self-gravitating system. It is concluded that a stable WH solution is acceptable for WH geometry.

We revisit the construction of the ${ \mathcal N }=1$ supersymmetric trinification models with gauge group U(3)_C × U(3)_L × U(3)_R in Type IIA string theory on ${{\mathbb{T}}}^{6}/({{\mathbb{Z}}}_{2}\times {{\mathbb{Z}}}_{2})$ orientifold with intersecting D6-branes. The non-trivial K-theory conditions and tadpole cancellation conditions severely restrict the number of allowed models even for the case of rectangular two-tori. Using a supervised search algorithm, we find a few four-family models where the highest wrapping number is 2. For these models, we present the complete particle spectra and the gauge coupling relations at the string-scale.

The axion-like particle (ALP) is one kind of the best-motivated new particles. We consider its production from the pseudoscalar mesonic decays $M\to M^{\prime} a$, with M being a pseudoscalar meson B or K. The upper limits on the flavor-conserving ALP–quark coupling parameter g_u are obtained by assuming the ALP to be an invisible particle. We find that the most severe constraint on g_u comes from the decay ${K}^{+}\to {\pi }^{+}\nu \bar{\nu }$ for 0.05 GeV ≤ M_a ≤ 0.35 GeV, while the decays ${B}^{+,0}\to {K}^{+,0}\nu \bar{\nu }$ and ${B}^{+,0}\to {\pi }^{+,0}\nu \bar{\nu }$ can also generate significant constraints.

We construct a doped holographic superconductor in the Gubser–Rocha model, and realize a superconducting dome in the middle of the temperature-doping phase diagram. It is worth noting that unlike in previous research, the profile of our dome shrinks inward near to zero temperature. From the numerical observation for the coupling dependence of the phase diagram, we find that the coupling between the two gauge fields plays a crucial role in the formation of the dome. We also analytically calculate the DC conductivity of the normal phase of the system in the momentum dissipation and obtain resistivity which is proportional to the temperature. The AC conductivity is calculated numerically.

We conducted a study on a simplified dark matter model that introduces a vector-like intermediate particle, facilitating exclusive interactions between dark matter and the top quark in the Standard Model. The analysis focused on the relic density of Dirac-type fermion dark matter and highlighted the complementary role of direct detection in constraining the dark matter model. Notably, in instances when dark matter mass is small, the tree-level two-body annihilation process experiences suppression. In such scenarios, the contributions of the three-body process ($\chi \bar{\chi }\to t\bar{b}{W}^{-}$) and the one-loop process ($\chi \bar{\chi }\to {gg}$) dominate the relic abundance. With regard to direct detection, calculations were performed for the two-loop contribution to the dark-matter–gluon interaction, yielding the corresponding spin-independent scattering cross section.

We solved the Dyson–Schwinger (DS) equations for a two-flavor system with symmetry to study its flavor mixing effects. Initially, we employed the point interaction model and bare vertex approximation to reveal the structure of the solutions. Using the point interaction model, the DS equations can be solved analytically, and we found that these solutions can be classified into three groups, each forming an ellipse. These solutions exhibit SO(2) symmetry, while the original SU(2) symmetry at the Lagrangian level is dynamically broken to SO(2), corresponding to the emergence of flavor mixing effects. However, this flavor mixing effect does not manifest in the final physical state. By utilizing the system's SO(2) symmetry, we can diagonalize the propagators of the DS equations, eliminating the flavor mixing effect but causing the originally degenerate masses at the Lagrangian level to split. These mass eigenstates have identical quantum numbers but different masses. If we can correspond these to quark particles of different generations, we can explain why the three generations of quarks have different masses and obtain the corresponding quark mass spectrum. Finally, we provide the corresponding numerical results using a more realistic interaction model.

We discuss the possibility of constructing an effective gauge field theory of the nucleon interactions based on the ideas of isotopic invariance as well as hypercharge invariance as a local gauge symmetry and spontaneous breaking of this symmetry. The constructed effective field theory predicts the structure of interactions of protons and neutrons with ρ- and σ-mesons, and with pi-mesons and photons, as well as interactions of these particles with each other. The Lagrangian of the theory consists of several parts involving dimension 4 and 5 gauge invariant operators. Feynman rules for physical degrees of freedom that follow on from the Lagrangian define the structure of diagrams for one-boson exchanges between nucleons, predicting the internucleon one-boson-exchange potential as well as nucleon scattering amplitudes. The range of applicability of the effective theory is discussed and estimates are made of the resulting coupling constants. The theory predicts the mass of the neutral ρ⁰-meson to be about 1 MeV larger than the mass of the charged mesons ρ^±. The vector ω-meson, which is a sterile particle with respect to the considered gauge group SU_I(2) × U_Y(1), can be added to the scheme via a gauge-invariant operator of dimension 5, as shown in the appendix.

By embedding the Standard Model effective field theory (SMEFT) in the more general Higgs effective field theory (HEFT), we expose correlations among the coefficients of the latter that, if found to be violated in future data, would lead to the experimental falsification of the SMEFT framework. These are derived from the necessary symmetric point of HEFT and analyticity of the SMEFT Lagrangian that allows the construction of the SMEFT expansion, as laid out by other groups, and properties at that point of the Higgs-flare function ${ \mathcal F }(h)$ coupling Goldstone and Higgs bosons, of the Higgs potential V(h) and of the Higgs-top quark coupling function ${ \mathcal G }(h)$.