We investigate the impact of including a dynamical charm quark on the properties of light hadrons. Our study compares the calculations performed on 2+1+1 flavor (highly improved staggered quark) ensembles at four lattice spacings to those on 2+1 flavor (clover fermion) ensembles at six lattice spacings, with both sets of ensembles employing the identical Symanzik gauge action. For the light, strange and charm flavor observables, we employ the same tadpole-improved clover fermion action. From numerical results for light and strange quark masses, pion and kaon decay constants, and $\Omega$ and $\Omega_{ccc}$ baryon masses, we find that the values obtained after continuum, chiral, and infinite-volume extrapolations are consistent within uncertainties. Even though the mixed action setup can introduce additional discretization effects, our calculation shows evidence that these effects can cancel with the discretization error in the unitary setup, resulting in better convergence in the continuum extrapolation.

We present a systematic classification of tree-level ultraviolet (UV) completions for dimension-nine SMEFT operators relevant to short-range neutrinoless double beta decay ($0\nu\beta\beta$). Using the SMEFT J-basis framework, we categorize distinct UV completions, including both all-boson and boson-fermion-boson topologies. A primary objective is the identification of minimal UV realizations, defined as the smallest set of genuine heavy degrees of freedom required to generate each operator. Out of the 505 unique mediator combinations identified, 440 are found to be minimal, with 12 cases necessitating only two distinct heavy species. While our findings reproduce the scalar- and fermion-mediated results in the literature, we extend earlier studies of vector representations by providing a complete compilation of 324 minimal UV completions featuring vector resonances within the short-range dim-9 SMEFT classification.

We investigate the impact of a circularly polarized laser field on the top quark decay process into a charged Higgs boson (t → bH⁺) within the type-I two Higgs doublet model. Our study aims to explore how an external electromagnetic field can modify key observables and potentially facilitate the experimental detection of the charged Higgs boson, addressing challenges related to missing energy in collider experiments such as the Large Hadron Collider. Employing the Dirac–Volkov formalism, we model the interaction between charged particles and the laser field and demonstrate that the presence of the laser can notably influence the decay branching ratios under suitable conditions. The analysis reveals that both the intensity and frequency of the laser field play a crucial role in determining the decay width. In particular, for a laser field strength of 3.8 × 10¹⁴ V cm⁻¹ and a photon energy of 0.117 eV, the branching ratio of the top quark decaying into a charged Higgs boson with mass in the range 80–150 GeV and a bottom quark reaches 0.97, surpassing the standard t → bW⁺ channel. These results suggest that strong electromagnetic fields can serve as an effective mechanism to enhance signals of new particles, offering promising avenues for experimental searches beyond the Standard Model.

We investigate the possibility of constraining primordial black holes (PBHs) with masses M_PBH ≳ 10¹⁵ g through Galactic diffuse synchrotron emissions. Due to Hawking radiation, these types of PBHs are expected to be stable sources of cosmic-ray (CR) electrons and positrons with energies below ${ \mathcal O }(10\,{\rm{MeV}})$. In many CR propagation models with diffusive re-acceleration characterized by a significant Alfvén velocity ${V}_{a}\sim { \mathcal O }(10)\,{\rm{km}}\,{{\rm{s}}}^{-1}$, the energies of the evaporated electrons/positrons can be further enhanced to ${ \mathcal O }(100)\,{\rm{MeV}}$ through their scattering with the Galactic random magnetic fields. Consequently, the observation of Galactic synchrotron emissions at frequencies above ∼20 MHz can provide useful constraints on the abundance of PBHs. Using the AMS-02 and Voyager-1 data on the boron-to-carbon nuclei flux ratio, we confirm that a significant Alfvén velocity V_a ∼ 20 km s⁻¹ is favored in several benchmark diffusive re-acceleration models. We show that, in this scenario, the observed low-frequency synchrotron emissions (from 22 MHz to 1.4 GHz) can provide stringent constraints on PBH abundance. The obtained conservative constraints are stronger than those derived from the Voyager-1 all-electron (electron plus positron) data by more than one order of magnitude for M_PBH ≳ 1 × 10¹⁶ g, and also stronger than our previous constraints derived from the AMS-02 positron data for M_PBH ≳ 2 × 10¹⁶ g.

In the SO(2, d) gauge theory formalism of AdS gravity established in Wang and Song (2021 Class. Quantum Grav. 38 205002), the dynamics of bulk gravity emerges from the vanishing of the boundary covariant anomaly for the SO(2, d) conservation law. In parallel with the known results on chiral anomalies, we establish the descendent structure of the holographic SO(2, d) anomaly. The corresponding anomaly characteristic class, bulk Chern–Simons like action as well as the boundary effective action are constructed systematically. The anomalous conservation law is presented both in the covariant and consistent formalisms. Due to the existence of the ruler field, not only the Bardeen–Zumino polynomial, but also the covariant and consistent currents are explicitly constructed.

In this paper, we study the phenomenology of a Dirac dark matter (DM) in the L_μ − L_τ model and investigate the neutrino oscillation behavior in the dark halo. Since DM couples to muon neutrino and tau neutrino with opposite sign couplings, it contributes effective potentials, ±A_χ, to the evolution equation of the neutrino flavor transition amplitude, which can be significant for high energy neutrino oscillations in a dense DM environment. We discuss neutrino masses, lepton mixing angles, Dirac CP phase, and neutrino oscillation probabilities in the dark halo using full numerical calculations. Results show that neutrinos can endure very different matter effects. When the potential A_χ becomes ultra-large, three neutrino flavors decouple from each other.

We consider the constraints of the purely leptonic decays of charged pseudoscalar mesons P (being π^±, K^±, ${D}_{(s)}^{\pm }$or B^±) and the gauge boson W, P → ℓν_ℓ, $P\to {\ell }{\nu }_{{\ell }}\nu \bar{\nu }$, P → ℓν_ℓe⁺e⁻ and W → ℓν_ℓ, on the leptophilic axion-like particles (ℓALPs, denoted as a) with masses in the range 10–100 MeV via the decays P → ℓν_ℓa and W → ℓν_ℓa, where a is assumed to escape the detector as missing energy or to decay into a pair of charged leptons. Comparing our numerical results with the latest experimental data, we find that some of these decay processes for P being π^± and K^± can give meaningful constraints on the flavor-diagonal coupling parameter g_ℓ/f_a with the ℓALP mass in the range considered in this paper. The strongest constraints come from the decay ${K}^{+}\to {\mu }^{+}{\nu }_{\mu }\nu \bar{\nu }$, which set upper limits of 7.60 × 10⁻²–1.03 × 10⁻¹ GeV⁻¹ on g_ℓ/f_a at the 90% C.L.

Similar to the covalent bond in chemical molecules induced by shared electrons, we proposed in Chen (2022 Commun. Theor. Phys. 74 125201) a hadronic covalent bond induced by shared light quarks to explain T_cc(3875) and the deuteron. In this paper we improve and extend this mechanism to explain Z_c(3900), which is bound by the shared light quark–antiquark pair along with sea quark–antiquark pairs from the vacuum. Our analysis is based on the following forward and backward reasoning: a hadronic molecule exists, iff the attraction between its components is strong enough, iff the wave functions of its components significantly overlap with each other, iff the Pauli principle is well satisfied among all the shared quarks and antiquarks. Additionally, X(3872) is so unique that we need to further consider the annihilation of the shared light quark–antiquark pair, just in line with the reasoning that the creation and annihilation of sea quark–antiquark pairs should be given equal consideration. Both the ‘creation’ and ‘annihilation’ molecular bonds exist only in the strong interaction, not in the electromagnetic interaction, and they provide a quasi-static low-energy platform for studying quantum chromodynamic confinement.

Both axion and dark photon dark matter are among the most promising candidates of dark matter. What we know with some confidence is that they exhibit a small velocity distribution δv $\lesssim$ v ~ 10^-3c. In addition, their mass is small, resulting in a long de Broglie wavelength and a high particle number density. Their phase space distribution contains many uncertainties, so they could give rise to either a coherent or noncoherent wave on the laboratory scale. In this paper, we demonstrated that a resonant cavity can enhance noncoherent axion-to-photon or dark photon-to-photon transitions, and the resulting power is the same as in the coherence case. The classical picture explanation is that a cavity can resonant with multiple different sources simultaneously. This aligns with the quantum perspective, where the cavity boosts dark matter particles transitioning into photons similarly to the Purcell effect. This effect increases the density of states near resonance, regardless of the coherence nature of dark matter. Certainly, the induced microwave signals in a cavity are also non-coherent, and in such case, a single-photon readout may be required.

We construct and study numerical solutions corresponding to generalized electrically charged half-monopole in Weinberg–Salam theory, denoted as Type I and Type II solutions. These solutions possess magnetic charge q_m = +2nπ/e (−2nπ/e) that is situated along the negative z-axis (positive z-axis) and electric charge q_e that depends on the electric charge parameter η, as well as net zero neutral charge. Other properties of this half-dyon configurations such as magnetic dipole moment and angular moment are studied. These solutions are closely related to the Cho–Maison monopole–antimonopole pair reported earlier but possess some distinctive features. Our results also show important implication that a full Cho–Maison monopole can undergo distortion and possesses an axially symmetric tear-drop shape.

For loops with UV divergences, assuming that the physical contributions of loops from UV regions are insignificant, a UV-free scheme method described by an equation is introduced to derive loop results without UV divergences in the calculations, i.e., a route of the analytic continuation ${{ \mathcal T }}_{{\rm{F}}}\to {{ \mathcal T }}_{{\rm{P}}}$ besides the traditional route ∞ − ∞ in the mathematical structure. This scheme provides a new perspective to an open question of the hierarchy problem of Higgs mass, i.e., an alternative interpretation without fine-tuning within the standard model.

We argue that the hypothesis that positive-parity charm meson resonances exhibit a compact tetraquark structure has some clear tension with recent lattice results for the S-wave πD system for an SU(3) flavor symmetric setting. In particular, we show that such a diquark–anti-diquark tetraquark scenario would call for the presence of a state in the flavor $[\overline{{\bf{15}}}]$ representation, not seen in the lattice analysis. Moreover, we show that analogous lattice data in the axial-vector channel are even more sensitive to the internal structure of these very interesting states.

We investigate the application of the on-shell unitarity method to compute the anomalous dimensions of effective field theory operators. We compute one-loop anomalous dimensions for the dimension-7 operator mixing in low-energy effective field theory (LEFT). The on-shell method significantly simplifies the construction of scattering amplitudes. By leveraging the correspondence between the anomalous dimensions of operator form factors and the double-cut phase-space integrals, we bypass the need for direct loop integral calculations. The resulting renormalization group equations derived in this work provide crucial insights into the scale dependence of the LEFT dimension-7 Wilson coefficients, which will aid in precision experimental fitting of these coefficients.

In 2021, LHCb collaboration reported a very narrow state in the D⁰D⁰π⁺ mass spectrum just below the D^*+D⁰ mass threshold. We consider the influence of the Castillejo–Dalitz–Dyson (CDD) pole in the scattering amplitude to derive a general treatment for the two-body final state interaction near its threshold. The line shape (or the energy dependent event distribution) are then obtained, where the parameters can be fixed by fitting to the experimental data on the D⁰D⁰π⁺ mass spectrum. Within our method the data are quite well reproduced. The pole structure in the complex energy plane indicates that the T_cc state has a large portion of elementary degree of freedom (e.g. the compact tetraquark component) inside its hadron wave function. The compositeness as a measure of molecule component in its wave function is predicted to be $0.2{3}_{-0.09}^{+0.40}$. Clearly, the non-molecular component takes a non-negligible or even dominant portion.

In this paper, we investigate three types of Feynman integrals up to any loop, including the massless banana integral, the one-mass banana integral, and the massless three-point multi-edge integral. Although these integrals are simple in topology, they are involved in many interesting processes, like heavy quarks production and decay, as well as massless quark gluon form factors. By using one-loop integration formulas recursively, we obtain the analytic results at any loop order. It turns out that the results are quite simple and compact. The calculation method used in this work is straightforward, and may be generalized to more general cases.

The abelian (p + 1)-form gauge field is inherently coupled to the p-brane worldvolume. After quantization, the corresponding p-form gauge transformation is associated with the local phase ambiguity of the p-brane wave functional. In essence, p-form gauge symmetry can be realized as a special construction of generic zero-form gauge symmetry in the functional space of p-brane configurations. The non-abelian generalization is straightforward in the functional space language. To simplify the analysis, we further introduce a toy model where the infinite dimensional functional space of p-brane configurations is replaced by a finite dimensional matrix space. After taking the symmetric trace in the matrix model, the original discussions of the p-form gauge symmetry can be inherited by the toy model.

We extend a well-known mass-gap equation for pure gluodynamics in global colour models (formulated in equal-time quantization in Coulomb gauge) to one in which gluons are split into two sets, each exhibiting different masses. If the theory is SU(N) × SU(M) with gluons in both groups having identical couplings (as suggested by Grand Unification arguments at large scales) it is immediate to see that different masses are generated for each subgroup. This global symmetry is not broken, but the split masses erase accidental symmetries that might be present due to the two couplings being the same at a large scale, such as SU(N × M) or similar. We also numerically explore a couple of low-dimensional examples of simple Lie groups, but in spite of the system of equations having a form that would seem to allow spontaneous symmetry breaking, it is not triggered for these groups whose algebra has no ideal, and the dispersion relations for the various gluons converge to the same form.

We present a calculation by including the relativistic and off-shell contributions to the interaction potentials between two spin-1/2 fermions mediated by the exchange of light spin-0 particles, in both momentum and coordinate spaces. Our calculation is based on the four-point Green's function rather than the scattering amplitude. Among the sixteen potential components, eight that vanish in the non-relativistic limit are shown to acquire nonzero relativistic and off-shell corrections. In addition to providing relativistic and off-shell corrections to the operator basis commonly used in the literature, we introduce an alternative operator basis that facilitates the derivation of interaction potentials in the coordinate space. Furthermore, we calculate both the long-range and short-range components of the potentials, which can be useful for future experimental analyses at both macroscopic and atomic scales.

The presence of background classical sources affects quantum field theory significantly in different ways. Neutrino oscillation is a phenomenon that confirms that neutrinos are massive fermions in nature, a celebrated result in modern physics. Neutrino oscillation plays an important role in many astrophysical observations. However, the interactions between the background classical sources with neutrinos are not often considered. In the present article, we show the effect of some classical sources, namely matter currents, electromagnetic waves, torsion, and gravitational waves on neutrino oscillation. It is shown explicitly that the above sources can change the helicity state of neutrinos during neutrino oscillation.

In this work, we investigate the thermodynamic variables of a harmonic oscillator in a conical geometry metric. Moreover, we introduce an external field in the form of a Wu–Yang magnetic monopole (WYMM) and an inverse square potential into the system and analyze the results. Using an analytical approach, we obtain the energy level and study the thermodynamics at finite temperature. Our findings demonstrate that thermodynamic variables, except for the specific heat and entropy, are influenced by the topological parameters, the strength of the WYMM, and the inverse square potential.

We use the Schrödinger–Newton equation to calculate the regularized self-energy of a particle using a regular self-gravitational and electrostatic potential derived in string T-duality. The particle mass M is no longer concentrated into a point but is diluted and described by a quantum-corrected smeared energy density resulting in corrections to the energy of the particle, which is interpreted as a regularized self-energy. We extend our results and find corrections to the relativistic particles using the Klein–Gordon, Proca and Dirac equations. An important finding is that we extract a form of the generalized uncertainty principle (GUP) from the corrected energy. This form of the GUP is shown to depend on the nature of particles; namely, for bosons (spin 0 and spin 1) we obtain a quadratic form of the GUP, while for fermions (spin 1/2) we obtain a linear form. The correlation we find between spin and GUP may offer insights for investigating quantum gravity.

We revisit the issue of whether the effective potential for the conformal factor of the metric, which is generated by quantized matter fields, possesses a non-vanishing vacuum expectation value (VEV) or not. We prove that the effective potential has a vanishing vacuum expectation value on the basis of a global GL(4) symmetry. We also account for why there seems to be two different effective potentials for the conformal factor in a theory, one of which gives rise to a vanishing VEV for the conformal factor, whereas the other has a non-vanishing VEV.

In this study, we investigated worldvolume fermions on the flavor brane in the D0–D4/D8 model, which is holographically equivalent to four-dimensional quantum chromodynamics with instantons or equivalently with a theta angle. The action involving the worldvolume fermions was obtained by the T-duality rules in string theory, and we accordingly derived their effective five-dimensional and canonical four-dimensional forms by using the systematic dimensional reduction and decomposition of the spinor. Subsequently, we used the AdS/CFT dictionary to evaluate the two-point correlation function as the spectral function for the worldvolume fermions and interpreted the fermions as baryons by analyzing their quantum number with the baryon vertex in holography. In this sense, the interacted action involving the worldvolume fermions and gauge field on the flavor brane was finally derived in holography, which describes the various interactions of mesons and baryons with instantons in the large-N limit. Therefore, this study provides a holographic picture to describe baryons and their interactions based on string theory, particularly in the presence of instantons or a theta angle.

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.

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 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.

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 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 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.

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.

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.

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.

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.