Standard Perturbation Theory Research Articles

Beyond 3×2-point cosmology: the integrated shear and galaxy 3-point correlation functions

We present the integrated 3-point correlation functions (3PCF) involving both the cosmic shear and the galaxy density fields. These are a set of higher-order statistics that describe the modulation of local 2-point correlation functions (2PCF) by large-scale features in the fields, and which are easy to measure from galaxy imaging surveys. Based on previous works on the shear-only integrated 3PCF, we develop the theoretical framework for modelling 5 new statistics involving the galaxy field and its cross-correlations with cosmic shear. Using realistic galaxy and cosmic shear mocks from simulations, we determine the regime of validity of our models based on leading-order standard perturbation theory with an MCMC analysis that recovers unbiased constraints of the amplitude of fluctuations parameter A s and the linear and quadratic galaxy bias parameters b 1 and b 2. Using Fisher matrix forecasts for a DES-Y3-like survey, relative to baseline analyses with conventional 3×2PCFs, we find that the addition of the shear-only integrated 3PCF can improve cosmological parameter constraints by 20–40%. The subsequent addition of the new statistics introduced in this paper can lead to further improvements of 10–20%, even when utilizing only conservatively large scales where the tree-level models are valid. Our results motivate future work on the galaxy and shear integrated 3PCFs, which offer a practical way to extend standard analyses based on 3×2PCFs to systematically probe the non-Gaussian information content of cosmic density fields.

Arbitrary (k, l) States-Solutions of the Dirac and Schrödinger Equations Interacting with Improved Spatially-Dependent Mass Coulomb Potential with an improved Coulomb-like tensor interaction Model for H-atoms from 3D-RNCS and 3D-NRNCS symmetries

The improved spatially dependent mass Coulomb potential with an improved Coulomb-like tensor interaction (ISDM(CP-CLTI)) model has been used to investigate the new bound state solutions of the deformed Dirac equation under the condition of spin symmetry and pseudospin symmetry in the 3D-relativistic noncommutative space (3D-RNCS) symmetries. The combination of spatially dependent mass Coulomb potential with Coulomb-like tensor interaction (ISDM(CP-CLTI)) and the terms (1/r^3 and 1/r^4 ) which are connected with the induced couplings (LΘ and L ̃Θ) produced from the topological defects of space-space gives ISDM(CP-CLTI) model. The new relativistic and non-relativistic energy eigenvalues for the hydrogen atoms (H-atoms) such as He+, Li+2, and Be3+ under the ISDM(CP-CLTI) model have been derived within the parameters of the parametric Bopp shift method and standard perturbation theory. The new values E_nc^sp(n,C,m_0,m_1,H,Θ,τ,χ,j,l,s,m) and E_nc^ps(n,C,m_0,m_1,H,Θ,τ,χ,j,l,l ̃,s ̃,m ̃) that we obtained appeared to be sensitive to the quantum atomic discrete quantum numbers (j,k,l,s,m,l ̃,s ̃,m ̃), the mixed potential depths (C/q,m_0,m_1,H), and noncommutativity parameters (NP) (Θ,σ,χ). Within the context of relativistic extended quantum mechanics, we have obtained certain significant special cases that, in our view, will be interesting to the specialized researcher. When we applied the three simultaneous limitations (Θ,σ,χ)→(0,0,0), NP we were able to retrieve the typical results of relativistic and non-relativistic cases in the literature. Our new model presented new physical features compared to other models known in the literature.

New relativistic and non-relativistic investigation of deformed Klein–Gordon and Schrödinger equations for new improved screened Kratzer potential in the framework of non-commutative space

The deformed Klein–Gordon equation has been solved in three-dimensional extended relativistic quantum mechanics (3D-ERQM) symmetries for the newly improved screened Kratzer potential (NISKP) model under the influence of the deformation space–space symmetries. The new relativistic energy eigenvalues were calculated using the parametric Bopp’s shift method and standard perturbation theory in addition to the approximation scheme suggested by Greene and Aldrich for the inverse square terms. The new relativistic energy eigenvalues of (LiH, HCl, CO and H2) molecules under the NISKP model were shown to be sensitive to the atomic quantum numbers ([Formula: see text]), mixed potential depths ([Formula: see text]), the screening parameter’s inverse [Formula: see text], control parameter [Formula: see text] and non-commutativity parameters ([Formula: see text], [Formula: see text], [Formula: see text]). In addition, we analyzed the non-relativistic energy values by applying the well-known transmission rules known in the literature. Furthermore, we studied many special cases useful to researchers in the framework of the new extended symmetries, such as the new screened cosine Kratzer potential, the new screened Kratzer potential (SKP), the new Kratzer potential, the new coshine Yukawa potential, the new coshine Yukawa potential, the new shifted improved SKP and the new shifted SKP. The study is further extended to calculate the mass spectra of mesons of the heavy quarkonium system such as charmonium [Formula: see text], bottomonium [Formula: see text], [Formula: see text] and light mesons [Formula: see text] and [Formula: see text] that have the quark and antiquark flavor within the framework of the NISKP model in 3D-ENRQM symmetries.

Effect of the non-commutativity of space on the improved Mobius square plus generalized Yukawa potentials of the Klein–Gordon and Schrödinger equations in 3D-RNCQS and 3D-NRNCQS symmetries

Under the influence of the deformation space-space symmetries, the improved Mobius square plus generalized Yukawa potentials (IMSGYPs) have been employed to solve the deformed Klien–Gordon equation in three-dimensional noncommutative relativistic quantum space (3D-RNCQS) symmetries. Combined with the approximation approach suggested by Greene and Aldrich, we also employ the parametric Bopp’s shift approach and standard perturbation theory to derive novel relativistic energy eigenvalues. The new relativistic energy eigenvalues of (N2, K2, NI, ScI, and RbH) diatomic molecules under the IMSGYPs were shown to be sensitive to the atomic quantum numbers ([Formula: see text]), the mixed potential depths ([Formula: see text]), the screening parameter’s inverse [Formula: see text] and non-commutativity parameters ([Formula: see text], [Formula: see text], [Formula: see text]). In addition, we analyzed the new non-relativistic energy values in three-dimensional noncommutative non-relativistic quantum space (3D-NRNCQS) symmetries, by applying the well-known mapping in the literature. Furthermore, we studied many special cases useful to researchers in the framework of the new extended symmetries, such as the newly generalized Mobius square potential, the newly generalized Yukawa potential, and the newly generalized Deng-Fan potential. The study is further extended to calculate the mass spectra of mesons of the heavy quarkonium system, such as [Formula: see text], bottomonium [Formula: see text], [Formula: see text] and light mesons [Formula: see text] and [Formula: see text], that have the quark and antiquark flavors within the framework of the IMSGYPs model in 3D-NRNCQS symmetries.

The Influence of Deformation Space-Space on High and Low Energy Spectra of Fermionic Particles and Spectra of Heavy Quarkonia with Improved Hulthén Plus Hyperbolic Exponential Inversely Quadratic Potential

In this work, the modified approximation to the centrifugal barrier term is applied to find approximate bound-state solutions of the deformed Dirac equation for the spin and pseudospin symmetries in a model with the improved hyperbolic Hulth´en and hyperbolic exponential inversely quadratic potentials (IHHEIQPs) using the parametric method of Bopp’s shift and the standard perturbation theory in the extended relativistic quantum mechanics (ERQM). Our results indicate that the new energy eigenvalues are highly sensitive to the potential parameters (ν1, A) and to the values of quantum atomicnumbers (j, k, l, m,̃︀ l, ̃︀ m, s, ̃︀s), range of the potential ν, and noncommutativity parameters (θ, σ, β). We found that the effect of a space-space deformation gives a correction in the energy spectrum, where the main energy term remains due to the effect of the hyperbolic Hulth´en and hyperbolic exponential inversely quadratic potentials known in the literature. The new nonrelativistic energies are obtained by applying the nonrelativistic limit to the relativistic spin-energy equation in the extended nonrelativistic quantum mechanics (ENRQM). The proposed potential model reduces to the improved Hulth´en and exponential inversely quadratic potentials as special cases in ERQM. The present results are applied for calculating the new mass spectra Mhiqpnc−nl of heavy mesons such as cc, bb, bc, bs, cs, and bq,q = (u, d) in ENRQM. It turns out that the values of masses come from the contribution of the mass spectra Mhiqpnl in NRQM, while the effect of a space-space deformation δMhiqpnc−nl is an infinitesimal correction as compared with Mhiqpnl. Our results seem to be significant and agree perfectly with the ones in the literature.

Propagation of Nonlinear Dust-Acoustic Waves in a Self-Gravitating Collision Magnetized Dusty Plasma in Earth’s Magnetosphere

The nonlinear propagation of different types of DANW acoustic dust nonlinear waves has been investigated in a magnetized dusty plasma consisting of negatively charged dust particles, Maxwellian electrons, and ions. Application of the standard reductive perturbation theory is used to derive the corresponding three-dimensional nonlinear a complex Ginzburg–Landau (3D-CGLE) equation which governs the dynamics of the dust-acoustic wave packets. The stationary analytical solutions of the CGLE are numerically analysed where the effect of the physical parameters of the dusty plasma model on the wave’s propagation is taken into account. It has been found that there can be a relationship between the appearance of soliton waves and electromagnetic waves, as well as between shock-like waves and periodic travelling waves. Expression of the importance of these findings is the cornerstone of explaining the true relationship between the propagation of nonlinear waves in the physics of space, for example, the Earth’s magnetic field.

The behaviour of the deformed Klein-Gordon and Schrödinger equations in 3D-RNCQS and 3D-NRNCQS symmetries under the improved Eckart plus a class of Yukawa potential

We report the new bound state solutions of novel non-central potentials, the improved Eckart plus an improved class of Yukawa potential (IECYP), in the deformation Klein-Gordon equation by using Bopp's shifts method and standard perturbation theory in the symmetries of three-dimensional relativistic noncommutative quantum space-space (3D-RNCQS). We apply a developed scheme to overcome centrifugal terms such as ( and ). The analytical expression of the energy spectra under the IECYP of (NI2, K2, VH, TiH, NiC, TiC, and CuLi) diatomic molecules in 3D-RNCQS and three-dimensional non-relativistic noncommutative quantum space-space (3D-NRNCQS) was obtained to be sensitive to the atomic quantum numbers , the mixed potential depths (), the screening parameter's inverse and non-commutativity parameters . We also discuss various special cases related to our considered combined potentials, which have utility for other physical and chemical systems, and the results are in excellent agreement with the existing literature. We have also studied the effect of deformation in space-space on thermodynamic quantities such as partition function, mean energy, free energy, specific heat, and entropy.

Perturbation theory remixed: Improved nonlinearity modeling beyond standard perturbation theory

We present a novel $n$EPT ($n$th-order Eulerian Perturbation Theory) scheme to model the nonlinear density field by the summation up to $n$th-order density fields in perturbation theory. The obtained analytical power spectrum shows excellent agreement with the results from all 20 Dark-Quest suites of $N$-body simulations spreading over a broad range of cosmologies. The agreement is much better than the conventional two-loop Standard Perturbation Theory and would reach out to $k_{\rm max}\simeq 0.4~h/{\rm Mpc}$ at $z=3$ for the best-fitting Planck cosmology, without any free parameters. The method can accelerate the forward modeling of the non-linear cosmological density field, an indispensable probe of cosmic mysteries such as inflation, dark energy, and dark matter.

Cosmological perturbation theory using generalized Einstein–de Sitter cosmologies

The separable analytical solution in standard perturbation theory for an Einstein--de Sitter (EdS) universe can be generalized to the wider class of such cosmologies (``generalized EdS,'' or gEdS) in which a fraction of the pressureless fluid does not cluster. We derive the corresponding kernels in both Eulerian perturbation theory (EPT) and Lagrangian perturbation theory, generalizing the canonical EdS expressions to a one-parameter family where the parameter can be taken to be the exponent $\ensuremath{\alpha}$ of the growing mode linear amplification $D(a)\ensuremath{\propto}{a}^{\ensuremath{\alpha}}$. For the power spectrum (PS) at one loop in EPT, the contribution additional to standard EdS is given, for each of the ``13'' and ``22'' terms, as a function of two infrared safe integrals. In the second part of the paper we show that the calculation of cosmology-dependent corrections in perturbation theory in standard [e.g., Lambda cold dark matter (LCDM)] models can be simplified, and their magnitude and parameter dependence better understood, by relating them to our analytic results for gEdS models. At second order the time dependent kernels are equivalent to the analytic kernels of the gEdS model with $\ensuremath{\alpha}$ replaced by a single redshift dependent effective growth rate ${\ensuremath{\alpha}}_{2}(z)$. At third order the time evolution can be conveniently parametrized in terms of two additional such effective growth rates. For the PS calculated at one-loop order, the correction to the PS relative to the EdS limit can be expressed in terms of just ${\ensuremath{\alpha}}_{2}(z)$, one additional effective growth rate function, and the four infrared safe integrals of the gEdS limit. This is much simplified compared to expressions in the literature that use six or eight redshift dependent functions and are not explicitly infrared safe. Using the analytic gEdS expression for the PS with $\ensuremath{\alpha}={\ensuremath{\alpha}}_{2}(z)$ gives a good approximation (to $\ensuremath{\sim}25%$) for the exact result.

Noncommutative Dirac and Schrödinger equations in the background of the new generalized Morse potential and a class of Yukawa potential with the improved Coulomb-like tensor potential as a tensor in 3D-ERQM and 3D-ENRQM symmetries

Relativistic and nonrelativistic quantum mechanics formulated in a noncommutative space–space have recently become the object of renewed interest. In the context of three-dimensional extended relativistic quantum mechanics (3D-ERQM) symmetries with arbitrary spin-orbit coupling quantum number [Formula: see text], we approximate to solve the deformed Dirac equation for a new suggested new generalized Morse potential and a class of Yukawa potential including improved Coulomb-like tensor interaction (N(GMP-CYP) plus ICLTI). In the framework of the spin and pseudospin (p-spin) symmetry, we obtain the global new energy eigenvalue which equals the energy eigenvalue in the usual relativistic QM as the main part plus three corrected parts produced from the effect of the spin-orbit interaction, the new modified Zeeman, and the rotational Fermi term, by using the parametric of the well-known Bopp’s shift method and standard perturbation theory using Greene–Aldrich approximation to nonlinear and exponential terms in the effective potential. The new values that we got appeared sensitive to the quantum numbers ([Formula: see text]), the mixed potential depths ([Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text]), the range of the potential [Formula: see text] and noncommutativity parameters ([Formula: see text],[Formula: see text],[Formula: see text]). We have studied the nonrelativistic limit of new spin symmetry under the N(GMP-ICYP) model, we will also treat some important special cases such as the new generalized Morse potential, the new class of Yukawa potential, the new Hellmann potential, the new inversely quadratic Yukawa potential, the new Hulthén potential and new Coulomb potential. Finally, we studied a case of composite systems.

The influence of noncommutativity on the energy spectra of bosonic particles in the framework of the DKGE with improved spatially-dependent mass including mixed scalar-vector Coulomb potentials in the ERQM symmetries

The bound state solutions of the deformed Klien-Gordon equation (DKGE) have been determined in the extended relativistic quantum mechanics ERQM symmetries using the improved spatially-dependent mass Coulomb potential with mixed scalar-vector Coulomb potentials (ISDM-SVCPs) model. The spatially-dependent mass Coulomb potential, as well as a combination of ((1/(r³)) and (1/(r⁴))), are included in the ISDM-SVCPs model, which is coupled with the coupling LΘ, which explains the interaction of the physical features of the system with the topological deformations of space-time. The new relativistic energy eigenvalues for the ISDM-CP have been derived using the parametric Bopp's shift method and standard perturbation theory. Quantum numbers (j,l,s,m), mixed potential depths (q/s_{c},m₀,m₁), and noncommutativity parameters (Θ,τ,χ) seemed to affect the new values we obtained. Within the framework of relativistic extended quantum mechanics, we have addressed certain significant particular instances that we hope will be valuable to the specialized researcher. In DKGE symmetries, we've also looked at the improved pure scalar Coulomb-like potential. The formulation of total energy was also discovered in the context of extended symmetries, which unified the energies of bosonic particles and antiparticles into a single mathematical formula. When the three simultaneous limits (Θ,τ,χ) were applied, we recovered the normal results of relativistic in the literature (0,0,0).

New bound state solutions of diatomic molecules and mass spectrum of heavy quarkonium with improved modified Kratzer potential plus screened Coulomb potential model in the framework of three-dimensional nonrelativistic noncommutative phase-space symmetries

Approximate analytical solutions of the modified Schrodinger equation have been obtained with a combination of two proposed potentials, called the improved modified Kratzer potential plus screened Coulomb (IMKSC) potential, using the approximation scheme to the centrifugal term for any l-states. We applied the generalized Bopp’s shift method and standard perturbation theory. IMKSC is a superposition of modified Kratzer potential plus screened Coulomb (MKSC) potential, and new terms appear as a result of the effects of the noncommutativity properties of space and phase potential on the MKSC potential. The new bound state energy eigenvalues for some diatomic molecules were obtained in terms of the discrete atomic quantum numbers (j,n,s,m), the infinitesimal NC phase-space parameters, the dissociation energy, the equilibrium internuclear separation, the depth of the potential, and the range of the potential. Furthermore, we have shown that the NC-Hamiltonian operator is the sum of the Hamiltonian operator of the MKSC potential, the perturbed spin-orbit, and the modified Zeeman operator. The new mass spectra of heavy quarkonium systems are calculated using the new energy spectrum expression. It appears as a combination of corresponding values in the literature plus an infinitesimal part that describes the deformation phase-space.

Strong-disorder renormalization group approach to the Anderson model using Rayleigh–Schrödinger perturbation theory

Previous work proposed a strong-disorder renormalization approach for the Anderson model, using it to calculate the density of states and the inverse participation ratio, Johri and Bhatt (2014). This is interesting because of the potential for expansion to higher dimensions and to interacting systems. The original proposal used a non-standard perturbation theory approach which avoided degeneracies. We implemented the same structure but with standard Rayleigh–Schrödinger perturbation theory. Here degeneracies do arise, and we consider two approaches, one in which renormalization is suppressed if degeneracy is present and a second in which the most common form of degeneracy is handled using standard degenerate perturbation theory. The version in which degeneracies are not handled performs similarly to the original proposal, and the addition of degeneracies provides some improvement.

Ab initio calculations of the 2p3/2→2s transition in He-, Li-, and Be-like uranium

The bound-state QED approach is applied to calculations of the $2p_{3/2} \rightarrow 2s$ transition energies in He-, Li-, and Be-like uranium. For U$^{90+}$ and U$^{89+}$, standard perturbation theory for a single level is employed, while the calculations of U$^{88+}$ have required its counterpart for quasidegenerate levels. The utilized approach merges the rigorous QED treatment up to the second order of perturbation theory with the higher-order electron-correlation contributions evaluated within the Breit approximation. The higher-order screened QED effects are estimated by means of the model-QED operator. The nuclear recoil, nuclear polarization, and nuclear deformation effects are taken into account as well. Along with the transition energies, their pairwise differences are calculated. The comprehensive analysis of the uncertainties due to uncalculated effects is carried out, and the most accurate theoretical predictions, which are in perfect agreement with available experimental data, are obtained.

Solid Electrolyte Interphase elastic instability in Li-ion battery anodes

Mechanical instabilities of the solid electrolyte interphase (SEI) layer can lead to battery degradation phenomena. Rubbery-based electrodes, during lithiation and delithiation, being subject to large volumetric changes impose large deformations to the SEI layer which, in turn, can lead to interphase mechanical instabilities. We investigate a typical instability which gives rise to the formation of wrinkling patterns. To account for large deformations, we consider the theory of nonlinear elasticity and we apply a standard perturbative theory to search for possible wavy solutions. By varying the geometrical and the constitutive parameters of interest we compute the critical radius of the electrode at which the instability can occur. We first describe the SEI as a homogeneous layer and we then incorporate in the model the well-established heterogeneity hypothesis of a bilayer SEI, whose inner and stiffer part includes mostly inorganic species whilst the outer and softer sublayer includes mostly the organic species. We show a nontrivial transition in passing from the former to the latter hypothesis and we derive general conditions where two theories can or cannot be considered both valid. Through our simulations, we then warn of the possible errors that a predicting model can exhibit by neglecting the microstructure of the shell layer. Furthermore, conditions for preventing the wrinkling patterns formation and/or for designing optimal artificial SEI can be deduced from some illustrative numerical examples.

Coherent forward scattering as a robust probe of multifractality in critical disordered media

We study coherent forward scattering (CFS) in critical disordered systems, whose eigenstates are multifractals. We give general and simple arguments that make it possible to fully characterize the dynamics of the shape and height of the CFS peak. We show that the dynamics is governed by multifractal dimensions D_1D1 and D_2D2, which suggests that CFS could be used as an experimental probe for quantum multifractality. Our predictions are universal and numerically verified in three paradigmatic models of quantum multifractality: Power-law Random Banded Matrices (PRBM), the Ruijsenaars-Schneider ensembles (RS), and the three-dimensional kicked-rotor (3DKR). In the strong multifractal regime, we show analytically that these universal predictions exactly coincide with results from standard perturbation theory applied to the PRBM and RS models.

New Bound State Solutions of 3-Dimensional Modified Eckart Potential Plus a New Modified Deformed Hylleraas Potential in RNCQM and NRNCQM Symmetries

Abstract: In this study, the solution of the deformed Klein-Gordon and deformed Schrödinger equations (DKGE and DSE for short) for the modified Eckart potential plus a new modified deformed Hylleraas potential (MEPNMDHP, for short) with the improved approximation to deal with the centrifugal term is investigated using Bopp’s shift and standard perturbation theory methods in the symmetries of relativistic noncommutative quantum mechanics RNCQM and nonrelativistic noncommutative quantum mechanics NRNCQM. To the best of our knowledge, this problem is examined in literature in usual RQM and NRQM. The new potential suggested describes some selected diatomic molecules such as the homogeneous diatomic molecules (N2 and H2) and heterogeneous diatomic molecules (HCl, HBr, SO, NO, and HI). The new values that we get appeared sensitive to the quantum numbers (j,s,m) in addition to the usual states’ numbers (n,l), the potential depths of the potential (V_0,V_1,V_2), the range of the potential α and noncommutativity parameters (Θ,σ,χ). We have highlighted three physical phenomena that automatically generate a result of the topological properties of non-commutativity. The first physical phenomenon is the perturbative spin-orbit coupling, the second is the magnetic induction, and the third is the rotational proper phenomenon. In both relativistic and nonrelativistic problems, we show that the corrections on the spectrum energy are smaller than the main energy in the ordinary cases of quantum field theory and quantum mechanics. In the new symmetries of NCQM, it is not possible to get the exact analytical solutions for l=0 and l≠0, so the approximate solutions are available. We have observed that the DKGE under the MEPNMDHP has a physical behavior similar to the Duffin-Kemmer equation that can describe the dynamic state of a particle with spin-1 in the symmetries of RNCQM. Four special cases; i.e., l wave is investigated in the context of DKGE and Schrödinger theories. The new relativistic and nonrelativistic energy for some potentials, such as only modified Eckart potential and only new modified Hylleraas potential, have also been obtained by varying some potential parameters. Keywords: Klein-Gordon equation, Schrödinger equation, Eckart potentials, Modified Hylleraas potential, Diatomic molecules, Noncommutative geometry, Star products, Bopp's shift method. PACs: 03.65. Ge; 03.65. Pm; 03.65.−w, 32.30 – r, 02.30. Gp.

Perturbation theory with dispersion and higher cumulants: Framework and linear theory

The authors develop an alternative perturbative approach to gravitational clustering and the large-scale structure. This approach is complementary to the effective field theory approach while being much more predictive and providing a more detailed understanding of mechanisms like the decoupling of nonlinear small-scale modes into dark matter halos where standard perturbation theory completely fails. This is an exciting step into the future of making predictions, especially in light of upcoming Stage IV experiments.

Perturbation theory with dispersion and higher cumulants: Nonlinear regime

The authors develop an alternative perturbative approach to gravitational clustering and the large-scale structure. This approach is complementary to the effective field theory approach while being much more predictive and providing a more detailed understanding of mechanisms like the decoupling of nonlinear small-scale modes into dark matter halos where standard perturbation theory completely fails. This is an exciting step into the future of making predictions, especially in light of upcoming Stage IV experiments.

Galaxy three-point correlation function in modified gravity

The next generation of galaxy surveys will provide highly accurate measurements of the large-scale structure of the Universe, allowing for more stringent tests of gravity on cosmological scales. Higher-order statistics are a valuable tool to study the non-Gaussianities in the matter field and to break degeneracies between modified gravity and other physical or nuisance parameters. However, understanding from first principles the behavior of these correlations is essential to characterize deviations from General Relativity (GR), and the purpose of this work. This work uses contemporary ideas of standard perturbation theory on biased tracers to characterize the three-point correlation function at tree level for modified gravity models with a scale-dependent gravitational strength, and applies the theory to two specific models [$f(R)$ and DGP] that are representative for Chameleon and Vainshtein screening mechanisms. Additionally, we use a multipole decomposition, which apart from speeding up the algorithm to extract the signal from data, also helps to visualize and characterize GR deviations.