Transverse momentum dependent parton distribution functions (TMDPDFs) provide a unique probe of the three-dimensional spin structure of hadrons. We construct spin-dependent quasi-TMDPDFs that are amenable to lattice QCD calculations and that can be used to determine spin-dependent TMDPDFs. We calculate the short-distance coefficients connecting spin-dependent TMDPDFs and quasi-TMDPDFs at one-loop order. We find that the helicity and transversity distributions have the same coefficient as the unpolarized TMDPDF. We also argue that the same is true for pretzelosity and that this spin universality of the matching will hold to all orders in $α_s$. Thus, it is possible to calculate ratios of these distributions as a function of longitudinal momentum and transverse position utilizing simpler Wilson line paths than have previously been considered.
Transverse-momentum-dependent parton distributions (TMDs) can be calculated from first principles by computing a related set of Euclidean lattice observables and connecting them via a factorization formula. This work focuses on the leading-power factorization formula connecting the lattice quasi-TMD and continuum Collins TMD for gluons. We calculate the one-loop gluon matching coefficient, which is known to be independent of spin and exhibits no mixing with quarks. We demonstrate that this coefficient satisfies Casimir scaling with respect to the quark matching coefficient at one-loop order. O…
Transverse-momentum-dependent parton distribution functions (TMDs) can be studied from first principles by a perturbative matching onto lattice-calculable quantities: so-called lattice TMDs, which are a class of equal-time correlators that includes quasi-TMDs and TMDs in the Lorentz-invariant approach. We introduce a general correlator that includes as special cases these two Lattice TMDs and continuum TMDs, like the Collins scheme. Then, to facilitate the derivation of a factorization relation between lattice and continuum TMDs, we construct a new scheme, the Large Rapidity (LR) scheme, inter…
We set up a formalism for calculating transverse-momentum-dependent parton distribution functions (TMDs) using the tools of saturation physics. By generalizing the quasi-classical Glauber-Gribov-Mueller/McLerran-Venugopalan approximation to allow for the possibility of spin-orbit coupling, we show how any TMD can be calculated in the saturation framework. This can also be applied to the TMDs of a proton by modeling it as a large "nucleus." To illustrate our technique, we calculate the quark TMDs of an unpolarized nucleus at large-x: the unpolarized quark distribution and the quark Boer-Mulders…
Using our formalism of parton virtuality distribution functions (VDFs) we establish a connection between the transverse momentum dependent distributions (TMDs) ${\cal F} (x, k_\perp^2)$ and quasi-distributions $Q(y,P_z)$ introduced recently by X. Ji for lattice QCD extraction of parton distributions $f(x)$. We build models for PQDs from the VDF-based models for soft TMDs, and analyze the $P_z$ dependence of the resulting PQDs. We observe a strong nonperturbative evolution of PQDs for small and moderately large values of $P_z$ reflecting the transverse momentum dependence of TMDs. Thus, the stu…
Matching relations for gluon TMDs up to one-loop accuracy
Università degli Studi di Pavia（帕维亚大学）；Istituto Nazionale di Fisica Nucleare, Sezione di Pavia（意大利国家核物理研究所帕维亚分部）；Università degli Studi di Cagliari（卡利亚里大学）；Istituto Nazionale di Fisica Nucleare, Sezione di Cagliari（意大利国家核物理研究所卡利亚里分部）
Generalized transverse momentum dependent parton distributions (GTMDs), as mother funtions of transverse momentum dependent parton distributions (TMDs) and generalized parton distributions (GPDs), encode the most general parton structure of hadrons. We calculate four twist-two time reversal odd GTMDs of pion in a scalar spectator model. We study the dependence of GTMDs on the longitudinal momentum fraction $x$ carried by the active quark and the transverse momentum $|\vec k_T|$ for different values of skewness $ξ$ defined as the longitudinal momentum transferred to the proton as well as the to…
We study the impact of non-zero (and apparently large) value of the nucleon mass $M$ on the shape of parton quasi-distributions $Q(y,p_3)$, in particular on its change with the change of the nucleon momentum $p_3$. We observe that the usual target-mass corrections induced by the $M$-dependence of the twist-2 operators are rather small. Moreover, we show that within the framework based on parametrizations by transverse momentum dependent distribution functions (TMDs) these corrections are canceled by higher-twist contributions. We identify a novel source of kinematic target-mass dependence of T…
We study the power corrections to Transverse Momentum Distributions (TMDs) by analyzing renormalon divergences of the perturbative series. The renomalon divergences arise independently in two constituents of TMDs: the rapidity evolution kernel and the small-b matching coefficient. The renormalon contributions have a non-trivial dependence on the Bjorken variable and the transverse distance. We discuss the consistency requirements for power corrections for TMDs and suggest inputs for the TMD phenomenology in accordance with this study. Both unpolarized quark TMD parton distribution function and…
We consider for the first time the QED corrections to the evolution of (un)polarized quark and gluon transverse-momentum-dependent distribution and fragmentation functions (TMDs in general). By extending their operator definition to QCD$\times$QED, we provide the mixed new anomalous dimensions up to ${\cal O}(α_sα)$ and the pure QED ones up to ${\cal O}(α^2)$. These new corrections are universal for all TMDs up to the flavor of the considered parton, i.e., the full flavor universality of TMD evolution found in pure QCD is broken in QCD$\times$QED by the presence of the electric charge. In addi…
The transverse momentum dependent parton distribution/fragmentation functions (TMDs) are essential in the factorization of a number of processes like Drell-Yan scattering, vector boson production, semi-inclusive deep inelastic scattering, etc. We provide a comprehensive study of unpolarized TMDs at next-to-next-to-leading order, which includes an explicit calculation of these TMDs and an extraction of their matching coefficients onto their integrated analogues, for all flavor combinations. The obtained matching coefficients are important for any kind of phenomenology involving TMDs. In the pre…
The description of the Drell-Yan (DY) transverse momentum spectrum requires matching of fixed order QCD calculations with soft gluon resummation up to all orders in the QCD coupling. It has been noticed in the literature that a consistent description of DY data in a wide kinematic regime from fixed-target to LHC energies is problematic. In this talk the predictions for transverse momentum spectrum of DY data coming from experiments in very different kinematic ranges (NuSea, R209, Phenix, LHC 8 TeV and 13 TeV center-of-mass energies $\sqrt{s}$) are calculated by applying transverse momentum dep…
We investigate the transverse momentum dependent parton distributions (TMDs) in the quasi-parton-distribution framework. The long standing hurdle of the so-called pinch pole singularity from the space-like gauge links in the TMD definitions can be resolved by the finite length of the gauge link along the hadron moving direction. In addition, with the soft factor subtraction, the quasi-TMD is free of linear divergence. We further demonstrate that the energy evolution equation of the quasi-TMD a.k.a. the Collins-Soper evolution, only depends on the hadron momentum. This leads to a clear matching…
Transverse momentum dependent distribution and fragmentation functions of unpolarized quarks inside unpolarized protons are extracted, for the first time, through a simultaneous analysis of semi-inclusive deep-inelastic scattering, Drell-Yan and $Z$ boson hadroproduction processes. This study is performed at leading order in perturbative QCD, with energy scale evolution at the next-to-leading logarithmic accuracy. Moreover, some specific choices are made to deal with low scale evolution around 1 GeV$^2$. Since only data in the low transverse momentum region are considered, no matching to fixed…
We study the transverse momentum dependent (TMD) parton distributions in the newly proposed quasi-parton distribution function framework in Euclidean space. A soft factor subtraction is found to be essential to make the TMDs calculable on lattice. We show that the quasi-TMDs with the associated soft factor subtraction can be applied in hard QCD scattering processes such as Drell-Yan lepton pair production in hadronic collisions. This allows future lattice calculations to provide information on the non-perturbative inputs and energy evolutions for the TMDs. Extension to the generalized parton d…
The parton quasi-distribution functions (QDFs) of Ji have been found by Radyushkin to be directly related to the transverse momentum distributions (TMDs), to the pseudo-distributions, and to the Ioffe-time distributions (ITDs). This makes the QDF results at finite longitudinal momentum of the hadron interesting in their own right. Moreover, the QDF-TMD relation provides a gateway to the pertinent QCD evolution, with respect to the resolution scale Q, for the QDFs. Using the Kwieciński evolution equations and well established parameterizations at a low initial scale, we analyze the QCD evolutio…
The theoretical description of Drell-Yan (DY) transverse momentum spectra over wide kinematic regions in energy, mass and transverse momentum requires not only fixed-order perturbative QCD calculations but also soft-gluon QCD resummations to all orders of perturbation theory. The latter are traditionally accomplished either by Parton Showers (PS) with Monte Carlo event generators or by (different versions of) analytical procedures. In this work we focus on issues involved in the matching of the fixed-order calculation and resummation, especially in the moderate to low mass and $p_{\bot}$ regio…
Matching of the quasi parton distribution functions between continuum and lattice is addressed using lattice perturbation theory specifically with Wilson-type fermions. The matching is done for nonlocal quark bilinear operators with a straight Wilson line in a spatial direction. We also investigate operator mixing in the renormalization and possible O(a) operators for the nonlocal operators based on a symmetry argument on lattice.
In recent years, the quasi parton distribution has been introduced for extracting the parton distribution functions from lattice QCD simulations. The quasi and standard distribution share the same perturbative collinear singularity and the renormalized quasi distribution can be factorized into the standard distribution with a perturbative matching factor. The quasi parton distribution is known to have power-law UV divergences, which do not exist in the standard distribution. We discuss in this talk the nonperturbative renormalization scheme for the power divergence. We also demonstrate the per…
We present an extraction of unpolarized partonic transverse momentum distributions (TMDs) from a simultaneous fit of available data measured in semi-inclusive deep-inelastic scattering, Drell-Yan and Z-boson production. To connect data at different scales, we use TMD evolution at next-to-leading logarithmic accuracy. The analysis is restricted to the low-transverse-momentum region, with no matching to fixed-order calculations at high transverse momentum. We introduce specific choices to deal with TMD evolution at low scales, of the order of 1 GeV$^2$. This could be considered as a first attemp…
The quasi parton distribution is a spatial correlation of quarks or gluons along the $z$ direction in a moving nucleon which enables direct lattice calculations of parton distribution functions. It can be defined with a nonperturbative renormalization in a regularization independent momentum subtraction scheme (RI/MOM), which can then be perturbatively related to the collinear parton distribution in the $\overline{\text{MS}}$ scheme. Here we carry out a direct matching from the RI/MOM scheme for the quasi-PDF to the $\overline{\text{MS}}$ PDF, determining the non-singlet quark matching coeffic…
The $x$-dependence of light-cone distribution amplitude (LCDA) can be directly calculated from a quasi distribution amplitude (DA) in lattice QCD within the framework of large-momentum effective theory (LaMET). In this paper, we study the one-loop renormalization of the quasi-DA in the regularization-independent momentum subtraction (RI/MOM) scheme. The renormalization factor for the quasi parton distribution function can be used to renormalize the quasi-DA provided that they are implemented on lattice and in perturbation theory in the same manner. We derive the one-loop matching coefficient t…
We complete the procedure of extracting parton distribution functions (PDFs) using large momentum effective theory (LaMET) at leading power accuracy in the hadron momentum. We derive a general factorization formula for the quasi PDFs in the presence of mixing, and give the corresponding hard matching kernel at $\mathcal O(α_s)$, both for the unpolarized and for the polarized quark and gluon quasi-PDFs. Our calculation is performed in a regularization-independent momentum subtraction scheme. The results allow us to match the nonperturbatively renormalized quasi-PDFs to normal PDFs in the presen…
The operator definition of generalised transverse-momentum-dependent (GTMD) distributions is exploited to compute for the first time the full set of one-loop corrections to the off-forward matching functions. These functions allow one to obtain GTMDs in the perturbative regime in terms of generalised parton distributions (GPDs). In the unpolarised case, non-perturbative corrections can be incorporated using recent determinations of transverse-momentum-dependent (TMD) distributions. Evolution effects for GTMDs closely follow those for TMDs and can thus be easily accounted for up to next-to-next…
A completely new strategy to calculate parton distribution functions on the lattice has recently been proposed. In this method, lattice calculable observables, called quasi distributions, are related to normal distributions. The quasi distributions are known to contain power-law UV divergences arise from a Wilson line in the non-local operator, while the normal distributions only have logatithmic UV divergences. We propose possible method to subtract the power divegence to make the matching of the quasi with the normal distributions well-defined. We also demonstrate the matching of the quasi q…
We investigate the non-perturbative contribution of instantons to current matching kernels used in the context of the large momentum effective theory (LaMET). We derive explicitly these contributions using first principle semi-classical calculus for the unpolarized and polarized quark parton distributions and the matching kernel, and show that they are part of a trans-series expansion. These contributions are substantial at current lattice matching momenta.
Within the framework of large momentum effective theory (LaMET), genenaralized parton distributions (GPDs) can be extracted from lattice calculations of quasi-GPDs through a perturbative matching relation, up to power corrections that are suppressed by the hadron momentum. In this paper, we focus on isovector quark GPDs, including the unpolarized, longitudinally and transversely polarized cases, and present the one-loop matching that connects the quasi-GPDs renormalized in a regularization-independent momentum subtraction (RI/MOM) scheme to the GPDs in MS scheme. We find that the matching coef…
Using soft collinear effective field theory, we derive the factorization theorem for the quasi-transverse-momentum-dependent (quasi-TMD) operator. We check the factorization theorem at one-loop level and compute the corresponding coefficient function and anomalous dimensions. The factorized expression is built from the physical TMD distribution, and a nonperturbative lattice related factor. We demonstrate that lattice related functions cancel in appropriately constructed ratios. These ratios could be used to explore various properties of TMD distributions, for instance, the nonperturbative evo…
We discuss the structure of the parton quasi-distributions (quasi-PDFs) $Q(y, P_3)$ outside the "canonical" $-1 \leq y \leq 1$ support region of the usual parton distribution functions (PDFs). Writing the $y^n$ moments of $Q(y, P_3)$ in terms of the combined $x^{n-2l} k_\perp^{2l}$-moments of the transverse momentum distribution (TMD) ${\cal F} (x,k_\perp^2)$, we establish a connection between the large-$|y|$ behavior of $Q(y,P_3)$ and large-$k_\perp^2$ behavior of ${\cal F} (x,k_\perp^2)$. In particular, we show that the $1/k_\perp^2$ hard tail of TMDs in QCD results in a slowly decreasing $\…
The factorization theorems for transverse momentum distributions of dilepton/boson production, recently formulated by Collins and Echevarria-Idilbi-Scimemi in terms of well-defined transverse momentum dependent distributions (TMDs), allows for a systematic and quantitative analysis of non-perturbative QCD effects of the cross sections involving these quantities. In this paper we perform a global fit using all current available data for Drell-Yan and $Z$-boson production at hadron colliders within this framework. The perturbative calculable pieces of our estimates are included using a complete …
Within the framework of large momentum effective theory (LaMET), the light-cone distribution amplitude of $B$-meson in heavy-quark effective theory (HQET) can be extracted from lattice calculations of quasidistribution amplitude through hard-collinear factorization formula. This quasiquantity can be renormalized in a regularization-independent momentum subtraction scheme (RI/MOM). In this work, we derive the matching coefficient which connects the renormalized quasiditribution amplitude in the RI/MOM scheme and standard LCDA in the $\overline{\textrm{MS}}$ scheme at one-loop accuracy. Our nume…
The perturbative procedure of matching was proposed to connect parton quasi-distributions that are calculable in lattice QCD to the corresponding light-cone distributions which enter physical processes. Such a matching procedure has so far been limited to the twist-2 distributions. Recently, we addressed the matching for the twist-3 PDF $g_T(x)$. In this work, we extend our perturbative calculations to the remaining twist-3 PDFs, $e(x)$ and $h_{L}(x)$. In particular, we discuss the non-trivialities involved in the calculation of the singular zero-mode contributions for the quasi-PDFs.
We discuss the physical nature of quasi-PDFs, especially the reasons for the strong nonperturbative evolution pattern which they reveal in actual lattice gauge calculations. We argue that quasi-PDFs may be treated as hybrids of PDFs and the rest-frame momentum distributions of partons. The latter is also responsible for the transverse momentum dependence of TMDs. The resulting convolution structure of quasi-PDFs necessitates using large probing momenta $p_3 \gtrsim 3$ GeV to get reasonably close to the PDF limit. To deconvolute the rest-frame distribution effects, we propose to use a method ba…
We propose a revised definition of quasi-distributions within the framework of large-momentum effective theory (LaMET) that improves convergence towards the large-momentum limit. Since the definition of quasi-distributions is not unique, each choice goes along with a specific matching function, we can use this freedom to optimize convergence towards the large-momentum limit. As an illustration, we study quasi-distributions with a Gaussian weighting factor that naturally suppresses long-range correlations, which are plagued by artifacts. This choice has the advantage that the matching functions…
We present the one-loop matching condition for the unpolarized and polarized generalized quark distributions in the non-singlet case. The matching condition links the quasi distributions defined in terms of spacelike correlators at finite nucleon momentum to the light-cone distributions, and is useful for extracting the latter from the former in a lattice QCD calculation. Our results show that at one-loop and leading power accuracy the matching for the light-cone generalized quark distribution $H$ ($\tilde H$) is non-trivial, whereas no matching is required for $E$ ($\tilde E$). Therefore, $E$…
We investigate the transport properties of a kinetic theory model that is tuned to describe the thermodynamic properties of QCD at zero chemical potential using a new formulation of the relaxation time approximation. In contrast to previous approaches, the latter is constructed to preserve the fundamental properties of the collision term of the Boltzmann equation for any energy-dependence of the relaxation time. A novel choice of matching conditions is implemented to ensure that the background mean-field depends only on the temperature even when the system is out of equilibrium. We provide a c…
The study of a quasi back-to-back isolated pair made of a heavy quarkonium, such as a J/psi or a Upsilon, and a photon produced in proton-proton collisions at the LHC, is probably the optimal way to get right away a first experimental determination of two gluon transverse-momentum-dependent distributions (TMDs) in an unpolarized proton, f1^g and h1^perp,g the latter giving the distribution of linearly polarized gluons. To substantiate this, we calculate the transverse-momentum-dependent effects that arise in the process under study and discuss the feasibility of their measurements.
We calculate the unpolarized transverse momentum dependent fragmentation function (TMDFF) at next-to-next-to-leading order (NNLO), evaluating separately TMD soft factor and TMD collinear correlator. For the first time the cancellation of spurious rapidity divergences in a properly defined individual TMD beyond the first non-trivial order is shown. This represents a strong check of the given TMD definition. We extract the matching coefficient necessary to perform the transverse momentum resummation at next-to-next-to-next-to-leading-logarithmic accuracy. The universal character of the soft func…
We derive one-loop matching condition for non-singlet quark distributions in transverse-momentum cut-off scheme, including unpolarized, helicity and transversity distributions. The matching is between the quasi-distribution defined by static correlation at finite nucleon momentum and the light-cone distribution measurable in experiments. The result is useful for extracting the latter from the former in a lattice QCD calculation.
Transverse Momentum Dependent (TMD) parton distributions obtained from the Parton Branching (PB) method are combined with next-to-leading-order (NLO) calculations of Drell-Yan (DY) production. We apply the MCatNLO method for the hard process calculation and matching with the PB TMDs. We compute predictions for the transverse momentum, rapidity and $ϕ^*$ spectra of Z-bosons. We find that the theoretical uncertainties of the predictions are dominated by the renormalization and factorization scale dependence, while the impact of TMD uncertainties is moderate. The theoretical predictions agree wel…
Recent developments showed that light cone parton distributions can be studied by investigating the large momentum limit of the so-called quasiparton distributions, which are defined in terms of spacelike correlators, and therefore can be readily computed on the lattice. These two distributions can be connected to each other by a perturbative factorization formula or matching condition that allows one to convert the latter into the former. Here we present the one-loop matching condition for the transversity generalized quark distribution in the nonsinglet case
