H. Q. Zheng
Studies on the dynamical properties of the $σ$ meson are reviewed and discussed. The important role of fundamental principles such as analyticity, unitarity and crossing symmetry played in the studies are stressed.
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The contributions of one-loop diagrams with dynamical vector mesons to masses and decay constants of pseudoscalar mesons are determined. Hereby, a relativistic Lagrangian for both the pseudoscalar-meson octet and the vector-meson nonet is used. The vector mesons are given in the antisymmetric tensor representation. Both the differences between static and dynamical vector mesons and the differences between calculations with and without vector mesons are studied as functions of the light quark mass.
In terms of an effective chiral theory of mesons it is shown quantitatively that in the limit of $m_{q}=0$, the masses of $ρ$, $ω$, $K^{*}(892)$, $m_ϕ$, $a_{1}$, $f_{1}(1286)$, $K_{1}(1400)$, and $f_{1}(1510)$ mesons originate from dynamical chiral symmetry breaking.
We study dynamically generated $a_0$ resonances by interactions between vector mesons including their coupling to channels of pseudoscalar mesons within coupled-channel approach. Both vector and pseudoscalar mesons are considered as $t$-channel exchanged mesons for calculating interactions between vector mesons. Analogous to $a_0(980)$ as a $K\bar K-πη$ dynamically generated state, there is an $a_0(1710)$ as a coupled channel dynamically generated state near $K^*\bar{K}^*$ threshold. The channels involved are $ρϕ,K^*\bar{K}^*,ρω, K\bar{K},πη$. This $a_0$ mainly decays to $ρω$, $πη$ and $K\bar{…
This paper is the SU(3) extension of the dynamically generated SU(2) linear $σ$ model Lagrangian worked out previously using dimensional regularization. After discussing the quark-level Goldberger-Treiman relations for SU(3) and the related gap equations, we dynamically generate the meson cubic and quartic couplings. This also constrains the meson-quark coupling constant to $g=2π/\sqrt3$ and determines the SU(3) scalar meson masses in a Nambu-Jona-Lasinio fashion. Finally we dynamically induce the U(3) pseudoscalar and scalar mixing angles in a manner compatible with data.
Within a formulation of a Dynamical AdS/QCD model we calculate the spectrum of light flavored mesons. The background fields of the model correspond to an IR deformed Anti de Sitter metric coupled to a dilaton field. Confinement comes as a consequence of the dilaton dynamics coupled to gravity. Additionally to the Regge-like spectrum of light- scalar, vector and higher spin mesons, we obtain the decay width of scalar mesons into two pions.
New equations are derived for meson and baryon Green's functions in a systematic method using the QCD Lagrangian and large N_c approximation as a starting point. The equations are shown to contain both confinement dynamics and chiral symmetry breaking. Linear Regge trajectories are demonstrated for the solutions.
Meson-Meson Molecular Binding Potentials in the Diabatic Dynamical Diquark Model
该研究在非绝热动力学双夸克模型中引入介子-介子分子势，计算混合束缚态并分析参数变化对态性质的影响，发现参数空间中存在可产生与χₙ₁(3872)、χₙ₀(3915)匹配的质量本征态及纯双介子分子态的区域。
In the Nambu--Jona-Lasinio (NJL-) model baryons are considered either as quark-diquark-composites or as soliton configurations of mesons. The construction of a hybrid model, which possesses a dynamical interplay between both pictures, is presented. The Bethe-Salpeter equation for diquarks and the Faddeev equation for diquark-quark bound states in the background of a soliton configuration is solved.
The elements of the theory of dynamical expansion of the theory of weak interaction working at the tree level, i.e. the model of dynamical analogy of Cabibbo-Kobayashi- Maskawa matrices, are given. The equation for mass difference of $K^0_1, K^0_2$ mesons or the length of $K^0, \bar K^0$ meson oscillations in the phase volume approach in the frame work of the standard model of the weak interactions and the model of dynamical analogy of Cabibbo-Kobayashi-Maskawa matrices is calculated (in this model the oscillations of $K^0, \bar K^0$ mesons arise at violation of strangeness by $B$ bosons). Com…
Two- and three-particle distribution amplitudes of heavy pseudoscalar mesons of well-defined geometric twist are introduced. They are obtained from appropriately parametrized vacuum-to-meson matrix elements by applying those twist projectors which determine the enclosed light-cone operators of definite geometric twist and, in addition, observing the heavy quark constraint. Comparing these distribution amplitudes with the conventional ones of dynamical twist we derive relations between them, partially being of Wandzura-Wilczek type; also sum rules of Burkhardt-Cottingham type are derived.The de…
We investigate the nonequilibrium evolution of the quark-meson model using two-particle irreducible effective action techniques. Our numerical simulations, which include the full dynamics of the order parameter of chiral symmetry, show how the model thermalizes into different regions of its phase diagram. In particular, by studying quark and meson spectral functions, we shed light on the real-time dynamics approaching the crossover transition, revealing e.g. the emergence of light effective fermionic degrees of freedom in the infrared. At late times in the evolution, the fluctuation-dissipatio…
We discuss the renormalization of the $1^{--}$ vector meson propagator within Resonance chiral theory at one loop. Using the particular form of the interaction Lagrangian we show that additional poles of the renormalized propagator corresponding to $1^{+-}$ degrees of freedom can be generated. We give a concrete example of such an effect.
I briefly review the Regge approach to the hadron spectrum and advocate a dynamical emergence of principal quantum number in the known spectrum of light non-strange mesons. Further it is shown how the linear radial trajectories with universal slope can be extended to heavy quarkonia and a qualitative string interpretation is given. After that I discuss a recently proposed non-string model leading to a natural appearance of linear Regge trajectories and explaining many mass relations.
The light scalar mesons below 1 GeV as tetraquark states are studied in the framework of the flux-tube model, the multi-body confinement instead of the additive two-body confinement is used. From the calculated results, we find that the light scalar mesons, $σ$, $κ$ could be well accommodated in the diquark-antidiquark tetraquark picture in the flux-tube model and they could be color confinement resonances. The mass of the first radial excited state of $[ud][\bar{u}\bar{d}]$ is 1019 MeV, which is close to the mass of $f_0(980)$. Whereas $a_0(980)$ can not be fitted in this interpretation.
We study meson-meson interactions using an extended $q^2\bar{q}^2(g)$ basis that allows calculating coupling of an ordinary meson-meson system to a hybrid-hybrid one. We use a potential model matrix in this extended basis which at quark level is known to provide a good fit to numerical simulations of a $q^2\bar{q}^2$ system in pure gluonic theory for static quarks in a selection of geometries. We use a combination of resonating group method formalism and Born approximation to include the quark motion using wave functions of a $q\bar{q}$ potential within a cluster. This potential is taken to be…
A simple, but fully-covariant model for describing neutral Vector Mesons, in both light and heavy sectors, is briefly illustrated. The main ingredients of our relativistic constituent model are i) an Ansatz for the Bethe-Salpeter vertex for Vector Mesons, and ii) a Mandelstam-like formula for the electromagnetic decay widths. The free parameters of our approach are fixed through a comparison with the valence transverse-momentum distribution, $n(k_\perp)$, obtained within phenomenological Light-Front Hamiltonian Dynamics models reproducing the mass spectra. Preliminary results for the transvers…
By using gauge/gravity duality, we calculate the spectral function of the heavy vector mesons with the presence of an intense magnetic field in a hot and dense medium. The results show that, a general conclusion, as the increases of magnetic field, chemical potential and temperature, the height of the peak of the spectral function decreases and the width increases. A nontrivial result is the change from the peak position of spectral function. We explain this non-trivial behavior by the interplay of the interaction between the two heavy quarks and the interaction between the medium with each of…
We compute the amplitudes for non-leptonic annihilation decays of $B$ mesons into two particles within the pQCD collinear approach. The end point divergences are regulated with the help of an infrared finite gluon propagator characterized by a non-perturbative dynamical gluon mass consistent with recent lattice simulations. The divergences at twist-3 are regulated by a dynamical quark mass. Our results fit quite well the existent data of $B^0\to D_s^-K^+$ and $B^0\to D_s^{-*}\bar{K}^+$ for the expected range of dynamical gluon masses. We also make predictions for the rare decays $\bar{B}^0\to …
The elements of the theory of vacuum oscillations and the model of dynamical expansion of the theory of weak interactions works at the tree level, i.e. the model of dynamical analogy of Cabibbo-Kobayashi- Maskawa matrices and its further development, are given. It is shown that the quarks and massive vector bosons must be structural and these structural particles (subparticles) must interact to generate quark and vector boson masses. In this case the problem of singularity cancellations does not arise in this model. It is also shown that, for self consistency of the theory, the weak decays of …
We show that the infrared dynamics of string modes dual to $q\bar q$ states within a Dynamical AdS/QCD model of coupled dilaton-gravity background gives the Reggelike spectrum of $f0's$ scalars and higher spin mesons consistent with experimental data. The pion mass and its trajectory were also described with a scale deformation of the metric and a rescaled string mass. The available experimental decay widths of the $S\to PP$ decays provided a complementary check of the proposed classification scheme for$f_0(980)$, $f_0(1370)$, $f_0(1500)$ and $f_0(1710)$ as radial excitations of $f_0(600)$. Fo…
In this talk I briefly review the theory of resonances dynamically generated from hadron-hadron scattering, sometimes referred to as "molecules"S. I give some classical examples of meson-meson and meson-baryon systems, as well as a few examples of different approaches to describe the interaction between hadrons. To conclude, I comment on a few recent works that suggest that some of the five new narrow $Ω_c$ states, recently discovered by the LHCb collaboration, can be interpreted as meson-baryon molecules.
We extend our recent computation of the low-energy limit of the linear O(4) Quark-Meson Model. The present analysis focuses on the transformation of the resulting effective action into a nonlinearly realized effective pion action, whose higher-derivative interaction terms are parametrized by so-called low-energy couplings. Their counterparts in the linear model are determined from the Functional Renormalization Group flow of the momentum-dependent four-pion vertex, which is calculated in a fully O(4)-symmetric approximation by including also momentum-dependent σπ interactions as well as σ self…
The SU(3) linear sigma model is dynamically generated in loop-order using the nonstrange-strange basis. Only self-consistent logarithmic divergent graphs are needed, with quadratic divergent graphs replaced by SU(3) mass-shell equal splitting laws. The latter lead to an eta'-eta mixing angle of 41.84 deg which is consistent with phenomenology. Finally this above SU(3) linear sigma model in turn predicts strong decay rates which are all compatible with data.
Spin Dependence of Meson Thresholds in the Diabatic Dynamical Diquark Model
该研究推广非绝热动力学双夸克模型，纳入重夸克自旋对称性对双介子对的影响，得到$J=0、1、2$态的显式相互作用势并应用于$χ_{c1}(3872)$等物理态，为奇特强子研究提供理论支持。
The fermion and meson sectors of the quark-level SU(2) linear sigma model are dynamically generated from a meson-quark Lagrangian, with the quark (q) and meson ($σ,π$) fields all treated as elementary, having neither bare masses nor expectation values. In the chiral limit, the masses are predicted to be $m_q = f_πg, m_π= 0, m_σ= 2m_q$, and we also find that the quark-meson coupling $g = 2π/sqrt{N_c}$, the three-meson coupling is $g' = m_σ^2/2f_π= 2gm_q$ and the four-meson coupling is $λ= 2g^2 = g'/f_π$, where $f_π\simeq 90$ MeV is the pion decay constant and $N_c = 3$ is the color number. By g…
The $ρρ$ scattering has been studied by two groups which both claimed a dynamical generated scalar meson, most likely to be $f_0(1370)$. Here we investigate the influence of coupled-channels of pseudoscalar mesons, i.e., $ππ$ and $\bar KK$, on this dynamical generated scalar state. With the coupled channel effect included, the pole and partial decay widths are found to be more close to PDG values for $f_0(1500)$.
We quantify the importance of dynamical spin effects in the holographic light-front wavefunctions of the pion, kaon, $η$ and $η^\prime$. Using a universal AdS/QCD scale and constituent quark masses, we find that such effects are maximal in the pion where they lead to an excellent simultaneous description of a wide range of data: the decay constant, charge radius, spacelike EM and transition form factors, as well as, after QCD evolution, both the parton distribution function and the parton distribution amplitude data from Fermilab. These dynamical spin effects lead up to a $30\%$ chance of find…
We present a study of Goldstone boson scattering based on the flavor SU(3) chiral Lagrangian formulated with vector mesons in the tensor field representation. A coupled-channel channel computation is confronted with the empirical s- and p-wave phase shifts, where good agreement with the data set is obtained up to about 1.2 GeV. There are two relevant free parameters only, the chiral limit value of the pion decay constant and the coupling constant characterizing the decay of the rho meson into a pair of pions. We apply a recently suggested approach that implements constraints from micro- causal…
We investigate the structure of the light meson with one dynamical gluon from the light-front QCD Hamiltonian, determined for their constituent quark-antiquark and quark-antiquark-gluon Fock sectors, together with a three-dimensional confinement. After fitting the light meson mass spectroscopy, the light-front wave functions provide a good quality description of the pion electromagnetic form factor, and the valence quark distribution functions following QCD scale evolution.
Starting with just one bare seed for each member of a scalar nonet, we investigate when it is possible to generate more than one hadronic state for each set of quantum numbers. In the framework of a simple model, we find that in the I=1 sector it is possible to generate two physical states with the right features to be identified with the a_0(980) and the a_0(1450). In the I=1/2 sector, we can generate a number of physical states with masses higher than 1 GeV, including one with the right features to be associated with the K_0^*(1470), but none which can be identified with the light kappa scal…
We present calculations of coherent photoproduction of vector mesons ($J/Ψ$ and $Υ$) with leading-order parton distribution functions to check new kinds of corrections of the DGLAP equations and nuclear modifications. The input gluon distribution of the proton is the dynamical parton model from the DGLAP equations with GLR-MQ-ZRS (Gribov-Levin-Ryskin, Mueller-Qiu, Zhu-Ruan-Shen) modifications. From comparison between several other gluon distribution models, we find that the dynamical gluon distribution fits with the results of meson photoproduction experiments in the high energy region. The ca…
The charmed-strange meson spectrum is calculated with the overlap valence fermions on 2+1 flavor domain wall dynamical configurations for $32^3\times 64$ lattices with a spatial size of 2.7 fm. Both charm and strange quark propagators are calculated with the overlap fermion action. The calculated scalar meson at 2304(22) MeV and axial-vector meson at 2546(27) MeV are in good agreement with the experimental masses of $D{s0}^*$(2317) and $D_{s1}$(2536).
Dynamical $\mathbf{O(4)}$-Symmetry in the Light Meson Spectrum within the Framework of the Regge Approach
本文利用Regge轨迹拟合2024年粒子数据，证实轻非奇异介子质量谱呈现类氢的动力学O(4)对称性，即质量依赖于l+n_r之和，并定性讨论了更受数据支持的弦状半相对论模型。
A three flavor linear sigma model with vector and axial-vector mesons is discussed. Preliminary results concerning on the symmetry breaking pattern, the question of parameterization, as well as the resulting meson masses are presented.
We study the coupling of a tetraquark system to an exchanged meson-meson channel, using a pure gluonic theory based four-quark potential {\em matrix} model which is known to fit well a large number of data points for lattice simulations of different geometries of a four-quark system. We find that if this minimal-area-based potential matrix replaces the earlier used simple Gaussian form for the gluon field overlap factor $f$ in its off-diagonal terms, the resulting $T$-matrix and phase shifts develop an angle dependence whose partial wave analysis reveals $D$ wave and higher angular momentum co…
$J/ψ$-meson photoproduction off the nucleon in a dynamical model
该研究构建含 Pomeron、介子交换、辐射及末态相互作用的 Hamiltonian 动力学模型，拟合 JLab 数据，揭示轻介子交换主导近阈值核子上的 $J/ψ$ 光生。
The influence of the phi-meson on the nucleon properties in the chiral soliton model is discussed.
We present the current status of a combined and simultaneous analysis of meson production reactions based on a dynamical coupled-channels (DCC) model, which is conducted at Excited Baryon Analysis Center (EBAC) of Jefferson Lab.
We investigate the quark condensate of the QCD vacuum within the instanton vacuum model. We calculate the meson-loop contributions to the dynamical quark mass and quark condensate to ${\cal O}(1/N_c)$-, ${\cal O}(m/N_c)$-, and ${\cal O}((m\ln m) /N_c)$-order corrections. We find that the meson (especially pion) loops provide substantial contributions to the dynamical quark mass and as a result to the quark condensate. The results indicate that the $1/N_c$ corrections should be reconsidered in the systematical way. The present results are consistent with those from chiral perturbation theory.
Proving Rho Meson be a Dynamical Gauge Boson of Hidden Local Symmetry
本文证明ρ介子作为隐藏局域对称性的动力学规范玻色子，其动能项由格拉斯曼模型的大N极限生成，外推至N=4时独立于参数a实现所有现象学结果，并建立大N动力学与实验的定量联系。
We construct a Schrodinger-like equation for the longitudinal wave function of a meson in the valence qq-bar sector, based on the 't Hooft model for large-N two-dimensional QCD, and combine this with the usual transverse equation from light-front holographic QCD, to obtain a model for mesons with massive quarks. The computed wave functions are compared with the wave function ansatz of Brodsky and De Teramond and used to compute decay constants and parton distribution functions. The basis functions used to solve the longitudinal equation may be useful for more general calculations of meson stat…
We present a detailed description of the extraction of the highly excited isovector meson spectrum on dynamical anisotropic lattices using a new quark-field construction algorithm and a large variational basis of operators. With careful operator construction, the combination of these techniques is used to identify the continuum spin of extracted states reliably, overcoming the reduced rotational symmetry of the cubic lattice. Excited states, states with exotic quantum numbers (0+-, 1-+ and 2+-) and states of high spin are resolved, including, for the first time in a lattice QCD calculation, sp…
Using a new quark-field construction algorithm and a large variational basis of operators, we extract a highly excited isovector meson spectrum on dynamical anisotropic lattices. We show how carefully constructed operators can be used to reliably identify the continuum spin of extracted states, overcoming the reduced cubic symmetry of the lattice. Using this method we extract, with confidence, excited states, states with exotic quantum numbers (0+-, 1-+ and 2+-) and states of high spin, including for the first time in lattice QCD, spin-four states.
We derive new bounds on the b-number form factor $F(q^2)$ of the B meson. (Revised version of hep-ph/9306214).
Although there are some experimental hints for the higher excited $ρ$ mesons, our knowledge of their properties is much poor theoretically. Based on our recent work about excited $ρ$ mesons [Phys.Rev.D104(2021)034013], we present the mass spectra and decay properties of the higher excited $ρ$ mesons with the modified Godfrey-Isgur quark model and the $^3P_0$ strong decay model, and compare our predictions with the experimental hints, which should be helpful to search for these higher excited $ρ$ mesons.
We study a modification of the five-dimensional description of dynamical electro-weak symmetry breaking inspired by the AdS/CFT correspondence. Conformal symmetry is broken in the low-energy region near the IR brane by a power-law departure from the pure AdS background. Such a modification--while not spoiling the identification of the IR brane with the scale of confinement-- has a dramatic effect on both the coupling of the first composite states to the standard model currents and their self-couplings. Chiral symmetry breaking can take place at a scale larger than the IR cut-off. This study sh…
The Anti-de Sitter Space/Conformal Field Theory (AdS/CFT) correspondence may offer new and useful insights into the non-perturbative regime of strongly coupled gauge theories such as Quantum Chromodynamics (QCD). We present an AdS/CFT-inspired model that describes the spectra of light mesons. The conformal symmetry is broken by a background dilaton field, and chiral symmetry breaking and linear confinement are described by a chiral condensate field. These background fields, along with a background glueball condensate field, are derived from a potential. We describe the construction of the pote…
The properties of $ρ$-mesons at finite temperature ($T$) are examined with an effective chiral lagrangian in which vector and axial-vector mesons are included as massive Yang-Mills fields of the chiral symmetry. It is shown that, at $T^2$ order, the effective mass is not changed but only the mixing effect in vector and axial-vector correlator appears.
Starting from a relativistic Lagrangian for pseudoscalar Goldstone bosons and vector mesons in the antisymmetric tensor representation, a one-loop calculation is performed to pin down the divergent structures that appear for the effective low-energy action at chiral orders Q^2 and Q^4. The corresponding renormalisation-scale dependences of all low-energy constants up to chiral order Q^4 are determined. Calculations are carried out for both the pseudoscalar octet and the pseudoscalar nonet, the latter in the framework of chiral perturbation theory in the limit of a large number of colours.
Properties of scalar mesons below 1.0 GeV have been studied by regarding them as hadronic molecular states. Using the effective Lagrangian approach, we have calculated their leptonic decays, strong decays and productions via the $ϕ$ meson radiative decays. Comparing our results with that given in the literature and the data, we conclude that it is difficult to arrange them in the same nonet if some of them are pure hadronic molecules.
In this work we study the implications of light-quark pionic matter at finite temperatures on the properties of open and hidden charm mesons. The meson-meson interactions are described by means of a chiral unitary approach accounting for coupled channels effects. The in-medium Lippmann-Schwinger Equations, which consider the change in self-energy that the mesons acquire from interacting with the surrounding pionic matter, are solved self-consistently, and the spectral functions of the mesons in the hot pion bath are obtained. It is observed that the charmed mesons develop a quite substantial p…
We discuss the sensitivity of nucleon properties (mass, magnetic moments and electromagnetic form factors) on the variation of the pseudoscalar meson masses in the context of the perturbative chiral quark model. The obtained results are compared to data and other theoretical predictions.
The flavour and glue structure of the light scalar mesons in QCD are probed by studying the couplings of the $σ(600)$ and $f_0(980)$ to the operators $\bar{q}q$, $θ_μ^μ$ and to two photons.
Although studied for many years the nature of the light scalar mesons remains controversial. Here we shall present a method, applicable for s-wave states located close to a threshold, that allows one to quantify the molecular part of a given state. When applied to the $f_0(980)$ a dominance of the molecular component is found. In the second part we show that requirements of field theoretic consistency and chiral symmetry, when applied to the scattering of light pseudo--scalars, naturally lead to the appearance of dynamical poles in the scalar sector. A program is proposed on how to further inv…
We present preliminary results for light, strange and charmed pseudoscalar meson physics from simulations using four flavors of dynamical quarks with the highly improved staggered quark (HISQ) action. These simulations include lattice spacings ranging from 0.15 to 0.06 fm, and sea-quark masses both above and at their physical value. The major results are charm meson decay constants f_D, f_{D_s} and f_{D_s}/f_D and ratios of quark masses. This talk will focus on our procedures for finding the decay constants on each ensemble, the continuum extrapolation, and estimates of systematic error.
The main motivation is to investigate meson properties in the quark model to understand the model applicability and generate possible improvements. Certain modifications to the model are suggested which have been inspired by fundamental QCD properties (such as running coupling or spin dependence of strong interactions). These modifications expand the limits of applicability of the constituent quark model and illustrate its weaknesses and strengths. The meson properties studied include meson spectra, decay constants, electromagnetic and electroweak form-factors and radiative transitions. The re…
We study the reactions $γγ\rightarrow π^0π^0$, $π^+π^-$, $K^0\bar{K}^0$, $K^+K^-$, $ηη$ and $π^0η$ based on a chiral Lagrangian with dynamical light vector mesons as formulated within the hadrogenesis conjecture. At present our chiral Lagrangian contains 5 unknown parameters that are relevant for the photon fusion reactions. They parameterize the strength of interaction terms involving two vector meson fields. These parameters are fitted to photon fusion data $γγ\rightarrow π^0π^0$, $π^+π^-, π^0η$ and to the decay $η\rightarrowπ^0γγ$. In order to derive gauge invariant reaction amplitudes in t…
We provide a glimpse of recent progress in meson physics made via QCD's Dyson-Schwinger equations with: a perspective on confinement and dynamical chiral symmetry breaking (DCSB); a pre'cis on the physics of in-hadron condensates; results for the masses of the π, σ, ρ, a_1 mesons and their first-radial excitations; and an illustration of the impact of DCSB on the pion form factor.
I explain the Sato-Lee (SL) model and its extension to the neutrino-induced pion production off the nucleon. Then I discuss applications of the SL model to incoherent and coherent pion productions in the neutrino-nucleus scattering. I mention a further extension of this approach with a dynamical coupled-channels model developed in Excited Baryon Analysis Center of JLab.
We obtain the light meson mass spectroscopy from the light-front quantum chromodynamics (QCD) Hamiltonian, determined for their constituent quark-antiquark and quark-antiquark-gluon Fock components, together with a three-dimensional confinement. The eigenvectors of the light-front effective Hamiltonian provide a good quality description of the pion electromagnetic form factor, decay constant, and the valence quark distribution functions following QCD scale evolution. We also show that the pion's gluon densities can be probed through the pion-nucleus induced $J/ψ$ production data. Our pion part…
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