Central exclusive production of $η_c$ and $χ_{c0}$ in the light-front k$_{\perp}$-factorization approach
Izabela Babiarz, Roman Pasechnik, Wolfgang Schäfer, Antoni Szczurek
备注：
6 pages, 3 figures, submision to SciPost, talk given by I. Babiarz at the XXVIII International Workshop on Depp-Inelastic Scattering and Related Subjects (DIS2021)
We study the exclusive production of $J^{PC}=0^{++}, 0^{--}$ charmonium states in proton-proton collisions at the LHC energies The $pp \to ppη_c$ reaction is discussed for the first time. We observe a substantial contribution from the nonperturbative domain of gluon virtualities, especially for $η_c$ production. To model the nonperturbative region better, we utilize models of the unintegrated gluon distribution based on parametrizations of the color dipole cross-section.
We discuss a necessary nonvalence contribution in timelike exclusive processes. Utilizing a Schwinger-Dyson type of approach, we relate the nonvalence contribution to an ordinary light-front wave function that has been extensively tested in the spacelike exclusive processes. An application to $K_{\ell3}$ decays provides encouraging results.
After reviewing a connection between quantum chromodynamics and contituent quark model pictures in the light-front quantization with some comparison and contrast to the ordinary equal-time bridge a la Bogoliubov-Valatin transformation, we discuss some newer development of the light-front quark model phenomenology in exclusive processes including the embedded state. The skewed parton distribution appears to be a good testing ground for our new effective treatment of the light-front nonvalence contributions in timelike exclusive processes.
Recent improvements of the hard scattering picture for exclusive reactions, namely the inclusion of both Sudakov corrections and the intrinsic transverse momentum dependence of the hadronic wave function, are reviewed. On account of these improvements the perturbative contribution to the pion's form factor can be calculated in a theoretically self-consistent way for momentum transfers as low about $2\,{\rm GeV}$. This is achieved at the expense of a substantial suppression of the perturbative contribution in the few GeV region. Eventual higher twist contributions are also discussed.\\
We calculate the twist three distribution f_\perp(x,k^\perp) contributing to Cahn effect in unpolarized semi-inclusive deep inelastic scattering. We use light-front Hamiltonian technique and take the state to be a dressed quark at one loop in perturbation theory. The 'genuine twist three' contribution comes from the quark-gluon interaction part in the operator and is explicitly calculated. f_\perp(x,k^\perp) is compared with f_1(x,k^\perp).
We discuss a necessary nonvalence contribution in timelike exclusive processes. Following a Schwinger-Dyson type of approach, we relate the nonvalence contribution to an ordinary light-front wave function that has been extensively tested in the spacelike exclusive processes. A complicate four-body energy denominator is exactly cancelled in summing the light-front time-ordered amplitudes. Applying our method to $K_{\ell3}$ and $D^0\to K^- \ell^+ ν_l$ where a rather substantial nonvalence contribution is expected, we find not only an improvement in comparing with the experimental data but also a…
While the particle-number-conserving convolution formalism established in the Drell-Yan-West reference frame is frequently used to compute exclusive amplitudes in the light-front quantization, this formalism is limited to only those frames where the light-front helicities are not changed and the good (plus) component of the current remains unmixed. For an explicit demonstration of such criteria, we present the relations between the current matrix elements in the two typical reference frames used for calculations of the exclusive amplitudes, i.e. the Drell-Yan-West and Breit frames and investig…
We study the light-front zero-mode contribution to the tensor form factors $T_i(i=1,2,3)$ for the exclusive rare $P\to V\ell^+\ell^-$ decays using a covariant fermion field theory model in $(3+1)$ dimensions. While the zero-mode contribution in principle depends on the form of the vector meson vertex $Γ^μ=γ^μ- (2k-P_V)^μ/D$, the three tensor form factors $T_i(i=1,2,3)$ are found to be free from the zero mode if the denominator $D$ contains the term proportional to the light-front energy or the longitudinal momentum fraction factor $(1/x)^n$ of the struck quark with the power $n>0$. Since the d…
We present the analysis of exclusive $0^{-}$$\to$$0^{-}$ semileptonic heavy meson decays using the constituent quark model based on the light-front quantization. Our model is constrained by the variational principle for the well-known linear plus Coulomb interaction motivated by QCD. Our method of analytic continuation to obtain the weak form factors avoids the difficulty associated with the contribution from the nonvalence quark-antiquark pair creation. Our numerical results for the heavy-to-heavy and heavy-to-light meson decays are in a good agreement with the available experimental data and…
We discuss a necessary nonvalence contribution in timelike exclusive processes. Following a Schwinger-Dyson type of approach, we relate the nonvalence contribution to an ordinary light-front wave function that has been extensively tested in the spacelike exclusive processes. A complicate multi-body energy denominator is exactly cancelled in summing the light-front time-ordered amplitudes. Applying our method to $K_{\ell3}$ and $D^0\to K^- \ell^+ ν_l$ where a rather substantial nonvalence contribution is expected, we find not only an improvement in comparing with the experimental data but also …
We consider the central exclusive production of $q\bar{q}$ pairs and Higgs boson at proton-proton collisions at LHC. The amplitude for the process is derived within the $k_{\perp}$-factorization approach and considered in different kinematical asymptotics, in particular, in the important high quark transverse momenta and massless quark limits. Quark helicity and spin-projection amplitudes in two different frames are shown in extenso. Rapidity distributions, quark jet $p_{\perp}$ distributions, invariant $q\bar{q}$ mass distributions, angular azimuthal correlations between outgoing protons and …
We study exclusive production of scalar $χ_{c0}\equiv χ_c(0^{++})$ and pseudoscalar $η_c$ charmonia states in proton-proton collisions at the LHC energies. The amplitudes for $gg \to χ_{c0}$ as well as for $gg \to η_c$ mechanisms are derived in the $k_{T}$-factorization approach. The $p p \to p p η_c$ reaction is discussed for the first time. We have calculated rapidity, transverse momentum distributions as well as such correlation observables as the distribution in relative azimuthal angle and $(t_1,t_2)$ distributions. The latter two observables are very different for $χ_{c0}$ and $η_c$ case…
The exclusive semileptonic decay Lambda_b -> Lambda_c + l nu_l is investigated including both radiative and first-order power corrections in the inverse heavy-quark mass, while the Isgur-Wise function is calculated within a light-front constituent quark model. It turns out that the dependence on the model parameters can be effectively constrained using recent lattice QCD results at low values of the recoil. Our final predictions for the exclusive semileptonic branching ratio, the longitudinal and transverse asymmetries, and the longitudinal to transverse decay ratio are: Br(Lambda_b -> Lambda_…
Numerical results for the calculation of the (rest-frame) $Q\bar{Q}$ potential, for light-front quantized $QCD_{2+1}$ on a $\perp$ lattice are presented. Both in the longitudinal as well as the $\perp$ spatial directions one obtains linear confinement. The resulting potential is almost rotational symmetric.
We consider central exclusive production of $gg$ dijets in proton-proton (proton-antiproton) collisions at LHC and Tevatron for different intermediate and final gluon polarisations. The amplitude for the process is derived within the $k_\perp$-factorization approach (with both the standard QCD and the Lipatov's effective three-gluon verticies) and is considered in various kinematical asymptotia, in particular, in the important limit of high-$p_\perp$ jets. Compared to earlier works we include emissions of gluons from different gluonic $t$-channel lines as well as emission of quark-antiquark di…
Exclusive $J/ψ$ photoproduction in photon-proton diffractive scattering: A light-front Hamiltonian approach
Institute of Modern Physics, Chinese Academy of Sciences（中国科学院近代物理研究所）；School of Nuclear Science and Technology, University of Chinese Academy of Sciences（中国科学院大学核科学与技术学院）；CAS Key Laboratory of High Precision Nuclear Spectroscopy, Institute of Modern Physics, Chinese Academy of Sciences（中国科学院高精度核谱学重点实验室，中国科学院近代物理研究所）；High Energy Accelerator Research Organization (KEK)（高能加速器研究机构（KEK））
An explicit relativistic light-front quark model is presented which gives the momentum transfer dependent form factors of weak hadronic currents among heavy pseudoscalar and vector mesons in the whole accessible kinematic region $ 0\leq q^2 \leq q^2_{max} $. For the numerical investigations of the $ B \to D^* l ν_l $, $ B \to ρl ν_l $, $ D\to K^* l ν_l $ and $ D \to ρl ν_l $ semileptonic decays the equal time wave functions corresponding to the updated version of the ISGW model are adopted. Using the available experimental information on branching fractions $ BR(B \to D^* l ν_l) $ and $ BR(B \…
We obtain the leading twist T-odd quark transverse momentum dependent parton distribution functions (TMDs) of the proton, namely the Sivers function, $f_{1T}^{\perp q}(x,p_\perp^2)$, and the Boer-Mulders function, $h_1^{\perp q}(x,p_\perp^2)$, in a light-front quark-diquark model constructed with the wave functions predicted by the soft-wall AdS/QCD. The gluon rescattering is crucial to predict a nonzero T-odd TMDs. We study the utility of a nonperturbative $SU(3)$ gluon rescattering kernel going beyond the usual approximation of perturbative $U(1)$ gluons. The spin asymmetries in semi-inclusi…
We investigate the total decay rate of the (ground state) $B_c$ meson within the framework of the relativistic constituent quark model formulated on the light-front (LF). The exclusive semileptonic (SL) and nonleptonic (NL) beauty and charm decays of the $B_c$ meson are described through vector and axial hadronic form factors, which are calculated in terms of the overlap of the parent and daughter LF wave functions. The latters are derived via the Hamiltonian LF formalism using as input the update version of the ISGW model. The inclusive SL and NL partial rates are calculated within a convolut…
Numerical results for the (rest-frame) $Q\bar{Q}$ potential in light-front quantized $QCD_{2+1}$ on a $\perp$ lattice are presented. Both in the longitudinal as well as the $\perp$ spatial directions one obtains linear confinement. The confinement mechanism in light-front QCD depends on the orientation of the external charges: for longitudinally (with respect to the boost direction in the infinite momentum frame) separated quarks, confinement arises from the instantaneous interaction piece in the Hamiltonian while for $\perp$ separated quarks, the confinement mechanism is similar to the one in…
We predict the twist-2 Transverse Momentum Dependent parton distribution functions (TMDs) of the pion, namely the unpolarized quark TMD, $f_{1}(x, k_\perp)$, and the transversely polarized quark TMD, also known as the Boer-Mulders function, $h^\perp_{1}(x, k_\perp)$, using a holographic light-front pion wavefunction with dynamical spin effects. These spin effects, in conjunction with gluon rescattering, are crucial to predict a non-zero holographic Boer-Mulders function. We investigate the use of a non-perturbative SU(3) gluon rescattering kernel, thus going beyond the usual approximation of p…
I review recent progress on understanding QCD dynamics involved in exclusive $B$ meson decays. Different frameworks, including light-cone sum rules, QCD factorization, perturbative QCD, soft-collinear effective theory, light-front QCD, are discussed. Results from lattice QCD are quoted for comparison. I point out the important issues in the above QCD methods, which require further investigation.
We investigate the exclusive rare $B_c\to D_{(s)}ν_{\ell}\bar{ν_{\ell}}$ and $B\to D_{(s)}\ell^+\ell^-$ ($\ell=e,μ,τ$) decays within the standard model and the light-front quark model constrained by the variational principle for the QCD motivated effective Hamiltonian. The form factors $f_{\pm}(q^2)$ and $f_T(q^2)$ are obtained from the analytic continuation method in the $q^+=0$ frame. While the form factors $f_+(q^2)$ and $f_T(q^2)$ are free from the zero-mode, the form factor $f_-(q^2)$ is not free from the zero-mode in the $q^+=0$ frame. We discuss the covariance(i.e. frame-independence) o…
Using the light-front quark model, we calculate the transition form factors, decay rates, and longitudinal lepton polarization asymmetries for the exclusive rare $B_s\to (K,η^{(\prime)})(\ell^+\ell^-,ν_{\ell}\bar{ν_{\ell}}$ ($\ell=e,μ,τ$) decays within the standard model, taking into account the $η-η'$ mixing angle. For the mixing angle $θ=-20^{\circ}$ ($-10^{\circ}$) in the octet-singlet basis, we obtain ${\rm BR}(B_s\to η\sumν_{\ell}\barν_{\ell})=1.1 (1.7)\times 10^{-6}$, ${\rm BR}(B_s\to ημ^+μ^-)=1.5 (2.4)\times 10^{-7}$, ${\rm BR}(B_s\to ητ^+τ^-)=3.8 (5.8)\times 10^{-8}$, ${\rm BR}(B_s\to …
The AdS/QCD holographic wave function of basis light-front quantization (BLFQ) for vector meson $J/ψ$ is applied in this manuscript. The exclusive production of $J/ψ$ in diffractive process is computed in dipole model with AdS/QCD holographic wave function. We use IP-Sat and IIM model in the calculation of the differential cross section of the dipole scattering off the proton. The prediction of AdS/QCD holographic wave function in BLFQ gives a good agreement to the experimental data.
We investigate the electroweak form factors and semileptonic decay rates of mesons using the constituent quark model based on the light-front degrees of freedom. Our results demonstrate the broader applicability of light-front approach including the timelike region of exclusive processes.
Protons and antiprotons at collider energies are a source of high energy Weizsaecker--Williams photons. This opens up a possibility to study at the LHC exclusive photoproduction of heavy vector mesons at energies much larger than possible at the HERA accelerator. We present selected results of our detailed studies of various distributions for the production of heavy quarkonia (e.g. rapidity, transverse momenta, azimuthal angles). We give predictions for LHC energies. Our calculations are based on modelling the photoproduction amplitudes in a k_\perp-factorisation approach which is checked agai…
We present an investigation of the total decay rate of the (ground state) B_c meson within the framework of the relativistic constituent quark model formulated on the light-front (LF). The exclusive semileptonic (SL) and nonleptonic (NL) beauty and charm decays of the B_c meson are described through vector and axial hadronic form factors, which are calculated in terms of a constituent quark model LF wave functions. The latter ones are derived via the Hamiltonian LF formalism using as input the update versions of the constituent quark model. The inclusive SL and NL partial rates are calculated …
Weak transition form factors among heavy pseudoscalar mesons are investigated within a relativistic quark model formulated on the light-front. It is shown that the light-front result derived in the time-like region for the matrix elements of the plus component of the weak vector current coincides with the spectator pole term of the quark triangle diagram. For the first time, the dependence of the form factors on the squared four-momentum transfer $q^2$ is calculated in the whole accessible kinematical region $0 \leq q^2 \leq q^2_{max}$. For the numerical investigations of the semileptonic $B \…
The AdS/CFT correspondence between string theory in AdS space and conformal field theories in physical space-time leads to an analytic, semi-classical model for strongly-coupled QCD which has scale invariance and dimensional counting at short distances and color confinement at large distances. One can use holography to map the amplitude describing the hadronic state in the fifth dimension of Anti-de Sitter space to the light-front wavefunctions of hadrons in physical space-time, thus providing a relativistic description of hadrons in QCD at the amplitude level. In particular, we show that ther…
A fundamental goal in QCD is to understand the non-perturbative structure of hadrons at the amplitude level--not just the single-particle flavor, momentum, and helicity distributions of the quark constituents, but also the multi-quark, gluonic, and hidden-color correlations intrinsic to hadronic and nuclear wavefunctions. A natural calculus for describing the bound-state structure of relativistic composite systems in quantum field theory is the light-front Fock expansion which encodes the properties of a hadrons in terms of a set of frame-independent n-particle wavefunctions. Light-front quant…
Kaon T-even transverse-momentum-dependent distributions and form factors in a self-consistent light-front quark model
Inha University（仁荷大学）；Kyungpook National University（庆北国立大学）；The University of Osaka（大阪大学）；Japan Atomic Energy Agency (JAEA)（日本原子力机构）；North Carolina State University（北卡罗来纳州立大学）
We obtain the leading-twist valence quark transverse-momentum-dependent parton distribution functions (TMD PDFs) for the proton within the basis light-front quantization (BLFQ) framework. Our results are consistent with lattice QCD calculations and our previous results for the collinear limit. We also obtain consistency with the Soffer-type bounds. Within our approach, we find that six T-even TMDs in the leading twist are all independent of each other, and previously found model-dependent relations do not hold. This is a promising sign that our results are representative of future, more extens…
We discuss the twist-three, unpolarized, chiral-odd, transverse momentum dependent parton distribution (TMD) $e^q(x,k_\perp)$ within a light-front model. We review a model-independent decomposition of this TMD, which follows from the QCD equations of motion and is given in terms of a leading-twist mass term, a pure interaction-dependent contribution, and singular terms. The leading-twist and pure twist-three terms are represented in terms of overlap of light-front wave functions (LFWFs), taking into account the Fock states with three valence quark ($3q$) and three-quark plus one gluon ($3q+g$)…
A natural calculus for describing the bound-state structure of relativistic composite systems in quantum field theory is the light-front Fock expansion which encodes the properties of a hadrons in terms of a set of frame-independent n-particle wavefunctions. Light-front quantization in the doubly-transverse light-cone gauge has a number of remarkable advantages, including explicit unitarity, a physical Fock expansion, the absence of ghost degrees of freedom, and the decoupling properties needed to prove factorization theorems in high momentum transfer inclusive and exclusive reactions. A numbe…
We analyze the exclusive pseudoscalar $D_{[s]}^+D_{[s]}^-$ pair production in $e^+e^-$ annihilations at $\sqrt{s}=10.6$ GeV using a non-factorized PQCD with the light-front wave function that goes beyond the peaking approximation. We compare our non-factorized analysis with the usual factorized analysis based on the peaking approximation in the calculation of the cross section for the heavy meson pair production. We also discuss the higher helicity contribution to the cross section. Our analysis provides a constraint on the size of quark transverse momentum inside the $D$ meson from the recent…
We investigate the Wigner distributions and generalized transverse momentum-dependent distributions (GTMDs) for $u$ and $d$ quarks in the proton by using light-front quark-diquark model. We consider the contribution of scalar and axial-vector diquark having spin-0 and spin-1 respectively. We take different polarization configurations of quark and proton to calculate the Wigner distributions. The Wigner distributions are studied in the impact-parameter space, momentum space and mixed space for $u$ and $d$ quarks in the proton. We also study the relation of GTMDs with longitudinal momentum fract…
We analyze the exclusive charmonium $J/ψ+η_c$ pair production in $e^+e^-$ annihilation using the nonfactorized perturbative QCD and the light-front quark model(LFQM) that goes beyond the peaking approximation. We effectively include all orders of higher twist terms in the leading order of QCD coupling constant and compare our nonfactorized analysis with the usual factorized analysis in the calculation of the cross section. We also calculate the quark distribution amplitudes, the Gegenbauer moments, and the decay constants for $J/ψ$ and $η_c$ mesons using our LFQM. Our nonfactorized result enha…
We extend the k_\perp-factorization formalism for exclusive photoproduction of vector mesons to the production of electroweak Z^0 bosons. Predictions for the gamma p \to Z^0 p and p p \to p p Z^0 reactions are given using an unintegrated gluon distribution tested against deep inelastic data. We present distributions in the Z^0 rapidity, transverse momentum of Z^0 as well as in relative azimuthal angle between outgoing protons. The contributions of different flavours are discussed. Absorption effects lower the cross section by a factor of 1.5-2, depending on the Z-boson rapidity. We also discus…
We investigate the magnetic dipole decays $V\to Pγ$ of various heavy-flavored mesons such as $(D,D^*,D_s,D^{*}_s,η_c, J/ψ)$ and $(B,B^*,B_s,B^*_s,η_b,Υ)$ using the light-front quark model constrained by the variational principle for the QCD-motivated effective Hamiltonian. The momentum dependent form factors $F_{VP}(q^2)$ for $V\to Pγ^*$ decays are obtained in the $q^+=0$ frame and then analytically continued to the timelike region by changing ${\bf q}_\perp$ to $i{\bf q}_\perp$ in the form factors. The coupling constant $g_{VPγ}$ for real photon case is then obtained in the limit as $q^2\to 0…
We investigate the electroweak form factors and semileptonic decay rates of pseudoscalar and vector mesons using the light-front constituent quark model. Our method of analytic continuation in the Drell-Yan-West (q^+=q^0+q^3=0) frame to obtain the weak form factors avoids the difficulty associated with the contributions from the nonvalence quark-antiquark pair creation. This method seems to provide a promising step forward to handle the exclusive processes in the timelike region. Our calculation reveals that the zero mode contribution is crucial to obtaining the correct values of the light-fro…
Even though the Drell-Yan-West formulation is the most rigorous and well-established framework to compute the exclusive processes, its utility has been limited only to the spacelike region because of the intrinsic kinematic constraint $q^+=0$. We present an explicit example demonstrating how one may obtain the necessary information (i.e. nonvalence or so called Z-graph contribution) in the timelike region of exclusive process without encountering a formidable task of direct calculation that has hindered so far the progress in this area. In the analysis of $q\bar{Q}$ bound state form factors us…
We present all the leading twist T-even TMDs in the light-front quark diquark model(LFQDM) and study the relations among them. The model contains both the scalar and vector diquarks with the light front wave functions modeled from the soft-wall AdS/QCD prediction. The $x-p^2_\perp$ factorization used in phenomenological extraction for TMDs is observed in this model. We present the results for the quark densities and the transverse shape of proton. The shape of the transversely polarized proton is shown to be non-spherical for nonzero transverse momentum. The scale evolution of both integrated …
A scenario is investigated in which the leading-twist pion distribution amplitude phi_pi (x) is approximated by the pion decay constant f_pi for all essential values of the light-cone fraction x. A model for the light-front wave function Psi(x,k_perp) is proposed that produces such a distribution amplitude and has a rapidly decreasing (exponential for definiteness) dependence on the light-front energy combination k_perp^2/x(1-x). It is shown that this model easily reproduces the fit of recent large-Q^2 BaBar data on the photon-pion transition form factor. Some aspects of scenario with flat pio…
We discuss central exclusive production (CEP) of the tensor $χ_c(2^{+})$ meson in proton-(anti)proton collisions at Tevatron, RHIC and LHC energies. The amplitude for the process is derived within the $k_t$-factorisation approach. Differential and total cross sections are calculated for several unintegrated gluon distributions (UGDFs). We compare exclusive production of all charmonium states $χ_c(0^+),\,χ_c(1^+)$ and $χ_c(2^+)$. Equally good description of the recent Tevatron data is achieved both with Martin-Ryskin phenomenological UGDF and UGDF based on unified BFKL-DGLAP approach. Unlike fo…
The light-front representation of quantum chromodynamics provides a frame-independent, quantum-mechanical representation of hadrons at the amplitude level, capable of encoding their multi-quark, hidden-color and gluon momentum, helicity, and flavor correlations in the form of universal process-independent hadron wavefunctions. The universality and frame-independence of the LCWF's thus allow a profound connection between diffractive dissociation, hard scattering exclusive processes such as elastic form factors, two-photon reactions, and heavy hadron decays. In this concluding talk of the ECT* I…
At leading twist the transverse-momentum dependent parton distributions of the pion consist of two functions, the unpolarized $f_{1,π}(x,\boldsymbol k^{\, 2}_\perp)$ and the Boer-Mulders function $h_{1,π}^{\perp}(x,\boldsymbol k^{\, 2}_\perp)$. We study both functions within a light-front constituent model of the pion, comparing the results with different pion models and the corresponding nucleon distributions from a light-front constituent model. After evolution from the model scale to the relevant experimental scales, the results for the collinear pion valence parton distribution function $f…
The transverse lattice approach to non-perturbative light-front hamiltonian QCD is described. Preliminary results on the pi-rho system are presented, at fixed DLCQ and Tamm-Dancoff cut-offs. A renormalised, approximately Lorentz covariant light-front hamiltonian is found to leading order of the colour-dielectric expansion, compatible with a massless pion. The pi light-front wavefunction is compared with experiment. Exclusive processes agree reasonably well, given the approximations, but inclusive processes, sensitive to higher Fock state structure, still exhibit large cut-off artifacts.
We study exclusive non-leptonic two-body $B_c\to(D_{(s)},η_c,B_{(s)})+F$ decays with $F$(pseudoscalar or vector meson) being factored out in QCD factorization approach. The non-leptonic decay amplitudes are related to the product of meson decay constants and the form factors for semileptonic $B_c$ decays. As inputs in obtaining the branching ratios for a large set of non-leptonic $B_c$ decays, we use the weak form factors for the semileptonic $B_c\to(D_{(s)},η_c,B_{(s)})$ decays in the whole kinematical region and the unmeasured meson decay constants obtained from our previous light-front quar…
We present the leading-twist quark transverse momentum-dependent parton distribution functions (TMDs) for the spin-1 target, such as the $ρ$-meson, in the light-front framework. Specifically, we predict the TMDs in the light-front holographic model and compare with the light-front quark model predictions. We obtain the TMDs using the overlap of the light-front wave functions. We evaluate the ${\bf k}_\perp$ moments upto second order and compare with the available theoretical predictions. Further, we analyze the leading-twist parton distribution functions (PDFs) of the $ρ$-meson in the light-fr…
