Renormalization-group approach to transverse-momentum dependent parton distribution functions in QCD
N. G. Stefanis, I. O. Cherednikov
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9 pages, 5 figures, invited plenary talk given by the first author at XIII International Conference on Selected Problems of Modern Theoretical Physics, Dubna, Russia, June 23-27, 2008
We discuss the renormalization of gauge-invariant transverse-momentum dependent (TMD), i.e., unintegrated, parton distribution functions (PDFs) and carry out the calculation of their anomalous dimension at one loop. We show that in the light-cone gauge, TMD PDFs contain UV divergences that may be attributed to the renormalization effect on a cusp-like junction point of the gauge contours at infinity. In order to eliminate the anomalous dimension ensuing from this cusp, we propose to use in the definition of the TMD PDFs, a soft counter term in terms of a path-ordered phase factor along a particular cusped contour extending to transverse light-cone infinity and comprising light-like and transverse segments. We argue that this additional factor is analogous to the "intrinsic" Coulomb phase factor found before in QED.
I discuss the physical interpretation of GPDs in the limit $ξ=0$, where the $t$ dependence contains information about the distribution of partons in the transverse plane. GPDs thus allow a simultaneous determination of the longitudinal momentum and transverse position of partons in the infinite momentum frame, which also provides a physical interpretation for Ji's angular momentum sum rule.
By working out the kinematics of double parton scattering at short relative transverse distances, we obtain an explicit link between the transverse centres of mass, of the two hard partonic interactions, and the contributions to the process, due to pairs of interacting partons generated by perturbative splitting. One my thus foresee the interesting possibility of discriminating experimentally between contributions to the double parton scattering cross section, due to interacting parton pairs originated by independent evolution, and contributions, due to interacting parton pairs generated by sp…
Recent developments showed that light-cone parton distributions can be studied by investigating the large momentum limit of the hadronic matrix elements of spacelike correlators, which are known as quasi parton distributions. Like a light-cone parton distribution, a quasi parton distribution also contains ultraviolet divergences and therefore needs renormalization. The renormalization of non-local operators in general is not well understood. However, in the case of quasi quark distribution, the bilinear quark operator with a straight-line gauge link appears to be multiplicatively renormalizabl…
Generalized parton distributions (GPDs) with purely transverse momentum transfer can be interpreted as Fourier transforms of the distribution of partons in impact parameter space. The helicity-flip GPD $E(x,0,-\Dps)$ is related to the distortion of parton distribution functions in impact parameter space if the target is not a helicity eigenstate, but has some transverse polarization. This transverse distortion can be used to develop an intuitive explanation for various transverse single spin asymmetries.
The scale factor $σ_{eff}$, which characterizes double parton collisions in high energy hadron interactions, is a direct manifestation of the distribution of the interacting partons in transverse space, in such a way that different distributions give rise to different values of $σ_{eff}$ in different double parton collision processes. We work out the value of the scale factor in a few reactions of interest, in a correlated model of the multi-parton density of the proton recently proposed.
We discuss the parton branching (PB) evolution equation for transverse momentum dependent (TMD) parton distribution functions (PDFs), especially we concentrate on the angular ordering constrain and its effect on the TMD distributions. We discuss application of the PB TMDs to precise prediction of Drell-Yan transverse momentum spectrum at LHC, stressing the role of angular ordering in the low transverse momentum region. We compare the PB implementation of the angular ordering condition with the Kimber-Martin-Ryskin-Watt (KMRW), both at the analytical and numerical level.
The cross section for transverse momentum dependent double parton scattering involves transverse momentum dependent double parton distributions (DTMDs). In the region of perturbative transverse momentum the DTMDs can be matched onto collinear double parton distributions. We present the framework and results for this matching, as well as the evolution equations for DTMDs in the region of large distance between the two partons. We discuss explicit results for one-loop matching coefficients and evolution kernels.
We derive the complete Wilson renormalization group equation which governs the evolution of the gluon distribution and other gluonic observables at low $x$ and arbitrary density.
We derive a nonperturbative transverse momentum distribution for partons using a potential model to describe the quark-quark interaction inside the proton. We use this distribution to calculate the diferential cross-section of $π^0$-production for intermediate values of transverse momentum in $p-p$ collisions at high energies. Assuming a variable string tension constant for the quark-quark potential we obtain a very good description of the experimental data at different energies. The corresponding values of the mean transverse momentum of the partons are essentially lower than those obtained u…
We study the impact of transverse-momentum dependent parton distributions on detailed features of multi-jet final states, focusing on angular jet correlations in DIS data.
I report on precision comparisons of theoretical predictions for Drell-Yan (DY) transverse momentum spectra from two approaches: the analytical resummation approach based on Collins-Soper-Sterman (CSS) formalism and the parton branching (PB) approach to the evolution of transverse momentum dependent (TMD) parton distributions.
Off-shell, transverse-momentum dependent splitting functions can be defined from the high-energy limit of partonic decay amplitudes. Based on these splitting functions, we construct Sudakov form factors and formulate a new parton branching algorithm. We present a first Monte Carlo implementation of the algorithm. We use the numerical results to verify explicitly momentum sum rules for TMD parton distributions.
We review the current status of collinear and Transverse Momentum Dependent densities based on the Parton Branching method. We investigate the performance of the PB-TMD evolution as well as PB-TMD parton shower with different configurations of the hard process in the four- and five-flavour schemes.
Transverse-momentum-dependent parton distribution functions are analyzed in semi-inclusive deep inelastic scattering at low transverse momentum using soft-collinear effective theory. The transverse-momentum-dependent parton distribution functions are defined on the lightcone without distorting the lightcone path nor adding additional soft Wilson lines. In this approach, the comparison between the integrated and unintegrated parton distribution functions becomes transparent. The procedure of computing radiative corrections in dimensional regularization is explained in detail, and the divergence…
We present a solution of the DGLAP evolution equations, written in terms of Sudakov form factors to describe the branching and no-branching probabilities, using a parton branching Monte Carlo method. We demonstrate numerically that this method reproduces the semi-analytical solutions. We show how this method can be used to determine Transverse Momentum Dependent (TMD) parton distribution functions, in addition to the usual integrated parton distributions functions. We discuss numerical effects of the boundary of soft gluon resolution scale parameter on the resulting parton distribution functio…
I give a brief historical account on the way I have been conducted to revive the concept of the transverse spin from previous studies on multiparton scatterings.
We propose the study of unpolarized transverse momentum dependent gluon parton distributions as well as the effect of linearly polarized gluons on transverse momentum and rapidity distributions of $J/ψ$ and $Υ$ production within the framework of transverse momentum dependent factorization employing color evaporation model (CEM) in unpolarized proton-proton collision. We estimate the transverse momentum and rapidity distributions of $J/ψ$ and $Υ$ at LHCb, RHIC and AFTER energies using TMD evolution formalism.
The Parton Branching (PB) approach describes the evolution of transverse momentum dependent (TMD) parton densities. We propose to extend the PB method by including TMD splitting functions, instead of the DGLAP splitting functions which assume strong ordering in transverse momentum. We present the evolution equations and their numerical solution, which is the first Monte Carlo implementation including TMD splitting functions.
Transverse-momentum spectra in hard processes are typically described either in terms of intrinsic transverse momentum of partons, or in terms of perturbative radiation. The relation between these descriptions is discussed for the example of semi-inclusive deep inelastic scattering, with special focus on the angular distribution of the observed hadron. This involves nontrivial theoretical issues, such as the proper definition of transverse-momentum dependent parton distributions, and has practical consequences for the description of pT spectra in phenomenology.
The conventional NLO parton model is enough for the description of total cross sections and one-particle distributions. In the case of two-particle correlations, the collinear approximation has failed, and it is necessary to account for the transverse momenta of initial partons. The different possibilities to do this are discussed.
The present knowledge on non-trivial relations between generalized parton distributions on the one hand and transverse momentum dependent distributions on the other is reviewed. While various relations can be found in the framework of spectator models, so far no model-independent non-trivial relations have been established.
Two factorization approaches have been proposed for single transverse spin asymmetries. One is the collinear factorization, another is the transverse-momentum-dependent factorization. They have been previously derived in a formal way by using diagram expansion at hadron level. If the two factorizations hold or can be proven, they should also hold when we replace hadrons with parton states. We examine these two factorizations at parton level with massless partons. It is nontrivial to generate these asymmetries at parton level with massless partons because the asymmetries require helicity-flip a…
We investigate the relations between transverse momentum dependent parton distributions (TMDs) and generalized parton distributions (GPDs) in a light-front quark-diquark model motivated by soft wall AdS/QCD. Many relations are found to have similar structure in different models. It is found that a relation between the Sivers function and the GPD $E_q$ can be obtained in this model in terms of a lensing function. The quark orbital angular momentum is calculated and the results are compared with the results in other similar models. Implications of the results are discussed. Relations among diffe…
We propose a consistent treatment of divergences emerging in the computation of transverse-momentum-dependent parton densities in leading $α_s$-order of QCD perturbation theory.
The transverse-momentum dependent quark distributions are studied in the limit of the large number of colors Nc. The leading orders of Nc are determined for all twist-2 distributions including the T-odd functions. The isospin structure of the dominant contributions is found.
We perform calculations of Z+jet cross-section taking into account the transverse momenta of the initial partons. Transverse Momentum Dependent (TMD) parton densities obtained with the Parton Branching method are used and higher order corrections are included via TMD parton showers in the initial state. The predictions are compared to measurements of forward Z+jet production of the LHCb collaboration at $\sqrt{s}=7$ TeV. We show that the results obtained in kT-factorization are in good agreement with results obtained from a NLO calculation matched with traditional parton showers. We also demon…
Transverse parton momentum dependent distribution functions (TMDs) of the nucleon are studied in a covariant model, which describes the intrinsic motion of partons in terms of a covariant momentum distribution. The consistency of the approach is demonstrated, and model relations among TMDs are studied. As a byproduct it is shown how the approach allows to formulate the non-relativistic limit.
We perform a calculation of inclusive $Z$ boson production in proton-lead collisions at the LHC taking into account the transverse momenta of the initial partons. We use the framework of $k_T$-factorization combining transverse momentum dependent parton distributions (TMDs) with off-shell matrix elements. In order to do it we need to construct appropriate TMDs for lead nuclei which is done using the parton branching method. Our computations are compared with data from CMS taken at $\sqrt{s}=5.02$ TeV. The results are in good agreement with the measurements especially the transverse momentum di…
Using partonic states we verify the transverse momentum dependent factorization for single transverse spin asymmetries in semi inclusive DIS at leading but nontrivial order of alpha_s. The factorization has been previously derived in a formal way by using diagram expansion at hadron level. We find with our partonic results that two relevant structure functions satisfy the factorization. Our results also satisfy the collinear factorization but with the perturbative coefficient different than that derived formally.
The proton's transverse polarization structure is examined in terms of the Lorentz-covariant Pauli-Lubanski vector in QCD. We find that there are contributions from leading, subleading, and next-to-subleading partonic contributions in the light-front system of coordinates. The subleading and next-to-subleading contributions are related to the leading one through Lorentz symmetry. And the leading contribution obeys a simple partonic angular momentum sum rule that gives a clear physical interpretation to a relation known previously.
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 have investigated the transverse charge density for longitudinal and transversely polarized $ρ$ meson in light-front quark model (LFQM). Charge densities are obtained from the elastic form factors of the $ρ$ meson calculated in LFQM including the zero-mode contributions. We have computed the different helicity matrix elements of the $ρ$ meson. In addition to this, we have also presented the results for the generalized parton distributions (GPDs) and impact-parameter dependent parton distribution functions (ipdpdfs) of the $ρ$ meson.
Our knowledge on the three-dimensional momentum structure of hadrons is encoded in the Transverse Momentum Dependent partonic distribution and fragmentation functions (TMDs). A brief and updated review of the TMDs and of the processes in which they might play a role is presented.
The hadron-energy evolution (Collins and Soper) equation for all the leading-twist transverse-momentum and spin dependent parton distributions is derived in the impact parameter space. Based on the result, we present resummation formulas for the spin structure functions in the semi-inclusive deep inelastic scattering.
Transverse-momentum-dependent parton distributions (TMDs) provide three-dimensional images of the partonic structure of the nucleon in momentum space. We made impressive progress in understanding TMDs, both from the theoretical and experimental point of view. This brief overview on TMDs is divided in two parts: in the first, an essential list of achievements is presented. In the second, a selection of open questions is discussed.
Longitudinal and transverse parton distributions for pion and nucleon are calculated from hadron vertexes obtained by a study of form factors within relativistic quark models. The relevance of the one-gluon-exchange dominance at short range for the behavior of the form factors at large momentum transfer and of the parton distributions at the end points is stressed.
We show how some parton distribution functions related to the transverse spin of nucleons can be extracted point by point from combinations of proton and deuteron observables. In particular, we present a determination of the valence and sea Sivers functions from the single-spin asymmetries measured by COMPASS.
The extension of the statistical parton distributions to include their transverse momentum dependence (TMD) is revisited by considering that the proton target has a finite longitudinal momentum. The TMD will be generated by means of a transverse energy sum rule. The new results are mainly relevant for electron-proton inelastic collisions in the low $Q^2$ region. We take into account the effects of the Melosh-Wigner rotation for the helicity distributions.
A range of issues pertaining to the use of Wilson lines in integrated and transverse-momentum dependent (TMD) parton distribution functions (PDF) is discussed. The relation between gauge invariance and the renormalization properties of the Wilson-line integrals is given particular attention. Using an anomalous-dimensions based analysis in the light-cone gauge, a generalized definition of the TMD PDFs is proposed, which employs a cusped Wilson line, and contains an intrinsic "Coulomb-like" phase.
We perform an analysis of transverse-momentum dependent parton distribution functions, making use of their renormalization properties in terms of their leading-order anomalous dimensions. We show that the appropriate Wilson line in the light-cone gauge, associated with such quantities, is a cusped one. To cancel the ensuing cusp anomalous dimension, we include in the definition of the transverse-momentum dependent parton distribution functions an additional soft counter term (gauge link) along that contour. We demonstrate that this is tantamount to an ``intrinsic (Coulomb) phase'', which accum…
The current status of the theoretical understanding of the transverse-momentum dependent parton densities (TMDs) is discussed. Special attention is payed to the difference between the operator definitions of TMDs proposed so far, the treatment of specific divergences, the geometry of the gauge links, and the role of the soft factors.
This talk provides a short overview of the phenomenology of transverse momentum depedent distribution and fragmention functions, focussing on the most recent phenomenological developments in the study of their scale evolution and energy depedence.
We consider the prompt photon production in pp collisions using, within the framework of perturbative QCD, a non-Gaussian distribution for the transverse momentum distribution of the partons inside the proton. Our description adopts the widely used in the literature factorization of the partonic momentum distribution into longitudinal and transverse components. It is argued that the non-Gaussian distribution of the intrinsic transverse momenta of the partons is dictated by the asymptotic freedom as well as the 3D confinement of the partons in the proton. To make this association more transpare…
If large-angle multigluon radiation gives significant contributions to parton showers in the LHC high-energy region, appropriate generalizations of parton branching methods are required for Monte Carlo simulations of exclusive high-multiplicity final states. We discuss the use in this context of transverse-momentum dependent kernels which factorize in the region of high energies. We give examples based on ep and p pbar multi-jet data, and point to possible developments for distributions associated with massive final states at the LHC.
The partonic transverse momentum, k_t distribution plays a crucial role in driving high-energy hadron interactions. If k_t is limited we have old fashioned Regge Theory. If k_t increases with energy the interaction may be described by perturbative QCD. We use BFKL diffusion in ln k_t, supplemented by a stronger absorption of low k_t partons, to estimate the growth of the mean transverse momenta <k_t> with energy. This growth reveals itself in the distribution of secondaries produced at the collider energies. We present a simple, BFKL-based, model to demonstrate the possible size of the effect.…
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 give a description of double parton scattering with measured transverse momenta in the final state, extending the formalism for factorisation and resummation developed by Collins, Soper and Sterman for the production of colourless particles. After a detailed analysis of their colour structure, we derive and solve evolution equations in rapidity and renormalisation scale for the relevant soft factors and double parton distributions. We show how in the perturbative regime, transverse momentum dependent double parton distributions can be expressed in terms of simpler nonperturbative quantities…
Recent experimental data on semi-inclusive deep-inelastic scattering from the HERMES collaboration allow us to discuss for the first time the flavor dependence of unpolarized transverse-momentum dependent distribution and fragmentation functions. We find convincing indications that favored fragmentation functions into pions have smaller average transverse momentum than unfavored functions and fragmentation functions into kaons. We find weaker indications of flavor dependence in the distribution functions.
We discuss the scale dependence of a skewed parton distribution of the pion obtained from a generalized light-cone wave function overlap formula. Using a simple ansatz for the transverse momentum dependence of the light-cone wave function and restricting ourselves to the case of a zero skewedness parameter, the skewed parton distribution can be expressed through an ordinary parton distribution multiplied by an exponential function. Matching the generalized and ordinary DGLAP evolution equations of the skewed and ordinary parton distributions, respectively, we derive a constraint for the scale …
Several methods to improve the parton-shower description of hard processes by an injection of matrix-element-based information have been presented over the years. In this article we study (re)weighting schemes for the first/hardest emission. One objective is to provide a consistent matching of the POWHEG next-to-leading order generator to the Pythia shower algorithms. Another is to correct the default behaviour of these showers at large transverse momenta, based on a comparison with real-emission matrix elements.
In this paper we study the matrix element of the transverse component of the bilocal vector current in the context of deep inelastic scattering. BJL limit of high energy amplitudes together with light-front current algebra imply the same parton interpretation for its matrix element as that of the plus component. On the other hand, the transverse component depends explicitly on the gluon field operator in QCD, appears as "twist three" and hence its matrix element has no manifest parton interpretation. In this paper we perform calculations in light-front time-ordering perturbative QCD for a dres…
Cross sections for physical processes that involve very different momentum scales in the same process will involve large logarithms of the ratio of the momentum scales when calculated in perturbation theory. One goal of calculations using parton showers is to sum these large logarithms. We ask whether this goal is achieved for the transverse momentum distribution of a Z-boson produced in hadron-hadron collisions when the shower is organized with higher virtuality parton splittings coming first, followed successively by lower virtuality parton splittings. We find that the virtuality ordered sho…
We present the construction of unintegrated double parton distribution functions which include dependence on transverse momenta of partons. We extend the formulation which was used to obtain the single unintegrated parton distributions from the standard, integrated parton distribution functions. Starting from the homogeneous part of the evolution equations for the integrated double parton distributions, we construct the unintegrated double parton distributions as the convolutions of the integrated double distributions and the splitting functions, multiplied by the Sudakov form factors. We show…
Different approaches to define transverse-momentum dependent parton distribution functions are considered from the point of view of their renormalization-group properties. The associated one-loop anomalous dimensions of these quantities are presented and compared to each other. We give arguments in favor of the "pure light-like" definition, and the use of the light-cone gauge.
In order to study twist-3 and transverse momentum dependent parton distributions, we use light-front time-ordered pQCD at order $α_s$ to calculate various distribution functions for a dressed quark target. This study enables us to investigate in detail the existing relations between twist-3 and transverse momentum dependent parton distributions. Our calculation shows explicitly that two versions of such relations, considered to be equivalent, occur in the literature which need to be distinguished. Moreover, we examine sum rules for higher twist distributions. While the Burkhardt-Cottingham sum…
The transverse momentum transfer correlation is introduced as a sensitive probe that can be used to discriminate between models for parton dynamics in low-x deep-inelastic scattering. Expectations for uncorrelated models and models with short-range or long-range correlations are discussed and confronted to results obtained from the LEPTO and ARIADNE Monte Carlo simulations programmes.
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…
Impact parameter dependent parton distributions are transversely distorted when one considers transversely polarized nucleons and/or quarks. This provides a physical mechanism for the T-odd Sivers effect in semi-inclusive deep-inelastic scattering. The transverse distortion can also be related to Ji's sum rule for the angular momentum carried by the quarks. The distortion of chirally odd impact parameter dependent parton distributions is related to chirally odd GPDs. This result is used to provide a decomposition of the quark angular momentum w.r.t. quarks of definite transversity. Chirally od…
