The use of the BFKL kernel improved by the inclusion of subleading terms generated by renormalization group (RG) analysis has been suggested to cure the instabilities in the behavior of the BFKL Green's function in the next-to-leading approximation (NLA). We test the performance of a RG-improved kernel in the determination of the amplitude of a physical process, the electroproduction of two light vector mesons, in the BFKL approach in the NLA. We find that a smooth behavior of the amplitude with the center-of-mass energy can be achieved, setting the renormalization and energy scales appearing in the subleading terms to values much closer to the kinematical scales of the process than in the approaches based on the unimproved kernel.
We calculate the amplitude for the forward electroproduction of two light vector mesons in next-to-leading order BFKL. This amplitude represents the first next-to-leading order amplitude ever calculated for a collision process between strongly interacting colorless particles.
The forward electroproduction of two light vector mesons is the first example of a collision process between strongly interacting colorless particles for which the amplitude can be written completely within perturbative QCD in the Regge limit with next-to-leading accuracy. This amplitude can be written as a convolution of two impact factors for the virtual photon to light vector meson transition with the BFKL Green's function. In this lecture we first describe how the relevant impact factor is calculated, then we perform the convolution with the BFKL Green's function and illustrate the numeric…
We test the performance of a RG-improved kernel in the determination of the amplitude of a physical process, the electroproduction of two light vector mesons,in the BFKL approach at the next-to-leading approximation (NLA). We find that a smooth behavior of the amplitude with the center-of-mass energy can be achieved, setting the renormalization and energy scales appearing in the subleading terms to values much closer to the kinematical scales of the process than in approaches based on unimproved kernels.
The amplitude for the forward electroproduction of two light vector mesons can be written completely within perturbative QCD in the Regge limit with next-to-leading accuracy, thus providing the first example of a physical application of the BFKL approach at the next-to-leading order. Recently, a numerical determination of the amplitude has been obtained in the case of equal photon virtualities, by using a definite representation for the amplitude and a definite optimization method for the perturbative series. Here, we study the main systematic effects in the previous determination, by consider…
The forward electroproduction of two light vector mesons is the first example of a collision process between strongly interacting colorless particles for which the amplitude can be written completely within perturbative QCD in the Regge limit with next-to-leading accuracy. In a previous paper we have given a numerical determination of the amplitude in the case of equal photon virtualities by using a definite representation for the amplitude and a definite optimization method for the perturbative series. Here we estimate the systematic uncertainty of our previous determination, by considering a…
We calculate the amplitude for the forward electroproduction of two light vector mesons in next-to-leading order BFKL. This amplitude is written as a convolution of two impact factors for the virtual photon to light vector meson transition with the BFKL Green's function. It represents the first next-to-leading order amplitude ever calculated for a collision process between strongly interacting colorless particles.
Hard scattering processes involving hadrons at small $x$ are described by a $k_T$-factorization formula driven by a BFKL gluon. We explore the equivalence of this description to a collinear-factorization approach in which the anomalous dimensions $γ_{gg}$ and $γ_{qg}/α_S$ are expressed as power series in $α_S \log (1/x)$, or to be precise $α_S/ω$ where $ω$ is the moment index. In particular we confront the collinear-factorization expansion with that extracted from the BFKL approach with running coupling included.
The proton structure function F2 is studied in the low x regime using BFKL evolution. The next to leading logarithmic (NLL) analysis requires the inclusion of running coupling effects which lead to off-diagonal terms in the evolution kernel. An all-orders resummation is used to improve the collinear behavior of the NLL BFKL result. We emphasize the theoretical uncertainties that appear throughout the analysis and give a comparison to the combined HERA data.
Exclusive electroproduction of hard dijets can be described within the collinear factorization. This process has clear experimental signature and provides one with an interesting alternative venue to test QCD description of hard diffractive processes and extract information on generalized nucleon parton distributions. In this work we present detailed leading-order QCD calculations of the relevant cross sections, including longitudinal momentum fraction distribution of the dijets and their azimuthal angle dependence.
We compute the exclusive electroproduction, $γ^* p \rightarrow V p$, of heavy quarkonia $V$ to NLO in the collinear factorisation scheme, which has been formally proven for this process. The inclusion of an off-shell virtuality $Q^2$ carried by the photon extends the photoproduction phase space of the exclusive heavy quarkonia observable to electroproduction kinematics. This process is relevant for diffractive scattering at HERA and the upcoming EIC, as well as at the proposed LHeC and FCC.
We present a detailed description of the Q^2 and x dependence of the structure functions F2 and FL as extracted from the Deep Inelastic Scattering data at HERA in the small Bjorken x region. Making use of a collinearly-improved BFKL equation at next-to-leading order and a treatment of the running of the coupling using non-Abelian physical renormalization together with the BLM scale choice allows us to reach low values of Q^2. We also provide some predictions for future lepton-hadron colliders.
We present the first evaluation of color dipole cross section from experimental data on diffractive photo- and electroproduction of vector mesons. The dipole-size and energy dependence of the found dipole cross section is consistent with expectations from the BFKL dynamics.
Motivated by the recent work of Brzemiński, Motyka, Sadzikowski and Stebel in arXiv:1611.04449, where forward Drell--Yan production is studied in proton-proton collisions at the LHC, we improve their calculation by introducing an unintegrated gluon density obtained in arXiv:1209.1353 and arXiv:1301.5283 from a fit to combined HERA data at small values of Bjorken $x$. This gluon density was calculated within the BFKL formalism at next-to-leading order with collinear corrections. We show that it generates a good description of the forward Drell--Yan cross section dependence on the invariant mass…
We study the contribution of the truncated BFKL series to the electroproduction process showing that a reliable description is obtained taking into account two orders in perturbation theory. Using the recent F2 logarithmic slope data as a constraint to the adustable parameters, the inclusive structure function is described in a wide range of the small x HERA kinematical region, consistent with the unitarity bound. We also extrapolate the extimates to the THERA region.
We show that definite polarization observables for the simplest electroproduction processes, $\ell+N\to \ell+B(1/2^\pm)+M(0^\pm)$, $B=Y(Λ, Σ$ or $Y_c$-hyperon, $Θ^+$-pentaquark) and $M=K$, $\bar{K}$ or $D$, are sensitive to the relative P-parity $π(B)$ of the NBM-system. The interference of the longitudinal and transversal amplitudes for the collinear regime of the processes $γ^*+N\to B+M $($γ^*$ is the virtual photon) - at any value of momentum transfer squared and excitation energy of the $BM$-system - generates model independent relations between analyzing powers (in unpolarized lepton scat…
The amplitude for the forward electroproduction of two light vector mesons can be written completely within perturbative QCD in the Regge limit with next-to-leading accuracy, thus providing the first example of a physical application of the BFKL approach at the next-to-leading order. We study in the case of equal photon virtualities the main systematic effects, by considering a different representation of the amplitude and different optimization methods of the perturbative series.
We calculate the rate for the deep-inelastic electroproduction of dijets at HERA. We study the weakening of the azimuthal (back-to-back) correlation between the jets, as $x$ decreases, to see whether it can be used to identify BFKL dynamics from conventional fixed-order QCD effects. We show how this may give information on the transverse momentum ($k_T)$ dependence of the gluon distribution of the proton.
We consider the process gamma^* N -> M_1 M_2 N' with a large rapidity gap between the two mesons M_1 and M_2. Within the QCD collinear approximation, the scattering amplitude may be written as a convolution of an impact factor describing the gamma^* -> M_1 transition and an amplitude describing the N -> M_2 N' collinear process.
I report on the recent proposal of a generalized small-x equation which, in addition to exact leading and next-to-leading BFKL kernels, incorporates renormalization group constraints in the relevant collinear limits.
We discuss exclusive vector meson electroproduction within the QCD collinear factorization framework. In Bjorken kinematics the amplitude factorizes in a convolution of the nonperturbative meson distribution amplitude and the generalized parton densities with the perturbatively calculable hard-scattering amplitudes, which are presently known to next-to-leading order (NLO). At small $x_{\rm B}$ NLO corrections are very large. It is related to appearance of BFKL type logarithms in the hard-scattering amplitudes, that calls for a resummation of these effects at higher orders. Here we report the f…
We investigate the stability under variation of the renormalization, factorization and energy scales entering the calculation of the cross section, at next-to-leading order in the BFKL formalism, for the production of Mueller-Navelet jets at the Large Hadron Collider, following the experimental cuts on the tagged jets. To find optimal values for the scales involved in this observable it is possible to look for regions of minimal sensitivity to their variation. We show that the scales found with this logic are more natural, in the sense of being more similar to the squared transverse momenta of…
We present a recent determination of an approximate expression for the O(alpha_s^3) contribution chi_2 to the kernel of the BFKL equation. This includes all collinear and anticollinear singular contributions and is derived using duality relations between the GLAP and BFKL kernels.
In the high-energy limit, we compute the gauge-invariant three-parton forward clusters, which in the BFKL theory constitute the tree parts of the NNLO impact factors. In the triple collinear limit, we obtain the polarized double-splitting functions. For the unpolarized and the spin-correlated double-splitting functions, our results agree with the ones obtained by Campbell-Glover and Catani-Grazzini, respectively. In addition, we compute the four-gluon forward cluster, which in the BFKL theory forms the tree part of the NNNLO gluonic impact factor. In the quadruple collinear limit we obtain the…
Relying on the collinear factorization approach, we demonstrate that H1 and ZEUS measurements of exclusive light vector meson and photon electroproduction cross sections can be simultaneously described for photon virtualities of $Q >2 GeV$. Our findings reveal that quark exchanges are important in this small $x_{Bj}$ region and that in leading order approximation the gluonic skewness ratio is much smaller than one.
We complete the calculation of the next-to-leading kernel of the BFKL equation, by disentangling its energy-scale dependent part from the impact factor corrections in large-k dijet production. Using the irreducible part previously obtained,we derive the final form of the kernel eigenvalue and of the hard Pomeron shift for various scales. We also discuss the scale changes,the physical equivalence of a class of scales, and how to use collinear safe ones.
In color dipole gBFKL dynamics, we describe the emerging gBFKL phenomenology of a subasymptotic energy dependence of the diffraction slope and discuss possibilities of testing the gBFKL predictions in exclusive photo- and electroproduction of vector mesons $V$ at HERA. A substantial shrinkage of the diffraction cone $γ^{*}p\rightarrow Vp$ processes from the CERN/FNAL to HERA range of energy $W$ is predicted. This subasymptotic shrinkage is faster than expected from the small slope of pomeron's Regge trajectory $α_{\Pom}'$. We point out that the diffraction slope is a scaling function of $(m_{V…
Progress in the understanding of the BFKL approach in the NLL approximation is reported. The study based on the iteration of the kernel using the exponentiation of the gluon Regge trajectory is reviewed in QCD and N=4 super Yang-Mills theories. Properties of the gluon Green's function in the high energy Regge limit for forward and non-forward scattering are considered. A novel representation of collinearly improved kernels is also presented.
We investigate the effect of the resummation of collinear double logarithms in the BFKL gluon Green function using the Monte Carlo event generator BFKLex. The resummed collinear terms in transverse momentum space were calculated in Ref. [1] and correspond to the addition to the NLO BFKL kernel of a Bessel function of the first kind whose argument contains the strong coupling and a double logarithm of the ratio of the squared transverse momenta of the reggeized gluons. We discuss how these additional terms improve the collinear convergence of the whole approach and reduce the asymptotic growth …
On the basis of a renormalization group analysis of the kernel and of the solutions of the BFKL equation with subleading corrections, we propose and calculate a novel expansion of a properly defined effective eigenvalue function. We argue that in this formulation the collinear properties of the kernel are taken into account to all orders, and that the ensuing next-to-leading truncation provides a much more stable estimate of hard Pomeron and of resummed anomalous dimensions.
Based on Regge-inspired arguments and counting rules, we formulate empirical models for zero-skewness generalized parton distributions (GPDs) ${\widetilde H}^{(3)}$ and ${\widetilde E}^{(3)}$ in the iso-vector sector. If a hypothetical $a_1/ρ_2$ master trajectory is taken into account, we find that the polarized deep inelastic structure function $g_1^{(3)}$, axial-vector form factor, pseudoscalar form factor, and lattice data are well described. Thereby, we use a symmetric valence scenario in which the `spin puzzle' in the iso-vector sector is trivially resolved. Utilizing a minimalist `hologr…
We determine an approximate expression for the O(alpha_s^3) contribution chi_2 to the kernel of the BFKL equation, which includes all collinear and anticollinear singular contributions. This is derived using recent results on the relation between the GLAP and BFKL kernels (including running-coupling effects to all orders) and on small-x factorization schemes. We present the result in various schemes, relevant both for applications to the BFKL equation and to small-x evolution of parton distributions.
We study the energy dependence of the saturation momentum in the context of the collinearly improved Leading and Next to Leading BFKL evolution, and in the presence of saturation boundaries. We find that the logarithmic derivative of the saturation momentum is varying very slowly with Bjorken-x, and its value is in agreement with the Golec-Biernat and Wusthoff model in the relevant x region. The scaling form of the amplitude for dipole-dipole or dipole-hadron scattering in the perturbative side of the boundary is given.
We study the decorrelation in azimuthal angle of Mueller-Navelet jets at hadron colliders within the BFKL formalism. We introduce NLO terms in the evolution kernel and present a collinearly-improved version of it for all conformal spins. We show how this further resummation has good convergence properties and is closer to the Tevatron data than a simple LO treatment. However, we are still far from a good fit. We offer estimates of these decorrelations for larger rapidity differences which should favor the onset of BFKL effects and encourage experimental studies of this observable at the LHC.
We investigate factorisation at small x using a variety of analytical and numerical techniques. Previous results on factorisation in collinear models are generalised to the case of the full BFKL equation, and illustrated in the example of a collinear model which includes higher twist terms. Unlike the simplest collinear model, the BFKL equation leads to effective anomalous dimensions containing higher-twist pieces which grow as a (non-perturbative) power at small x. While these pieces dominate the effective splitting function at very small x they do not lead to a break-down of factorisation in…
We investigate the impact of so called kinematic constraint on gluon evolution at small $x$. Implanting the constraint on the real emission term of gluon ladder diagram, we obtain an integro-differential form of BFKL equation. Later we solve the equation analytically using the method of characteristics. We sketch the Bjorken x and transverse momentum $k_t^2$ dependence of our solution of unintegrated gluon distributions $f(x,k_t^2)$ in kinematic constraint supplemented BFKL equation and contrasted the same with original BFKL equation. Then we extract the collinear gluon density $xg(x,Q^2)$ fro…
We show that in hard exclusive electroproduction, ep-->eVp, the leading-twist hard-scattering coefficient for the production of a transversely polarized vector meson V vanishes to all orders of perturbation theory. This implies that this process cannot be used to measure the skewed transversity distribution of quarks in a hadron. In contrast, a recent calculation obtained a non-zero value at NLO. We show that this calculation is incorrect because it failed to include the necessary collinear subtractions. Our method of proof also applies to other processes whose hard-scattering coefficients are…
We present a systematic formalism for the derivation of the kernel of the BFKL equation from that of the GLAP equation and conversely to any given order, with full inclusion of the running of the coupling. The running coupling is treated as an operator, reducing the inclusion of running coupling effects and their factorization to a purely algebraic problem. We show how the GLAP anomalous dimensions which resum large logs of x can be derived from the running-coupling BFKL kernel order by order, thereby obtaining a constructive all-order proof of small x factorization. We check this result by ex…
We bring evidence that the recently discovered property of natural stability of the high-energy resummation is directly connected to the fragmentation mechanism of heavy hadrons. As a phenomenological support, we provide predictions for differential distributions sensitive to heavy-hadron tags, calculated at the next-to-leading logarithmic level of the hybrid high-energy/collinear factorization (NLL/NLO), as implemented in the JETHAD multimodular code. We show that the stabilizing mechanism is encoded in gluon channels of both heavy-flavor collinear fragmentation functions extracted from data …
We calculate the amplitude for exclusive electroweak production of a pseudoscalar $D_s$ or a vector $D^*_s$ charmed strange meson on an unpolarized nucleon, through a charged current, in leading order in $α_s$. We work in the framework of the collinear QCD approach where generalized gluon distributions factorize from perturbatively calculable coefficient functions. We include both $O(m_c)$ terms in the coefficient functions and $O(M_D)$ mass term contributions in the heavy meson distribution amplitudes. We show that this process may be accessed at future electron-ion colliders.
We have calculated the leading order amplitude of hard diffractive electroproduction of two pions in lepton nucleon scattering. At the leading twist level a pion pair can be produced only in an isospin one or zero state. We have shown that isoscalar states are produced dominantly for x_{Bj}>0.3 and with an invariant mass of the two pions close to the threshold (S-wave) and in the f_2 resonance region (D-wave). These isoscalar pion pairs are dominantly produced by two collinear gluons. Comparing the production of charged and neutral pion pairs as a function of x_{Bj} and m_pipi one can get info…
We calculate the eigenvalues of the next-to-leading kernel for the BFKL equation in the adjoint representation of the gauge group $SU(N_c)$ in the N=4 supersymmetric Yang-Mills model. These eigenvalues are used to obtain the high energy behavior of the remainder function for the 6-point scattering amplitude with the maximal helicity violation in the kinematical regions containing the Mandelstam cut contribution. The leading and next-to-leading singularities of the corresponding collinear anomalous dimension are calculated in all orders of perturbation theory. We compare our result with the kno…
We study, within QCD collinear factorization and including BFKL resummation at the next-to-leading order, the production of Mueller-Navelet jets at LHC with center-of-mass energy of 7 TeV. The adopted jet vertices are calculated in the approximation of small aperture of the jet cone in the pseudorapidity-azimuthal angle plane. We consider several representations of the dijet cross section, differing only beyond the next-to-leading order, to calculate a few observables related with this process. We use various methods of optimization to fix the energy scales entering the perturbative calculatio…
We review one-particle inclusive production of heavy-flavored hadrons in a framework which resums the large collinear logarithms through the evolution of the FFs and PDFs and, at the same time, retains the full dependence on the heavy-quark mass without additional theoretical assumptions. The main focus is on the production of D^* mesons in deep inelastic electron--proton scattering at HERA. We show results, neglecting for the time being the heavy-quark mass terms, for deep inelastic D^* meson production at finite transverse momenta. Work to implement this process in the above mentioned massiv…
Due to multiple possible polarizations hard exclusive production of tensor mesons by virtual photons or in heavy meson decays offers interesting possibilities to study the helicity structure of the underlying short-distance process. Motivated by the first measurement of the transition form factor $γ^*γ\to f_2(1270)$ at large momentum transfers by the BELLE collaboration we present an improved QCD analysis of this reaction in the framework of collinear factorization including contributions of twist-three quark-antiquark-gluon operators and an estimate of soft end-point corrections using light-c…
We consider the electroproduction of two vector mesons with a large rapidity gap between them on a nucleon target in the process $ γ^* N \to ρ_1 ρ_2 N'$. We calculate the Born term for this process within the collinear factorization framework. The resulting scattering amplitude may be represented as a convolution of an impact factor describing the $γ^* \to ρ_1$ transition and an amplitude describing the $N\to ρ_2 N'$ transition. The latter amplitude is analogous to deeply virtual electroproduction of a meson, the virtual photon being replaced by two gluon (Pomeron) exchange . The long distance…
Radiative effects in the electroproduction of photons in polarized $ep$-scattering are calculated with the next-to-leading (NLO) accuracy. The contributions of loops and two photon emission were presented in analytical form. The covariant approach of Bardin and Shumeiko was used to extract the infrared divergence. All contributions to the radiative correction were presented in the form of the correction to the leptonic tensor thus allowing for further applications in other experiments, e.g., deep inelastic scattering. The radiative corrections (RC) to the cross sections and polarization asymme…
We link the recent computation beyond leading twist of the impact factor of the transition gamma*T -> rhoT performed in the light-cone collinear approach, to the dipole picture by expressing the hard part of the process through its Fourier transform in coordinate space. We show that in the Wandzura-Wilczek approximation the impact factor up to twist 3 factorises in the wave function of the photon combined with the distribution amplitudes of the rho-meson and the colour dipole scattering amplitude with the t-channel gluons. We show also that beyond the Wandzura-Wilczek approximation, the hard c…
The solution to the impact-parameter dependent Balitsky-Kovchegov equation with the collinearly improved kernel is studied in detail. The solution does not present the phenomenon of Coulomb tails at large impact parameters that have affected previous studies. The origin of this behaviour is explored numerically. It is found to be linked to the fact that this kernel suppresses large daughter dipoles. Solutions based on a physics motivated form of the initial condition are used to compute predictions for structure functions of the proton and the exclusive photo- and electroproduction of vector m…
We propose the inclusive hadroproduction of a heavy-light dijet system, as a new channel for the investigation of high energy QCD. We build up a hybrid factorization that incorporates a partial next-to-leading BFKL resummation inside the standard collinear description of observables. We present a detailed analysis of different observables: cross-section summed over azimuthal angles and differential in rapidity and azimuthal distribution. The stability that these distributions show under higher-order corrections motivates our interest in future studies, doable at new generation colliding machin…
By making use of the hybrid collinear and high-energy factorization, where the BFKL resummation of leading and next-to-leading energy logarithms is combined with the standard description in terms of collinear parton densities, we compare predictions for Mueller-Navelet jet rapidity and angular differential rates with data collected by CMS at $\sqrt{s} = 7$ TeV. We provide an evidence that the study of azimuthal distributions, calculated as a Fourier sum of correlation moments and embodying the high-energy signal coming from all conformal-spin modes, permits us to overcome the well-known issues…
We propose the study of the inclusive hadroproduction of a heavy-flavored jet in association with a light jet, as a probe channel of strong interactions at high energies. We build up a hybrid factorization that encodes genuine high-energy effects, provided by a partial next-to-leading BFKL resummation, inside the standard collinear structure of the cross section. We present a detailed analysis of different distributions, shaped on kinematic ranges typical of experimental analyses at the Large Hadron Collider, and differential in rapidity, azimuthal angle and transverse momentum. The fair stabi…
Two relevant points related to the application of the BFKL formalism to phenomenology are discussed. First, we have presented a set of observables characterizing multi-jet configurations event by event (average transverse momentum, average azimuthal angle, average ratio of jet rapidities) which can be used to find distinct signals of BFKL dynamics at the LHC. A numerical analysis has been shown using the Monte Carlo event generator BFKLex, modified to include higher-order collinear corrections in addition to the transverse-momentum implementation of the NLO kernel. We require to have two tagge…
Gluon GTMDs in the exclusive electroproduction of heavy-quark pairs
Università di Cagliari（卡利亚里大学）；INFN, Sezione di Cagliari（意大利国家核物理研究所，卡利亚里分部）；Van Swinderen Institute for Particle Physics and Gravity, University of Groningen（格罗宁根大学范斯温德恩粒子物理与引力研究所）
We calculate the transverse momentum dependent gluon-to-gluon splitting function within kT-factorization [EPJC 78 (2018) 174], generalizing the framework of [NPB 175 (1980) 27; NPB 427 (1994) 475] and extending our previous works [JHEP 01 (2016) 181; PRD 94 (2016) 114013] for the quark splittings. While existing versions of kT factorized evolution equations contain already a gluon-to-gluon splitting function i.e. the leading order BFKL kernel or the CCFM kernel, the obtained splitting function has the important property that it reduces to the leading order BFKL kernel in the low z limit, to th…
