Transverse-momentum-dependent (TMD) gluon distributions have different operator definitions, depending on the process under consideration. We study that aspect of TMD factorization in the small-x limit, for the various unpolarized TMD gluon distributions encountered in the literature. To do this, we consider di-jet production in hadronic collisions, since this process allows to be exhaustive with respect to the possible operator definitions, and is suitable to be investigated at small x. Indeed, for forward and nearly back-to-back jets, one can apply both the TMD factorization and Color Glass Condensate (CGC) approaches to compute the di-jet cross-section, and compare the results. Doing so, we show that both descriptions coincide, and we show how to express the various TMD gluon distributions in terms of CGC correlators of Wilson lines, while keeping Nc finite. We then proceed to evaluate them by solving the JIMWLK equation numerically. We obtain that at large transverse momentum, the process dependence essentially disappears, while at small transverse momentum, non-linear saturation effects impact the various TMD gluon distributions in very different ways. We notice the presence of a geometric scaling regime for all the TMD gluon distributions studied: the "dipole" one, the Weizsäcker-Williams one, and the six others involved in forward di-jet production.
It has recently been realized that the Balitsky-JIMWLK equations have serious shortcomings as equations to be used in small-x evolution near the unitarity limit. A recent generalization of the Balitsky equations has been given which corrects these shortcomings. In this paper we present an equivalent discussion, but in terms of the JIMWLK equation where we show that a new (fourth order functional derivative) term should be included. We also present a stochastic version of the new equation which, however, has some unusual mathematical aspects which are not as yet well understood.
The JIMWLK equation with a "daughter dipole" running coupling is solved numerically starting from an initial condition given by the McLerran-Venugopalan model. The resulting Wilson line configurations are then used to compute the spectrum of gluons comprising the glasma inital state of a high energy heavy ion collision. The development of a geometrical scaling region makes the spectrum of produced gluons harder. Thus the ratio of the mean gluon transverse momentum to the saturation scale grows with energy. Also the total gluon multiplicity increases with energy slightly faster than the saturat…
When JIMWLK evolution really matters: the example of incoherent diffraction
University of Jyväskylä（于韦斯屈莱大学）；Helsinki Institute of Physics（赫尔辛基物理研究所）；European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*)（核物理及相关领域理论研究中心（ECT*））；Fondazione Bruno Kessler（布鲁诺·凯斯勒基金会）
We investigate the question whether Lipatov's high energy effective action is capable to reproduce quark and gluon propagators which resum interaction with a strong background field within high energy factorization. Such propagators are frequently employed in calculations within the Color Glass Condensate formalism, in particular when considering scattering of a dilute projectile on a dense target nucleus or nucleon. We find that such propagators can be obtained from the high energy effective action, if a special parametrization of the gluonic field is used, first proposed by Lipatov in the or…
The small-$x$ evolution of protons is determined from numerical solutions of the JIMWLK equations, starting from an initial condition at moderate $x$ for a finite size proton. The resulting dipole amplitude is used to calculate the total reduced cross section $σ_r$ and charm reduced cross section $σ_{rc}$, as well as diffractive vector meson production. We compare results to experimental data from HERA and discuss fundamental problems arising from the regime sensitive to non-perturbative physics. We emphasize that information on the gluonic content of the proton, gluon spatial distributions an…
Within the effective theory for the Color Glass Condensate, we study multi-particle production with rapidity correlations in proton-nucleus collisions at high energy. The high-energy evolution responsible for such correlations is governed by a generalization of the JIMWLK equation which describes the simultaneous evolution of the (strong) nuclear color fields in the direct amplitude and the complex conjugate amplitude. This functional equation can be used to derive ordinary evolution equations for the cross-sections for particle production (a generalization of the Balitsky hierarchy). However,…
In the high energy limit of hadron collisions, the evolution of the gluon density in the longitudinal momentum fraction can be deduced from the Balitsky hierarchy of equations or, equivalently, from the nonlinear Jalilian-Marian-Iancu-McLerran-Weigert-Leonidov-Kovner (JIMWLK) equation. The solutions of the latter can be studied numerically by using its reformulation in terms of a Langevin equation. In this paper, we present a comprehensive study of systematic effects associated with the numerical framework, in particular the ones related to the inclusion of the running coupling. We consider th…
We construct the effective Hamiltonian which governs the renormalization group flow of the gluon distribution with increasing energy and in the leading logarithmic approximation. This Hamiltonian defines a two-dimensional field theory which involves two types of Wilson lines: longitudinal Wilson lines which describe gluon recombination (or merging) and temporal Wilson lines which account for gluon bremsstrahlung (or splitting). The Hamiltonian is self-dual, i.e., it is invariant under the exchange of the two types of Wilson lines. In the high density regime where one can neglect gluon number f…
Soft components of the light cone wave-function of a fast moving projectile hadron is computed in perturbation theory to third order in QCD coupling constant. At this order, the Fock space of the soft modes consists of one-gluon, two-gluon, and a quark-antiquark states. The hard component of the wave-function acts as a non-Abelian background field for the soft modes and is represented by a valence charge distribution that accounts for non-linear density effects in the projectile. When scattered off a dense target, the diagonal element of the S-matrix reveals the Hamiltonian of high energy evol…
Many-body QCD in leading high energy Regge asymptotics is described by the Balitsky-JIMWLK hierarchy of renormalization group equations for the x evolution of multi-point Wilson line correlators. These correlators are universal and ubiquitous in final states in deeply inelastic scattering and hadronic collisions. For instance, recently measured di-hadron correlations at forward rapidity in deuteron-gold collisions at the Relativistic Heavy Ion Collider (RHIC) are sensitive to four and six point correlators of Wilson lines in the small x color fields of the dense nuclear target. We evaluate the…
We study multi-particle production with rapidity correlations in proton-nucleus collisions at high energy in the Color Glass Condensate framework. The high-energy evolution responsible for such correlations is governed by a generalization of the JIMWLK equation describing the simultaneous evolution of the strong nuclear color fields in the direct amplitude and the complex conjugate amplitude. This functional equation can be used to derive ordinary evolution equations for the cross-sections for particle production, but the ensuing equations appear to be too complicated to be useful in practice,…
The conventional and linearly polarized Weizsäcker-Williams gluon distributions at small x are defined from the two-point function of the gluon field in light-cone gauge. They appear in the cross section for dijet production in deep inelastic scattering at high energy. We determine these functions in the small-x limit from solutions of the JIMWLK evolution equations and show that they exhibit approximate geometric scaling. Also, we discuss the functional distributions of these WW gluon distributions over the JIMWLK ensemble at rapidity $Y\sim 1/α_s$. These are determined by a 2d Liouville acti…
We propose a new method to implement the running coupling constant in the JIMWLK equation, imposing the scale dependence on the correlation function of the random noise in the Langevin formulation. We interpret this scale choice as the transverse momentum of the emitted gluon in one step of the evolution and show that it is related to the "Balitsky" prescription for the BK equation. This slows down the evolution speed of a practical solution of the JIMWLK equation, bringing it closer to the x-dependence inferred from fits to HERA data. We further study our proposal by a numerical comparison of…
We reinterpret the JIMWLK/KLWMIJ evolution equation as the QCD Reggeon field theory (RFT). The basic "quantum Reggeon field" in this theory is the unitary matrix $R$ which represents the single gluon scattering matrix. We discuss the peculiarities of the Hilbert space on which the RFT Hamiltonian acts. We develop a perturbative expansion in the RFT framework, and find several eigenstates of the zeroth order Hamiltonian. The zeroth order of this perturbation preserves the number of $s$ - channel gluons. The eigenstates have a natural interpretation in terms of the $t$ - channel exchanges. Study…
The high-energy evolution of Wilson line operators, which at leading order is described by the Balitsky-JIMWLK equations, receives large radiative corrections enhanced by single and double collinear logarithms at next-to-leading order and beyond. We propose a method for resumming such logarithmic corrections to all orders, at the level of the Langevin formulation of the JIMWLK equation. The ensuing, collinearly-improved Langevin equation features generalized Wilson line operators, which depend not only upon rapidity (the logarithm of the longitudinal momentum), but also upon the transverse siz…
We generalize the JIMWLK evolution equation to include the small-$x$ evolution of helicity distributions. To derive helicity JIMWLK, we use the method devised by A.H. Mueller for the usual unpolarized JIMWLK, and apply it to the helicity evolution equations derived earlier. We obtain a functional evolution equation for the generalized JIMWLK weight functional. The functional now is a sum of a helicity-independent term and a helicity-dependent term. The kernel of the new evolution equation consists of the standard eikonal leading-order JIMWLK kernel plus new terms containing derivatives with re…
In the Color Glass Condensate(CGC) effective theory, the physics of valence gluons with large longitudinal momentum is reflected in the distribution of color charges in the transverse plane. Averaging over the valence degrees of freedom is effected by integrating over classical color charges with some quasi probability weight functional $W[{\mathbf{j}}]$ whose evolution with rapidity is governed by the JIMWLK equation. In this paper, we reformulate this setup in terms of effective quantum field theory on valence Hilbert space governed by the reduced density matrix $\hatρ$ for hard gluons, whic…
We demonstrate that the Balitsky-JIMWLK equations describing the high-energy evolution of the n-point functions of the Wilson lines (the QCD scattering amplitudes in the eikonal approximation) admit a controlled mean field approximation of the Gaussian type, for any value of the number of colors Nc. This approximation is strictly correct in the weak scattering regime at relatively large transverse momenta, where it reproduces the BFKL dynamics, and in the strong scattering regime deeply at saturation, where it properly describes the evolution of the scattering amplitudes towards the respective…
We determine the distribution of linearly polarized gluons of a dense target at small $x$ by solving the B-JIMWLK rapidity evolution equations. From these solutions we estimate the amplitude of $\sim \cos 2ϕ$ azimuthal asymmetries in DIS dijet production at high energies. We find sizeable long-range in rapidity azimuthal asymmetries with a magnitude in the range of $v_2=\langle\cos 2ϕ\rangle \sim 10\%$.
NLO evolution of the Jalilian-Marian-Iancu-McLerran-Weigert-Leonidov-Kovner (JIMWLK) equation with massless quarks was derived a few years ago. We make a step further to compute the evolution kernels focusing on the effects due to finite quark masses. To this goal, the light-cone wave function of a fast moving dilute hadronic projectile is computed up to ${\cal O}(g^3)$ in QCD coupling constant. Compared with the massless case, a new IR divergence emerges, which is eventually canceled by a mass dependent counter term. Our results extend the theoretical tools used in physics of gluon saturation…
We continue the study of hadronic scattering amplitudes at high energy by systematically including nonlinear effects of finite partonic density in hadronic wave function as well as the effects of multiple rescatterings in the scattering process. In this paper we derive expressions for a single inclusive gluon production amplitude and multigluon inclusive production amplitudes when the rapidities of all observed gluons are not very different. We show that at leading order these observables exhibit a semiclassical structure. Beyond the semiclassical result, we find that the gluon emission has so…
We develop the formalizm for calculating semi inclusive observables at high energy in the JIMWLK/KLWMIJ approach. We consider several examples including diffractive processes, elastic and inclusive over the target degrees of freedom, scattering with fixed total transverse momentum transfer and inclusive gluon production. We discuss evolution of these observables with respect to various rapidity variables involved in their definitions (total rapidity, rapidity gap, width of diffractive interval etc.). We also discuss the dipole model limit of these observables and derive closed simple (as oppos…
We introduce a Wilson line distribution function bar{W}_tau(v) to study gluon saturation at small Feynman x_F, or large tau=ln(1/x_F). This new distribution can be obtained from the distribution W_tau(alpha) of the Color Glass Condensate model and the JIMWLK renormalization group equation. bar{W}_tau(v) is physically more relevant, and mathematically simpler to deal with because of unitarity of the Wilson line v. A JIMWLK equation is derived for bar{W}_tau(v); its properties are studied. These properties are used to complete Mueller's derivation of the JIMWLK equation, though for bar{W}_tau(v)…
We study the inclusion of running coupling corrections into the non-linear small-x JIMWLK and BK evolution equations by resumming all powers of alpha_s N_f in the evolution kernels. We demonstrate that the running coupling corrections are included in the JIMWLK/BK evolution kernel by replacing the fixed coupling constant alpha_s in it with alpha_s (1/r_1^2) alpha_s (1/r_2^2) / alpha_s (1/R^2), where r_1 and r_2 are transverse distances between the emitted gluon and the harder gluon (or quark) off of which it was emitted to the left and to the right of the interaction with the target. In the fo…
We develop an approach for calculating the inclusive multigluon production within the JIMWLK high energy evolution. We give a formal expression of multigluon cross section in terms of a generating functional for arbitrary number of gluons $n$. In the dipole limit the expression simplifies dramatically. We recover the previously known results for single and double gluon inclusive cross section and generalize those for arbitrary multigluon amplitude in terms of Feynman diagramms of Pomeron - like objects coupled to external rapidity dependent field $s(η)$. We confirm the conclusion that the AGK …
We write down explicit expressions for the $x$-evolution (equivalent to energy or rapidity evolution) of $2 n$ ($n = 1, 2, ...$) Wilson lines using the JIMWLK equation and the Color Glass Condensate formalism. We investigate the equation in the weak gluon field limit (linear regime) by expanding the Wilson lines in powers of the gluon field and show that it reduces to the BJKP equation describing the evolution of a state of $2 n$ Reggeized gluons with energy. We also make available for download a {\it Mathematica} program which provides this expression for any value of $n$.
In this work we extend the JIMWLK formalism to the two-source problem. The S-matrix for the forward scattering can be written in a double functional integral representation which involves the usual functional integral for the gluon field and the spin path integral for the external color sources. Modifications needed in the light-cone gauge are discussed. Using our source term we compute the produced gluon field and discuss the duality of the high energy evolution kernel in the pA collision.
We further discuss the QCD Reggeon field theory (RFT) as it emerges from the JIMWLK/KLWMIJ evolution equation and beyond. We give an explicit expression for the calculation of scattering amplitude in terms of the eigenstates of the RFT Hamiltonian. We point out that the spectrum of RFT is doubly degenerate, the degeneracy being related to the spontaneous breaking of the Dense-Dilute Duality symmetry of RFT. The degeneracy is between the "almost white" states (the Yang sector) which contain a small number of gluons, and "almost black" states (the Yin sector). The excitations above the Yang vacu…
We use the Color Glass Condensate (CGC) framework to study the production of forward heavy quark-antiquark pairs in unpolarized proton-nucleus or proton-proton collisions in the small-x regime. In the limit of nearly back-to-back jets, the CGC result simplifies into the transverse-momentum dependent (TMD) factorization approach. For massless quarks, the TMD factorization formula involves three unpolarized gluon TMDs: the Weizsäcker-Williams gluon distribution, the adjoint-dipole gluon distribution, and an additional one. When quark masses are kept non-zero, three new gluon TMDs appear -- each …
Resummations of soft gluon emissions play an important role in many applications of QCD, among them jet observables and small x saturation effects. Banfi, Marchesini, and Smye have derived an evolution equation for non-global jet observables that exhibits a remarkable analogy with the BK equation used in the small x context. Here, this analogy is used to generalize the former beyond the leading N_c approximation. The result shows striking analogy with the JIMWLK equation describing the small x evolution of the color glass condensate. A Langevin description allows numerical implementation and p…
The high-energy factorization and the associated B-JIMWLK or BK evolution equations are presented, using the example of DIS structure functions. The necessity of taking gluon saturation into account is discussed, and also the various approximations underlying high-energy factorization. The appearance of large NLL corrections in such a framework with or without gluon saturation is recalled, and their physical origin is explained. Finally, old and new results are presented about the resummation of some of those large corrections, related to kinematical approximations.
We derive a high energy factorization theorem for inclusive gluon production in A+A collisions. Our factorized formula resums i) all order leading logarithms (g^2 \ln(1/x_{1,2}))^n of the incoming partons momentum fractions, and ii) all contributions (g ρ_{1,2})^n that are enhanced when the color charge densities in the two nuclei are of order of the inverse coupling-- ρ_{1,2}\sim g^{-1}. The resummed inclusive gluon spectrum can be expressed as a convolution of gauge invariant distributions W[ρ_{1,2}] from each of the nuclei with the leading order gluon number operator. These distributions ar…
We derive the effective action of the hard large-x valence charges up to fourth order in their density. Such non-Gaussian weight functionals contribute at leading order in N_c to the connected two-gluon production diagrams which determine di-hadron correlations. The corresponding diagrams are not necessarily (highly) suppressed by the density of valence charges since their infrared divergences differ from those obtained in a Gaussian theory. Therefore, it appears prudent to include such higher dimensional operators when determining initial ensembles for JIMWLK evolution of higher n-point funct…
JIMWLK equation tells how gauge invariant higher order Wilson line correlators would evolve at high energy. In this article we present a convenient integro-differential form of this equation, for 2n-tuple correlator, where all real and virtual terms are explicit. The `real' terms correspond to splitting (say at position z) of this 2n-tuple correlator to various pairs of 2m-tuple and (2n+2-2m)-tuple correlators whereas `virtual' terms correspond to splitting into pairs of 2m-tuple and (2n-2m)-tuple correlators. Kernels of virtual terms with m=0 (no splitting) and of real terms with m=1 (splitti…
We examine both analytically and numerically the validity of factorization for the double dipole scattering amplitude T^{(2)} which appears on the right hand side of the BK--JIMWLK equation. We demonstrate that, if one uses a dilute object (e.g., a proton in DIS) as the initial condition, the correlation in the transverse plane induced by the leading order BFKL evolution is generally strong, resulting in a violation of the mean field approximation T^{(2)} \approx TT even at zero impact parameter by a factor ranging from 1.5 to O(10) depending on the relative size of the scatterers and rapidity…
The transverse momentum dependent gluon density obtained with CCFM evolution is determined from a fit to the latest combined HERA structure function measurements.
Recently, Kovner and Lublinsky proposed new small-x QCD evolution equations valid when the gluon density inside the target is low. The key element of their construction is the Wess-Zumino term which ensures the non-commutativity of valence charges. In this paper we clarify the origin and significance of this term by showing that it can be naturally incorporated in the effective theory of Color Glass Condensate. We also reexamine the renormalization group description in the high density (JIMWLK) regime.
It is known the single transverse spin asymmetry in semi-inclusive deep inelastic scattering can be factorized by a twist-3 distribution function $T_F$, which contains a gluon field strength tensor. With transverse gluon included in the power expansion, we find the gluon field strength tensor can be recovered definitely in soft-gluon-pole contribution at leading order of $α_s$ expansion. This conclusion holds in Feynman and light-cone gauges.
The computation of observables in high energy QCD involves an average over stochastic semiclassical small-x gluon fields. The weight of various configurations is determined by the effective action. We introduce a method to study fluctuations of observables, functionals of the small-x fields, which does not explicitly involve dipoles. We integrate out those fluctuations of the gluon field under which a given observable is invariant. Thereby we obtain the effective potential for that observable describing its fluctuations about the saddle point. We determine explicitly the effective potential fo…
We combine soft and collinear matrix elements to define transverse momentum dependent parton distributions (TMDs) for gluons, free from rapidity divergences. We establish a factorization theorem at next-to-leading order for the Higgs boson transverse momentum ($q_T$) spectrum, and use it to derive evolution equations for gluon TMDs. The evolution for all gluon TMDs is driven by a universal kernel, i.e. the same for all polarizations. In the region of intermediate $q_T$ we match the unpolarized, helicity and linearly polarized gluon distributions onto PDFs. We calculate the resummed Higgs trans…
We derive the analog of the Wandzura-Wilczek relation for the light-cone distributions of polarized gluons in a transversely polarized nucleon. The short distance cross-section is entirely due to the intrinsic transverse momenta of gluon in the nucleon, in complete analogy to the quark case. Numerical estimate for the double spin transverse asymmetries at RHIC are presented.
We study transverse-momentum-dependent (TMD) gluon distributions and factorization theorems for the gluon-initiated semi-inclusive processes at hadron colliders. Gauge-invariant TMD gluon distributions are defined, and their relations to the integrated (Feynman) parton distributions are explored when the transverse momentum is large. Through explicit calculations, soft-collinear factorization is verified at one-loop order for scalar particle production. Summation over large double logarithms is made through solving the Collins-Soper equation. We reproduce the known result in the limit that the…
We consider the production of high transverse momentum gluons in the McLerran-Venugopalan model of nuclear structure functions. We explicitly compute the high momentum component in this model. We compute the nuclear target size $A$ dependence of the distribution of produced gluons.
Back-to-back dijet cross-sections in deeply inelastic scattering (DIS) at small $x_{\rm Bj}$ are suppressed by many-body multiple scattering and screening effects arising from gluon saturation at high parton densities. They are similarly sensitive in these kinematics to large Sudakov logarithms from soft gluon radiation. Uncovering novel physics in this DIS channel therefore requires understanding the interplay of the two phenomena. In this work, we compute the small $x_{\rm Bj}$ inclusive dijet DIS cross-section in back-to-back kinematics at next-to-leading order (NLO) in the Color Glass Cond…
We investigate the gluon distribution functions and their contributions to the Higgs boson production in pp collisions in the transverse momentum dependent factorization formalism. In addition to the usual azimuthal symmetric transverse momentum dependent gluon distribution, we find that the azimuthal correlated gluon distribution also contributes to the Higgs boson production. This explains recent findings on the additional contribution in the transverse momentum resummation for the Higgs boson production as compared to that for electroweak boson production processes. We further examine the s…
We present a simple gluon production picture which is based on the McLerran-Venugopalan model and gluon BFKL evolution in relativistic heavy ion collision. Results for the multiplicity and transverse energy distribution in both the central and forward rapidity regions for gluon production in early stages of heavy ion collisions at the LHC are given. Finally, we provide a general qualitative discussion of the consequences of the forward rapidity behavior of produced gluons.
The transverse symmetry transformations associated with the normal symmetry transformations in gauge theories are introduced, which at first are used to reproduce the transverse Ward-Takahashi identities in the Abelian theory QED. Then the transverse symmetry transformations associated with the BRST symmetry and chiral transformations in the non-Abelian theory QCD are used to derive the transverse Slavnov-Taylor identities for the vector and axial-vector quark-gluon vertices, respectively. Based on the set of normal and transverse Slavnov-Taylor identities, an expression of the quark-gluon ver…
We study how gluons carrying linear polarization inside an unpolarized hadron contribute to the transverse momentum distribution of Higgs bosons produced in hadronic collisions. They modify the distribution produced by unpolarized gluons in a characteristic way that could be used to determine whether the Higgs boson is a scalar or a pseudoscalar particle.
We study resummation of transverse-momentum-related large logarithms generated from soft-gluon radiations in soft-collinear effective field theory. The anomalous dimensions of the effective quark and gluon currents, an important ingredient for the resummation, are calculated to two-loop order. the result at next-to-leading-log reproduces that obtained using the standard method for deep-inelastic scattering, Drell-Yan process, and Higgs production through gluon-gluon fusion. We comment on the extension of the calculation to next-to-next-to-leading logarithms.
We utilise the fact that the CCFM equation in the single loop approximation can be diagonalised by the Fourier-Bessel transform. The analytic solution of the CCFM equation for the moments $f_ω(b,Q)$ of the scale dependent gluon distribution is obtained, where $b$ is the transverse coordinate conjugate to the transverse momentum of the gluon. The unintegrated gluon distributions obtained from this solution are analysed. It is shown how the approximate treatment of the exact solution makes it possible to express the unintegrated gluon distributions in terms of the integrated ones. The correspond…
We calculate the transverse-momentum $Q_T$ spectrum of the dilepton in the transversely polarized Drell-Yan process on the basis of the factorization theorem in QCD. We take into account universal logarithmically enhanced corrections in edge region of phase space by resumming multiple soft-gluon emissions to all orders in the small $Q_T$ region.
We discuss the infrared limit for soft gluon kt-resummation and relate it to physical observables such as the intrinsic transverse momentum and the high energy limit of total cross-sections.
The inclusive hadronic production of $η_Q$($η_c$ or $η_b$) pair is proposed to extract the transverse momentum dependent(TMD) gluon distribution functions. We use nonrelativistic QCD(NRQCD) for the production of $η_Q$. Under nonrelativistic limit TMD factorization for this process is assumed to make a lowest order calculation. For unpolarized initial hadrons, unpolarized and linearly polarized gluon distributions can be extracted by studying different angular distributions.
The back-to-back $J/ψ$ and $π$ associated production at $e^+ e^-$ colliders is proposed to detect the gluon transverse momentum dependent(TMD) fragmentation functions. TMD factorization is assumed for this process. With spinless pion, unpolarized and linearly polarized gluon TMD fragmentation functions can be defined. It is found at parton level the hadronic tensor can be described by four structure functions. As a result, there are three independent angular distributions, of which a $\cos{2ϕ}$ azimuthal asymmetry is sensitive to the linearly polarized gluon fragmentation function.
