We discuss production of two pairs of $c \bar c$ in proton-proton collisions at the LHC. Both double-parton scattering (DPS) and single-parton scattering (SPS) contributions are included in the analysis. Each step of DPS is calculated within $k_t$-factorization approach. The conditions how to identify the DPS contribution are presented. The discussed mechanism leads to the production of pairs of mesons: each containing $c$ quarks or each containing $\bar c$ antiquarks. We discuss corresponding production rates and some differential distributions for $(D^0 D^0$ + $\bar D^0 \bar D^0)$ production. Within large theoretical uncertainties the predicted DPS cross section is fairly similar to the cross section measured recently by the LHCb collaboration. The best description is obtained with the Kimber-Martin-Ryskin (KMR) unintegrated gluon distribution. The contribution of SPS, calculated in the high-energy approximation, turned out to be rather small. Finally, we emphasize significant contribution of DPS mechanism to inclusive charmed meson spectra measured recently by ALICE, ATLAS and LHCb.
In the present paper we discuss how to maximize the double-parton scattering (DPS) contribution in four-jet production by selecting kinematical cuts. Here both single-parton and double-parton scattering effects are calculated in the $k_T$-factorization approach, following our recent developments of relevant methods and tools. Several differential distributions are shown and discussed in the context of future searches for DPS effects, such as rapidity of jets, rapidity distance, and azimuthal correlations between jets. The dependences of the relative DPS amount is studied as function of those o…
We perform a first study of 4-jet production in a complete high-energy factorization (HEF) framework. We include and discuss contributions from both single-parton scattering (SPS) and double-parton scattering (DPS). The calculations are performed for kinematical situations relevant for two experimental measurements (ATLAS and CMS) at the LHC. We compare our results to those reported by the ATLAS and CMS collaborations for different sets of kinematical cuts. The results of the HEF approach are compared with their counterparts for collinear factorization. For symmetric cuts the DPS HEF result is…
It is argued that while the scale dependence of the parton distributions in the fixed flavor factorization scheme is governed by three active flavors, the scale dependence of the running coupling should nevertheless be better governed by a variable number of active flavors.
We show that the momentum sum rule is a necessary condition for factorization of double parton distributions into a product of two single parton distributions for small values of the parton momentum fractions x and large enough values of the evolution scale Q. This is a somewhat surprising result since the momentum sum rule involves integration over all values of the momentum fraction. In essence, the momentum sum rule provides a proper relation between the double and single parton distributions, which is necessary for the small x factorization at large $Q$.
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 outline the basic properties of a pertubative QCD factorization formalism that maintains exact over-all kinematics in both the initial and final states. Such a treatment requires the use of non-perturbative factors that depend on all components of parton four-momentum. These objects are referred to as parton correlation functions. We describe the complications faced in defining parton correlation functions and discuss recent progress. Emphasis is placed on the need for precise operator definitions in a complete derivation of factorization.
I review the basics of the collinear factorization theorem applied primarily to deep inelastic scattering (DIS) involving forward parton distributions (PDFs) and the extensions of this theorem for exclusive processes probing non-forward parton distributions (GPDs), the generalized distribution amplitudes (GDAs) and the transition distribution amplitudes (TDAs). These QCD factorization theorem is an important tool in the description of hard processes in QCD. Whenever valid, it permits to represent the cross section or the scattering amplitude for such a process as a convolution in partonic mome…
We investigate parton distributions in the virtual photon target, both polarized and unpolarized, up to the next-leading order (NLO) in QCD. Parton distributions can be predicted completely up to NLO, but they are factorization-scheme-dependent. We analyze parton distributions in several factorization schemes and discuss their scheme dependence. Particular attentions are paid to the axial anomaly effect on the first moments of the polarized quark parton distributions, and also to the large-$x$ behaviors of polarized and unpolarized parton distributions.
It is shown that the difference between the c-quark proton structure functions calculated in the k_T-factorization approach using different unintegrated gluon distribution functions is the same order as the difference between results obtained in the parton model and in the k_T-factorization approach.
We calculate the three-parton twist-3 contribution to the $B\toπ$ transition form factors in the $k_T$ factorization theorem. Since different mesons are involved in the initial and final states, two(three)-parton-to-three(two)-parton amplitudes do not vanish. It is found that the dominant contribution arises from the diagrams with the additional valence gluon attaching to the leading-order hard gluon. Employing the three-parton meson distribution amplitudes from QCD sum rules, we show that this subleading piece amounts only up to few percents of the form factors at large recoil of the pion. Th…
Spin-dependent parton distributions in the polarized virtual photon are investigated in QCD up to the next-to-leading order (NLO). In the case $Λ^2 \ll P^2 \ll Q^2$, where $-Q^2$ ($-P^2$) is the mass squared of the probe (target) photon, parton distributions can be predicted completely up to the NLO, but they are factorization-scheme-dependent. Parton distributions are analyzed in four different factorization schemes and their scheme dependence are discussed. Particular attentions are paid to the axial anomaly effect on the first moments of quark parton distributions, and also to the large-$x$…
A factorization algorithm for a patron shower model based on the evolution of momentum distributions proposed in a previous work is studied. The scaling violation of initial state parton distributions is generated using parton showers to an accuracy of the next-to-leading logarithmic (NLL) order of quantum chromodynamics (QCD) using the information from only splitting functions and initial parton distributions at some fixed low energy. In the algorithm proposed in this paper, the total momentum of the initial state partons is conserved in any factorization scheme for mass singularities. As an …
We investigate spin-dependent parton distributions in the polarized virtual photon target in perturbative QCD up to the next-to-leading order (NLO). In the case $Λ^2 \ll P^2 \ll Q^2$, where $-Q^2$ ($-P^2$) is the mass squared of the probe (target) photon, parton distributions can be predicted completely up to NLO, but they are factorization-scheme-dependent.We analyze parton distributions in six different factorization schemes and discuss their scheme dependence. We study, in particular, the QCD and QED axial anomaly effects on the first moments of parton distributions to see the interplay bet…
I review the treatment of high-energy QCD in Minkowski space, with an emphasis on factorization theorems as extensions of the operator product expansion. I discuss how the factorization properties of high-energy cross sections and amplitudes lead to evolution equations that resum large logarithms for two-scale problems.
We report on a first study of 4-jet production in a high energy factorization (HEF) framework \cite{Kutak:2016mik,Kutak:2016ukc}. We include and discuss contributions from both single-parton scattering (SPS) and double-parton scattering (DPS) and compare to the measured data. The DPS HEF result is considerably smaller than the one obtained with collinear factorization. The mechanism leading to this difference is of kinematical nature. In contrast to the collinear approach, the HEF approach nicely describes the distribution of the $ΔS$ variable, which involves all four jets and their angular co…
We present a preliminary study of both Single and Double Parton Scattering contributions to the inclusive 4-jet production in the kt-factorization framework at Leading Order and $E_{CM} = 7$ TeV. We compare our results to collinear results in the literature and to the ATLAS and CMS data at 8 and 7 TeV respectively. We also discuss the importance of double parton scattering for relatively soft cuts on the jet transverse momenta and find out that symmetric cuts do not quite suit well to kt-factorization predictions, because of a kinematic effect suppressing the double parton scattering contribut…
We outline arguments for the cancellation of soft singularities in transition probabilities and parton distributions with incoming partons and Wilson lines, and with observed jets or heavy colored particles in the final states. We show the cancellation of Glauber gluons and divergent phases, relating finite remainders to corrections that arise from restrictions on final states in factorized cross sections.
The transverse component of the axial-vector correlation function of quark fields is a natural starting object for lattice calculations of twist-3 nucleon parton distribution functions. In this work we derive the corresponding factorization expression in terms of twist-2 and twist-3 collinear distributions to one-loop accuracy. The results are presented both in position space, as the factorization theorem for Ioffe-time distributions, and in momentum space, for the axial-vector quasi- and pseudodistributions.
The p_T-spectra for direct J/psi and psi' in hadroproduction at Tevatron energy have been calculated based on NRQCD formalism and fragmentation approximation in the collinear parton model and k_T-factorization approach. We have described the CDF data and obtained a good agreement between the predictions obtained in the parton model and k_T-factorization approach. We performed the calculations using the relevant leading order in alpha_s hard amplitudes and the equal values of the color-octet long-distance matrix elements for the both models.
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…
We show that both the k_T- and collinear factorization for the DIS structure functions can be obtained by consecutive reductions of the Compton scattering amplitude. Each of these reductions is an approximation valid under certain assumptions. In particular, the transitions to the k_T- factorization is possible when the virtualities of the partons connecting the perturbative and non-perturbative blobs are space-like. Then, if the parton distribution has a sharp maximum in k_T, the k_T factorization can be reduced to the collinear factorization.
We discuss the single diffractive production of $c \bar c$ pairs and charmed mesons at the LHC. For a first time we propose a $k_t$-factorization approach to the diffractive processes. The transverse momentum dependent diffractive parton distributions are obtained from standard (collinear) diffractive parton distributions used in the literature. In this calculation the transverse momentum of the pomeron is neglected with respect to transverse momentum of partons entering the hard process. We also perform the first evaluation of the cross sections at the LHC using the diffractive transverse mom…
We discuss charm production at LHC. The production of single $c \bar c$ pairs is calculated in the $k_t$-factorization approach. We use several unintegrated gluon distributions from the literature. Differential distributions for several charmed mesons are presented and compared to recent results of the ALICE and LHCb collaborations. Some missing strength can be observed. Furthermore we discuss production of two $c \bar c$ pairs within a simple formalism of double-parton scattering (DPS). Surprisingly large cross sections, comparable to single-parton scattering (SPS) contribution to $c \bar c$ …
We compare the numerical predictions for heavy quark production in high energy hadron collisions of the conventional QCD parton model and the $k_T$-factorization approach (semihard theory). The total production cross sections, one-particle rapidity and $p_T$ distributions as well as two-particle correlations are considered. The difference in the predictions of the two approaches is not very large, while the shapes of the distributions are slightly different.
We present first results from a new parton shower event generator, Deductor. Anticipating a need for an improved treatment of parton color and spin, the structure of the generator is based on the quantum density matrix in color and spin space. So far, Deductor implements only a standard spin-averaged treatment of spin in parton splittings. Although Deductor implements an improved treatment of color, in this paper we present results in the standard leading color approximation so that we can compare to the generator Pythia. The algorithms used incorporate a virtuality based shower ordering param…
In the present paper, we check and study the validity of the \textsc{KaTie} parton level event generator, by calculating the inclusive electron-proton (ep) dijet and the Proton-proton (p-p) Drell-Yan electron-pair productions differential cross sections in the $k_t$- and collinear factorization frameworks. The Martin-Ryskin-Watt (MRW) unintegrated parton distribution functions (UPDFs) are used as the input UPDFs. The results are compared with those of ZEUS ep inclusive dijet and ATLAS p-p Drell-Yan electron-pair productions, experimental data. The \textsc{KaTie} parton level event generator ca…
The factorization of short-distance partonic cross-sections from universal long-distance kinematic distributions is fundamental to phenomenology at hadron colliders. It has been predicted however that observables sensitive to momenta transverse to the direction of an energetic parton cannot be factorized in the usual way, even at high energies. It should be possible to study this factorization breaking using Z+jet production in high-energy proton-proton collisions by studying azimuthal correlations between a Z boson and associated charged hadrons. A plan to perform this measurement with data c…
We prove $k_T$ factorization theorem in perturbative QCD (PQCD) for exclusive processes by considering $πγ^*\to γ(π)$ and $B\toγ(π) l\barν$. The relevant form factors are expressed as the convolution of hard amplitudes with two-parton meson wave functions in the impact parameter $b$ space, $b$ being conjugate to the parton transverse momenta $k_T$. The point is that on-shell valence partons carry longitudinal momenta initially, and acquire $k_T$ through collinear gluon exchanges. The $b$-dependent two-parton wave functions with an appropriate path for the Wilson links are gauge-invariant. The …
We prove collinear factorization theorem for the process $πγ^*\toπ$ at the twist-3 level in the covariant gauge by means of the Ward identity, concentrating on the two-parton case. It is shown that soft divergences cancel and collinear divergences are grouped into the pseudo-scalar and pseudo-tensor two-parton twist-3 pion distribution amplitudes. The delicate summation of a complete set of diagrams for achieving factorization in momentum, spin, and color spaces is emphasized. The proof is then extended to the exclusive semileptonic decay $B\toπl\barν$, assuming the hard scale to be of $O(\sqr…
We present first results for the $2 \to 4$ single-parton scattering $g g \to c \bar c c \bar c$ subprocess for the first time fully within the $k_t$-factorization approach. In this calculation we have used the Kimber-Martin-Ryskin unintegrated gluon distribution which effectively includes some class of higher-order gluon emissions, and an off-shell matrix element squared calculated using recently developed techniques. The results are compared with our earlier result obtained within the collinear-factorization approach. Only slightly larger cross sections are obtained than in the case of the co…
Single transverse-spin asymmetries have been studied intensively both in experiment and theory. Theoretically, two factorization approaches have been proposed. One is by using transverse-momentum-dependent factorization and the asymmetry comes from the so called Sivers function. Another is by using collinear factorization where the nonperturbative effect is parameterized by a twist-3 hadronic matrix element. However, the factorized formulas for the asymmetries in the two approaches are derived at hadron level formally by diagram expansion, where one works with various parton density matrices o…
We study chiral-odd quark-antiquark correlation functions suitable for lattice calculations of twist-three nucleon parton distribution functions $h_L(x)$ and $e(x)$, and also the twist-two transversity distribution $δq(x)$. The corresponding factorized expressions are derived in terms of the twist-two and twist-three collinear distributions to one-loop accuracy. The results are presented both in position space, as the factorization theorem for Ioffe-time distributions, and in momentum space, for quasi- and pseudo-distributions. We demonstrate that the twist-two part of the $h_L$ quasi(pseudo)-…
In the present work, it is intended to calculate the unintegrated parton distribution functions (UPDFs) of the $^6{Li}$ nucleus, which depend not only on the longitudinal momentum fraction $x$ (the Bjorken variable) and the factorization scale $μ$ of partons, but also on their transverse momentum ($k_t$). Therefore, the KMR and MRW procedures are applied to generate the $k_t$-dependent parton distributions from the familiar integrated parton densities (PDFs), which were determined in our recent related work, using the constituent quark exchange model (CQEM) for the $^6Li$ nucleus. Then, the re…
In a large-momentum nucleon state, the matrix element of a gauge-invariant Euclidean Wilson line operator accessible from lattice QCD can be related to the standard light-cone parton distribution function through the large-momentum effective theory (LaMET) expansion. This relation is given by a factorization theorem with a non-trivial matching coefficient. Using the operator product expansion we prove the large-momentum factorization of the quasi-parton distribution function in LaMET, and show that the more recently discussed Ioffe-time distribution approach also obeys an equivalent factorizat…
In this paper, the twist-3 two parton corrections in charmless $B\to PP$ decays are shown to be factorizable under the QCD factorization approach. The factorizability of the twist-3 two parton corrections is constructed on the following findings. Under the energetic meson limit, the pseudoscalar distribution amplitude for a light pseudoscalar meson is allowed to be non-constant by the equations of motion for the quark. The non-constant pseudoscalar distribution amplitude is then used to regularize the end-point divergences in the hard spectator corrections at twist-3 order. By retaining the mo…
I report on progress in the calculation of one-loop amplitudes with a space-like gluon. Amplitudes with space-like partons are relevant for factorization procedure s that require initial-state partons with non-vanishing transverse momentum components.
The unintegrated parton distribution functions (UPDFs) of $^3H$, $^3He$ and $^4He$ nuclei are generated to calculate their structure functions (SFs) in the $k_t$-factorization approach. The Kimber-Martin-Ryskin (KMR) formalisn is applied to evaluate the double-scale UPDFs of these nuclei from their single-scale parton distribution functions (PDFs), which can be obtained from the constituent quark exchange model (CQEM). Afterwards, these SFs are used to calculate the European Muon Collaboration (EMC) ratios of these nuclei. The resulting EMC ratios are then compared with the available experimen…
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…
Global perturbative QCD analyses, based on large data sets from electron-proton and hadron collider experiments, provide tight constraints on the parton distribution function (PDF) in the proton. The extension of these analyses to nuclear parton distributions (nPDF) has attracted much interest in recent years. nPDFs are needed as benchmarks for the characterization of hot QCD matter in nucleus-nucleus collisions, and attract further interest since they may show novel signatures of non- linear density-dependent QCD evolution. However, it is not known from first principles whether the factorizat…
Motivated by the need to correct the potentially large kinematic errors in approximations used in the standard formulation of perturbative QCD, we reformulate deeply inelastic lepton-proton scattering in terms of gauge invariant, universal parton correlation functions which depend on all components of parton four-momentum. Currently, different hard QCD processes are described by very different perturbative formalisms, each relying on its own set of kinematical approximations. In this paper we show how to set up formalism that avoids approximations on final-state momenta, and thus has a very ge…
The end-point contributions in the quark longitudinal momentum fraction of the virtual photon ($γ^*$) to vector meson ($V$) impact factor to the diffractive electroproduction amplitude can be factorized in terms of a generalized parton evolution of the target parton distribution. The result is used to model the helicity amplitudes $γ^* p \to V p$ in terms of small $x$ generalized parton distributions.
Factorization scheme analysis of $F_2^γ(x,Q^2)$ in the next-to-leading order QCD is revisited. It is emphasized that the presence of the inhomogeneous term in the evolution equations for quark distribution functions of the photon implies subtle but important difference in the way factorization mechanism works in photon-hadron and photon-photon collisions as compared to the hadronic ones. It is argued that none of the existing NLO analyses of $F_2^γ(x,Q^2)$ takes this difference properly into account. The source of the ensuing incompleteness is traced back to the misinterpretation of the behavi…
We perform a NLO QCD analysis of single-diffractive dijet production in proton-antiproton collisions. By comparing the ratio of single- and non-diffractive cross sections to data from the Tevatron, the rapidity-gap survival probability is determined as a function of the momentum fraction of the parton in the antiproton. Assuming Regge factorization, this probability can be interpreted as a suppression factor for the diffractive structure function measured in deep-inelastic scattering at HERA. In contrast to the observations for photoproduction, the suppression factor in proton-antiproton colli…
We discuss production of two pairs of $c \bar c$ in proton-proton collisions at the LHC. Both double-parton scattering (DPS) and single-parton scattering (SPS) contributions are included in the analysis. Each step of DPS is calculated within $k_t$-factorization approach, i.e. effectively including next-to-leading order corrections. The conditions how to identify the DPS contribution are presented. The discussed mechanism unavoidably leads to the production of pairs of mesons: $D_i D_j$ (each containing $c$ quarks) or $\bar D_i \bar D_j$ (each containing $\bar c$ antiquarks). We calculate corre…
We study large-$p_T$ three-photon production at the LHC at the center-of-mass energy $\sqrt{s}=8$ TeV. We use the LO approximation of the parton Reggeization approach consistently merged with the real NLO corrections. For numerical calculations use the parton-level generator KaTie and modified KMR-type unintegrated parton distribution functions. We find good agreement between our predictions and data with the same accuracy as in the NNLO calculations based on the collinear parton model of QCD. At higher energies ($\sqrt{s}=13$ and 27 TeV) parton Reggeization approach predicts larger cross sect…
In this work we consider how a parton distribution function, with an explicit transverse momentum dependence can be properly defined in a regularization-scheme independent manner. We argue that by considering a factorized form of the transverse momentum dependent spectrum for the production of a heavy lepton pair in Drell-Yan reaction, one should first split the relevant soft function into two boost invariant contributions. When those soft contributions are added to the pure collinear contributions, well-defined hadronic matrix elements emerge, i.e., the transverse momentum dependent distribut…
I show that the forms of the Collins-Soper-Sterman and renormalization-group equations for the evolution of transverse-momentum-dependent (TMD) parton densities in QCD follow from the structure of TMD factorization. A derivation does not need to directly use detailed properties of the operator definition of the TMD parton densities.
In the present work, it is intended to calculate and study the single and double differential cross-sections of the prompt single-photon production as a function of produced single-photon transverse momentum and rapidity, in the high-energy P-P(P) colliders, such as LHC and TEVATRON. The differential cross-sections of prompt single-photon production are calculated in the $k_t$-factorization frameworks using various angular ordering unintegrated parton distribution functions (UPDF), namely the Kimber et al. and Martin et al. procedures. These scheme-dependent UPDF are generated in the leading a…
We investigate the inclusive hadroproduction of a heavy quarkonium ($J/ψ$ or $Υ$), in association with a light-flavored jet, as a testing ground for the semi-hard regime of QCD. Our theoretical setup is the hybrid high-energy and collinear factorization, where the standard collinear approach is supplemented by the $t$-channel resummation of leading and next-to-leading energy logarithms à la BFKL. We present predictions for rapidity and azimuthal-angle differential distributions, hunting for stabilizing effects of the high-energy series under higher-order corrections. Our reaction represents an…
We discuss production of $c \bar c$-pairs within $k_T$-factorization approach (off-shell initial partons) with unintegrated parton distribution functions (uPDFs). We present a consistent prescription which merges the standard leading-order (LO) $k_T$-factorization calculations for this process with tree-level next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) matrix elements. For the first time we include in this framework 2 $\to$ 3 and 2 $\to$ 4 processes with extra partonic emissions for single particle distributions as well as for correlation observables. The use of the KM…
We show that the correct way to extract parton distribution functions from the reduced Ioffe-time distribution (RITD), a ratio of the Ioffe-time distribution for a moving hadron and a hadron at rest, is through a factorization formula. This factorization exists because, at small distances, forming the ratio does not change the infrared behavior of the numerator, which is factorizable. We illustrate the effect of such a factorization by applying it to results in the literature.
Recently, the ATLAS data of isolated three-photon production showed that the next-to-leading order (NLO) collinear factorization is not enough to describe experimental data. Therefore, one needs to calculate the cross section beyond the NLO, and as showed later, these data can be well described by the NNLO calculation within the collinear factorization framework. However, it is shown that the $k_t$-factorization can be quite successful in describing exclusive and high energy collision processes, henceforth we decided to calculate isolated three-photon production within this framework. In this …
We present a short overview on transverse momentum dependent parton distribution and fragmentation functions, giving their partonic interpretation and ways to access them. We then discuss the issue of their universality and its connection to factorization in perturbative QCD.
We compare recent, seemingly different, approaches to TMD-factorization (due to Echevarria, Idilbi, and Scimemi and to Collins), and show that they are the same, apart from an apparent difference in their definition of the \MSbar{} renormalization scheme.
In the field theoretical description of hadronic scattering processes, single transverse-spin asymmetries arise due to gluon initial and final state interactions. These interactions lead to process dependent Wilson lines in the operator definitions of transverse momentum dependent parton distribution functions. In particular for hadron-hadron scattering processes with hadronic final states this has important ramifications for possible factorization formulas in terms of (non)universal TMD parton distribution functions. In this paper we will systematically separate the universality-breaking part…
