The valence-quark distribution function of the pion has been of interest for decades; particularly, the profile it should adopt when $x\to1$ (the large-$x$ behavior) is subject of a long-standing debate. In the light-front holographic QCD (LFHQCD) approach, this behavior is controlled by the so called reparametrization function, $w_τ(x)$, which is not fully determined from first principles. We show that, owing to the flexibility of $w_τ(x)$, the large-$x$ profile $u^π(x)\sim (1-x)^{2}$ can be contained within the LFHQCD formalism. This is in contrast with a previous LFHQCD study~(Guy F. de Teramond et al., Phys. Rev. Lett., 120(18), 2018) in which $u^π(x)\sim (1-x)^{1}$ is found instead. Given our observations, augmented by perturbative QCD and recent lattice QCD results, we state that the large-$x$ exponent of "2" cannot be excluded.
Using the light-front holographic model, we study the transverse momentum dependent parton distributions (TMDs) for the case of pion. At leading twist, the unpolarized parton distribution function $ f_{1π}(x,\bfk^{2}) $ and the Boer-Mulders function $ h_{1π}^{\bot}(x,\bfk^{2}) $ are obtained for pion. We calculate both the functions using the light-front holographic model with spin improved wave function and compare the predicted results with available results of other models. In order to provide inputs in predicting future experimental data, a LO evolution is performed from model scale to exp…
Two-parton correlations in the pion are investigated in terms of double parton distribution functions. A Poincaré covariant Light-Front framework has been adopted. As non perturbative input, the pion wave function obtained within the so-called soft-wall AdS/QCD model has been used. Results show how novel dynamical information on the structure of the pion, not accessible through one-body parton distribution, are encoded in double parton distribution functions.
We study the Wigner distributions of the pion using a holographic light-front pion wavefunction with dynamical spin effects to reveal its multidimensional structure.
The holographic QCD prediction for the pion distribution amplitude (DA) $ϕ_{hol}(u)$ is used to compute the pion spacelike electromagnetic form factor $F_π(Q^2)$ within the QCD light-cone sum rule method. In calculations the pion's renormalon-based model twist-4 DA, as well as the asymptotic twist-4 DA are employed. Obtained theoretical predictions are compared with experimental data and with results of the holographic QCD.
We use detailed balance for a hadron composed of quark and gluon Fock states to obtain parton distributions in the proton and pion on the basis of a simple statistical model.
We report that the inclusion of dynamical spin effects in the pion holographic light-front wavefunction leads to a remarkable improvement in describing pion observables (pion mean charge radius, decay constant, spacelike electromagnetic form factor) without the need to invoke higher Fock state contributions and/or a special AdS/QCD mass scale for the pion.
Two-parton correlations in the pion, a non perturbative information encoded in double parton distribution functions, are investigated in the Nambu and Jona-Lasinio model. It is found that double parton distribution functions expose novel dynamical information on the structure of the pion, not accessible through one-body parton distributions, as it happens in several estimates for the proton target and in a previous evaluation for the pion, in a light-cone framework. Expressions and predictions are given for double parton distributions corresponding to leading-twist Dirac operators in the quark…
We calculate skewed parton distributions $H^I$ of a pion within a non-local NJL model with momentum dependent constituent quark mass $M(k)$. In the forward limit $H^I$ correspond to parton distributions, whereas for $t<0$, $H^I$ is related to the pion electromagnetic form-factor
We present a thorough analysis of unpolarized Drell-Yan (DY) pair production in pion-nucleus scattering. On the nucleus side, we use nuclear parton distributions along with parametrisations of the nucleon partonic transverse distribution available in the literature. Partonic longitudinal and transverse distributions of the pion are those obtained in a recent calculation in a Nambu-Jona Lasinio (NJL) framework, with Pauli-Villars regularization. The scale of the NJL model is fixed by a comparison between NLO predictions based on NJL evolved pion distributions and rapidity differential DY cross …
We report that the pion radius, decay constant, EM and transition form factors can simultaneously and accurately be predicted in light-front holographic QCD with a universal AdS/QCD mass scale, provided dynamical spin effects are taken into account.
We derive the lowest-order $q\bar q$ valence light-front wave function for the pion with Lz=0 and |Lz|=1, which reproduces its valence parton distribution and the electromagnetic form factor consistent with data.
We present a new, comprehensive global analysis of parton-to-pion fragmentation functions at next-to-leading order accuracy in QCD. The obtained results are based on the latest experimental information on single-inclusive pion production in electron-positron annihilation, lepton-nucleon deep-inelastic scattering, and proton-proton collisions. An excellent description of all data sets is achieved, and the remaining uncertainties in parton-to-pion fragmentation functions are estimated based on the Hessian method. Extensive comparisons to the results from our previous global analysis are performe…
Using light-front holography, we predict simultaneously the pion decay constant and the pion charge radius by taking into account (higher twist) dynamical spin effects whose relative importance is constrained by QCD.
We use phenomenological nonlocal Lagrangians, which lead to non trivial forms for the quark propagator, to describe the pion. We define a procedure, based on the Dyson-Schwinger equations, for the calculation of the pion parton distributions at low Q^2. The obtained parton distributions fulfill all the wishful properties. Using a convolution approach we incorporate the composite character of the constituent quarks in the formalism. We evolve, using the Renormalization Group, the calculated parton distributions to the experimental scale and compare favorably with the data and draw conclusions.
We present a new analysis to extract pion parton distribution functions (PDFs) within the framework of the statistical model. Starting from the statistical model first developed for the spin-1/2 nucleon, we extend this model to describe the spin-0 pion. Based on a combined fit to both the pion-induced Drell-Yan data and the pion-induced $J/Ψ$ production data, a new set of pion PDFs has been obtained. The inclusion of the $J/Ψ$ production data in the combined fit has provided additional constraints for better determining the gluon distribution in the pion. We also compare the pion PDFs obtained…
We show that the holographic Schrödinger equation of light-front chiral QCD, together with the 't Hooft equation of (1+1)-dimensional QCD in the large $N_c$ limit, can simultaneously describe pion spectroscopy as well as the pion decay constant, charge radius, electromagnetic form factor, photon-to-pion transition form factor, Parton Distribution Function (PDF) and Distribution Amplitude (DA). Furthermore, the chiral-limit constraints, as encoded in the Gell-Mann-Oakes-Renner (GMOR) relation, are satisfied.
We study the generalized parton distributions (GPDs) of pion for non-zero skewness by considering the leading $| q \bar{q} \rangle$ Fock state component. Inspired from AdS/QCD light-front wave functions, we calculate the pion GPDs. By taking the Fourier transforms we obtain the results for impact-parameter dependent parton distribution functions (ipdpdf) and for GPDs in longitudinal boost-invariant space. We also calculate the charge density and gravitational form factor of the pion. The pion unpolarized transverse momentum distribution (TMD) have also been calculated. The results provide rich…
We present a model for the structure of the pion. Based on ideas of a recently developed statistical model of the nucleon, we assume the pion to be a gas of partons. The finite-size corrections (FSC) are incorporated through two parameters. Using the same two FSC parameters for the proton and pion we reproduce quantitatively the data on $π^- N \to μ^+ μ^- X$ Drell-Yan production and valence quark distribution of the pion.
We present an extensive study on generalized parton distributions of pions. In particular, we present the results for the pion-photon transition distribution amplitudes in the Nambu - Jona-Lasinio model. We discuss the phenomenology of such objects and give explicit estimates for cross sections. The spin structure of the proton is also studied: we discuss the results for the T-odd transverse momentum distributions in quark models.
We study the low $Q^2$ wave-functions of pions and kaons as an expansion in terms of hadron-like Fock state fluctuations. In this formalism, pion and kaon wave-functions are related one another. Consequently, the knowledge of the pion structure allows the determination of parton distributions in kaons. In addition, we show that the intrinsic (low $Q^2$) sea of pions and kaons are different due to their different valence quark structure. Finally, we analize the feasibility of a method to extract kaon's parton distribution functions within this approach and compare with available experimental da…
We calculate the leading twist valence quark distribution in the pion in the franework of QCD sum rules with nonlocal condensates. Particular attention has been paid to the correct account of the bilocal power corrections.
We present pion and kaon parton distribution functions from a global QCD analysis of the experimental data within the framework of dynamical parton model. We use the DGLAP equations with parton-parton recombination corrections and the valence input of uniform distribution which maximizes the information entropy. At our input scale $Q_0^2$, there are no sea quark and gluon distributions. All the sea quarks and gluons of the pion and the kaon are completely generated from the parton splitting processes. The mass-dependent parton splitting kernel is applied for the strange quark distribution in t…
We set up a framework for the study of the power-suppressed three-parton contribution to the pion electromagnetic form factor in the $k_T$ factorization theorem. It is first shown that the gauge dependence proportional to parton transverse momenta from the two-parton Fock state and the gauge dependence associated with the three-parton Fock state cancel each other. After verifying the gauge invariance, we derive the three-parton-to-three-parton $k_T$-dependent hard kernel at leading order of the coupling constant, and find that it leads to about 5% correction to the pion electromagnetic form fa…
The partonic flux originated from a heavy nucleus is not the mere sum of the fluxes coming from the individual nucleons. There are various effects that give rise to modifications. One of these effects is here investigated i.e. the presence of a cloud of virtual pions co-moving with the nucleus. It is found that the contribution of these virtual particles to the total parton flux should be rather small even if one includes the contribution of resonances.
We account for dynamical spin effects in the holographic light-front wavefunction of the pion in order to predict its mean charge radius, decay constant, spacelike electromagnetic form factor, twist-2 Distribution Amplitude and the photon-to-pion transition form factor. Using a universal fundamental AdS/QCD scale of 523 MeV and a constituent quark mass of 330 MeV, we find a remarkable improvement in describing all observables.
Using constituent quark model constraints we calculate the gluon and sea-quark content of pions solely in terms of their valence density (fixed by πN Drell-Yan data) and the known sea and gluon distributions of the nucleon, using the most recent updated valence-like input parton densities of the nucleon. The resulting small-x dynamical QCD predictions for g^π(x,Q^2) and \bar{q}^π(x,Q^2) are unique and parameter free. Simple analytic parametrizations of the resulting parton distributions of the pion are presented in LO and NLO. These results and parametrizations will be important, among other t…
We investigate the quark Wigner distributions of the pion to reveal the multidimensional picture of pion. We have used the spin improved wave functions of pion deduced from the light-front holographic model of mesons. By using the Fock-state overlap representation, the Wigner distributions of an unpolarized, longitudinally polarized and transversely polarized quark inside the pion are calculated. We have presented the results of transverse Wigner distributions in impact-parameter space as well as in momentum space. In order to understand the role of skewness which gives the longitudinal moment…
The quark generalised parton distributions of the pion are calculated from lightcone wavefunctions in transverse lattice gauge theory at large N_c. The pion effective size is found to decrease with increasing momentum transfer. An analytic ansatz, consistent with finite boundstate lightcone energy conditions, is given for the lightcone momentum dependence of the wavefunctions. This leads to simple, universal predictions for the behaviour of the distributions near the endpoints, complementing numerical DLCQ data.
Using ideas from Light Front Holography, we discuss the calculation of the nucleon helicity-independent generalized parton distributions of quarks in the zero skewness case.
Pion parton distribution functions and pion-nucleus induced $J/ψ$ production in extended light-front holographic QCD
Institute of Modern Physics, Chinese Academy of Sciences（中国科学院近代物理研究所）；State Key Laboratory of Heavy Ion Science and Technology（重离子物理国家重点实验室）；University of Chinese Academy of Sciences（中国科学院大学）
Generalized Parton Distributions of the pion are evaluated in chiral quark models with the help of double distributions. As a result the polynomiality conditions are automatically satisfied. In addition, positivity constraints, proper normalization and support, sum rules, and soft pion theorems are fulfilled. We obtain explicit expressions holding at the low-energy quark-model scale, which exhibit no factorization in the t-dependence. The crucial QCD evolution of the quark-model distributions is carried out up to experimental or lattice scales. The obtained results for the Parton Distribution …
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…
Holographic soft-wall model is successful in the phenomenology of hadrons. Here with the use of generalized parton distributions (GPDs) obtained from AdS/QCD, perturbative effects are entered into the formalism. Perturbations are incorporated in the formalism through the evolution of GPDs according to the DGLAP like equations. Evolved proton GPDs are compared with a phenomenological model to show that we can get good improvements of the holographic model. It seems that combining the holographic soft-wall model with perturbative effects to some extent, gives the correct physics of GPDs.
Off-forward structure functions of the pion are investigated in twist-two and twist-three approximation. A simple model is used for the pion, which allows to introduce finite size effects, while preserving gauge invariance. Results for the imaginary parts of the gamma^* pi -> gamma^* pi off-forward amplitude and of the structure functions are presented. Generalized Callan-Gross relations are obtained.
The generalized parton distributions of the pion are studied within different light-front approaches for the quark-hadron and quark-photon vertices, exploring different kinematical regions in both the valence and non-valence sector. Moments of the generalized parton distributions which enter the definition of generalized form factors are also compared with recent lattice calculations.
We evaluate Generalized Parton Distributions of the pion in two chiral quark models: the Spectral Quark Model and the Nambu-Jona-Lasinio model with a Pauli-Villars regularization. We proceed by the evaluation of double distributions through the use of a manifestly covariant calculation based on the alpha representation of propagators. As a result polynomiality is incorporated automatically and calculations become simple. In addition, positivity and normalization constraints, sum rules and soft pion theorems are fulfilled. We obtain explicit formulas, holding at the low-energy quark-model scale…
We calculate the off-forward structure function of the pion within a simple model where the size of the pion is accounted for using a momentum cut-off. Twist-two and twist-three generalized parton distributions are extracted. Relations between twist-three and twist-two contributions are obtained, the origin of which is not kinematical as they differ from those arising from the Wandzura-Wilczek approximation.
Pion generalised parton distributions are calculated within the framework of the Nambu--Jona-Lasinio model using different regularization schemes, including the proper time regularization scheme, the three dimensional momentum cutoff scheme, the four dimensional momentum cutoff scheme, and the Pauli-Villars regularization scheme. Furthermore, we check the theoretical constraints of pion generalised parton distributions required by the symmetries of quantum chromodynamics in different regularization schemes. The diagrams of pion parton distribution functions are plotted, in addition, we evaluat…
We argue that the collinear factorization of the fragmentation functions in high energy hadron and nuclei collisions breaks down at transverse momenta kT < Qs/g due to high parton densities in the colliding hadrons and/or nuclei. We calculate, at next-to-leading order in projectile parton density and to all orders in target parton density, the double-inclusive cross section for production of a pair of gluons in the scalar J^(PC)=0^(++) channel. Using the low energy theorems of QCD we find the inclusive cross section for pi-meson production.
An explicit evaluation of the two pion transverse momentum dependent parton distributions at leading twist is presented, in the framework of the Nambu-Jona Lasinio model with Pauli-Villars regularization. The transverse momentum dependence of the obtained distributions is generated solely by the dynamics of the model. Using these results, the so called generalized Boer-Mulders shift is studied and compared with recent lattice data. The obtained agreement is very encouraging, in particular because no additional parameter has been introduced. A more conclusive comparison would require a precise …
We study the leading-twist quasi parton distribution function (quasi-PDF) and quasi generalize parton distribution (quasi-GPD) of the pion meson by using a spectator model. We consider the case the quasi functions are defined via inserting the matrix $γ^3$ in the spacial correlation functions. We obtain the analytical expressions for the quasi-PDF and quasi-GPD. The numerical results for them are calculated from the parameters obtained by fitting the model results to the known parametrization. Particularly, we investigate the hadron momentum dependence of the quasi functions as well as compare…
In this article, we calculate the contribution of the higher-twist Feynman diagrams to the large-$p_T$ inclusive single pion production cross section in photon-photon collisions in case of the running coupling and frozen coupling approaches within holographic QCD. We compare the resummed higher-twist cross sections with the ones obtained in the framework of the frozen coupling approach and leading-twist cross section. Also, we show that in the context of frozen coupling approach a higher-twist contribution to the photon-photon collisions cross section is normalized in terms of the pion electro…
In this review, we discuss the results on the parton-to-pion fragmentation functions obtained in a combined NLO fit to data of single-inclusive hadron production in electron-positron annihilation, proton-proton collisions, and lepton-nucleon deep-inelastic scattering. A more complete discussion can be found in Ref. [1].
We study pion in the holographic model of Hirn and Sanz which contains two Yang-Mills fields defined in the background of the sliced AdS space. The infrared boundary conditions imposed on these fields generate the spontaneous breaking of the chiral symmetry down to its vector subgroup. Within the framework of this model, we get an analytic expression for the pion form factor and a compact result for its radius. We also extend the holographic model to include Chern-Simons term which is required to reproduce the appropriate axial anomaly of QCD. As a result, we calculate the anomalous form facto…
During the INT-18-3 workshop, we presented an analysis of unpolarized Drell-Yan pair production in pion-nucleus scattering with a particular focus into the pion Transverse Momentum Distributions in view of the future Electron Ion Collider. The transverse distributions of the pion calculated in a Nambu--Jona-Lasinio framework, with Pauli-Villars regularization, were used. The pion Transverse Momentum Distributions evolved up to next-to-leading logarithmic accuracy is then tested against the transverse momentum spectrum of dilepton pairs up to a transverse momentum of 2 GeV. We found a fair agre…
Parton distributions for the pion are studied in a chiral quark model characterized by a quark propagator for which a spectral representation is assumed. Electromagnetic and chiral symmetry constraints are imposed through the relevant Ward-Takahashi identities for flavoured vertex functions. Finiteness of the theory, requires the spectral function to be non-positive definite. Straightforward calculation yields the result that the pion structure function becomes one in the chiral limit, regardless of the details of the spectral function. LO and NLO evolution provide a satisfactory description o…
Parton distributions of pseudoscalar pi,K and eta mesons obtained within the NJL model using the Pauli-Villars regularization method are analyzed in terms of LO and NLO evolution, and the valence sea quark and gluon parton distributions for the pion are obtained at Q^2 = 4 GeV^2 and compared to existing parametrizations at that scale. Surprisingly, the NLO order effects turn out to be small compared to the LO ones. The valence distributions are in good agreement with experimental analyses, but the gluon and sea distributions come out to be softer in the high-x region and harder in the low-x re…
We determine the valence parton distribution function of the pion by performing a new analysis of data for the Drell-Yan process $π^- N\to μ^+μ^-X$. Compared to previous analyses, we include next-to-leading-logarithmic threshold resummation effects in the calculation of the Drell-Yan cross section. As a result of these, we find a considerably softer valence distribution at high momentum fractions $x$ than obtained in previous next-to-leading order analyses, in line with expectations based on perturbative-QCD counting rules or Dyson-Schwinger equations.
We present here an extension of our model for the pion, which we used previously to calculate its diagonal structure function, to the off-forward case. The imaginary part of the off-forward gamma^* pi -> gamma^* pi scattering amplitude is evaluated in the chiral limit (m_pi=0) and related to the twist-two and twist-three generalised parton distributions H, H3, H3tilde. Non-perturbative effects, linked to the size of the pion and still preserving gauge invariance, are included. Remarkable new relations between H, H3 and H3tilde are obtained and discussed.
Using a holographic dual model of QCD, we compute the pion electromagnetic form factor F_pi(Q^2) in the spacelike momentum transfer region, as well as pion couplings to vector mesons g_rho^(n) pi pi. Spontaneous and explicit chiral symmetry breaking are intrinsic features of this particular holographic model. We consider variants with both ``hard-wall'' and ``soft-wall'' infrared cutoffs, and find that the F_pi(Q^2) data tend to lie closer to the hard-wall model predictions, although both are too shallow for large Q^2. By allowing the parameters of the soft-wall model (originally fixed by obse…
We consider a general formalism to compute inclusive polarised and unpolarised cross sections within pQCD and the factorisation scheme, taking into account parton intrinsic motion in distribution and fragmentation functions, as well as in the elementary dynamics. Surprisingly, the intrinsic partonic motion, with all the correct azimuthal angular dependences, produces a strong suppression of the transverse single spin asymmetry arising from the Collins mechanism. As a consequence, and in contradiction with earlier claims, the Collins mechanism is unable to explain the large asymmetries found in…
We extend the investigation of the Generalized Parton Distribution for a charged pion within a fully covariant constituent quark model, in two respects: (i) calculating the tensor distribution and (ii) adding the treatment of the evolution, needed for achieving a meaningful comparison with both the experimental parton distribution and the lattice evaluation of the so-called generalized form factors. Distinct features of our phenomenological covariant quark model are: (i) a 4D Ansatz for the pion Bethe-Salpeter amplitude, to be used in the Mandelstam formula for matrix elements of the relevant …
In a model of hadrons in which the hadron wave function at low $Q^2$ is represented by a hadron-like Fock state expansion, we calculate the $π$ and $K$ parton distribution functions by comparing to experimental data. We show that the model gives an accurate description of available data and, in addition, gives a clear prediction of the initial flavors to be considered at the low $Q^2$ scale where perturbative evolution starts
