In this paper we developed approach based on the BFKL evolution in $\ln\Lb Q^2\Rb$. We show that the simplest diffusion approximation with running QCD coupling is able to describe the HERA experimental data on the deep inelastic structure function with good $χ^2/d.o.f. \approx 1.3$. From our description of the experimental data we learned several lessons; (i) the non-perturbative physics at long distances started to show up at $Q^2 = 0.25\,GeV^2$; (ii) the scattering amplitude at $Q^2 = 0.25\,GeV^2$ cannot be written as sum of soft Pomeron and the secondary Reggeon but the Pomeron interactions should be taken into account; (iii) the Pomeron interactions can be reduced to the enhanced diagrams and, therefore, we do not see any needs for the shadowing corrections at HERA energies; and (iv) we demonstrated that the shadowing correction could be sizable at higher than HERA energies without any contradiction with our initial conditions.
We propose a phenomenological study of the next-to-leading Balitsky-Fadin-Kuraev-Lipatov (BFKL) approach applied to the data on the proton structure function $F_2$ measured at HERA in the small-$x_{Bj}$ region.
We perform a BFKL-NLL analysis of forward jet production at HERA which leads to a good description of data over the full kinematical domain. We also predict the azimuthal angle dependence of Mueller-Navelet jet production at the Tevatron and the LHC using the BFKL NLL formalism.
We show that the next-to-leading-order renormalization-group-improved asymptotically-free BFKL Pomeron provides a good fit to HERA data on virtual photoproduction at small x and large Q^2. The leading discrete Pomeron pole reproduces qualitatively the Q^2 dependence of the HERA data for x ~10^{-3}, and a fit using the three leading discrete singularities reproduces quantitatively the Q^2 and x dependence of the HERA data for x < 10^{-2}. This fit fixes the phase for all the BFKL wavefunctions at a chosen infrared scale.
In the framework of semihard QCD approach, with the emphasis on the BFKL evolution of gluon distributions, we calculate the differential cross sections of inclusive $D^{*\pm}$ mesons electroproduction at HERA and confront our results with available ZEUS experimental data.
The general properties of the Lipatov or BFKL equation are reviewed. Modifications to the infrared region are proposed. Numerical predictions for the deep-inelastic electron-proton structure functions at small $x$ are presented and confronted with recent HERA measurements.
The BFKL equation and the kT-factorization theorem are used to obtain predictions for F2 in the small Bjorken-x region over a wide range of Q**2. The dependence on the parameters, especially on those concerning the infrared region, is discussed. After a background fit to recent experimental data obtained at HERA and at Fermilab (E665 experiment), we find that the predicted, almost Q**2 independent BFKL slope lambda >= 0.5 appears to be too steep at lower Q**2 values. Thus there seems to be a chance that future HERA data can distinguish between pure BFKL and conventional field theoretic renorma…
We present a BFKL phenomenology of structure functions in the framework of the generalized BFKL equation for the dipole cross section proposed by us recently. We show that the HERA measurements of excitation of charm at $Q^{2} \lsim $10 GeV$^{2}$ allow determination of the intercept of the BFKL pomeron. A comparison with the experimental data suggests a large contribution from the nonperturbative cross section for colour dipoles of large size $r$.
We show that next-leading logarithmic (NLL) Balitsky-Fadin-Kuraev-Lipatov (BFKL) effects can be tested by the forward-jet cross sections recently measured at HERA. For dσ/dx, the NLL corrections are small which confirms the stability of the BFKL description. The triple differential cross section dσ/dxdk_T^2dQ^2 is sensitive to NLL effects and opens the way for an experimental test of the full BFKL theoretical framework at NLL accuracy.
In the framework of semihard (k$_T$ factorization) QCD approach, we consider the differential cross sections of $D^{*\pm}$ meson production at HERA. The consideration is based on BFKL and CCFM gluon distributions. We find that in the case of BFKL LO gluon distribution the theoretical results are sensitive to the Pomeron intercept parameter $Δ$. We present a comparison of the theoretical results with available ZEUS experimental data.
In the framework of the semi-hard ($k_t-$factorization) approach, we analyze the various charm production processes in the kinematic region covered by the HERA experiments
Recently the H1 and ZEUS collaborations have presented cross sections for DIS events with a forward jet. The BFKL formalism is able to produce an excellent fit to these data. The extracted intercept of the hard pomeron suggests that when all higher order corrections are taken into account the cross section will still rise very rapidly as expected for low $x$ dynamics.
Elastic production of vector mesons $γ^{*} N\to V N$ is the pomeron-exchange dominated diffractive reaction with much potential of probing the BFKL pomeron. The BFKL pomeron can conveniently be described in terms of the dipole cross section which is a solution of the generalized BFKL equation. In this paper we discuss, how the energy and $Q^{2}$ dependence of elastic production of vector mesons at HERA will allow scanning the dipole cross section as a function of dipole size $r$. We show that determinaton of the intercept of the BFKL pomeron requires measuring the $ρ^{0}$ and $J/Ψ$ production …
We suggest a formula interpolating between the known asymptotic regimes of the BFKL equation as the approximate solution of that equation. The parameters appearing in this interpolation are fitted to the data on deep inelastic scattering in a wide range of the kinematical variables. Care is taken of the large-$x$ domain as well, outside the HERA kinematical region. The boundaries and the interface between various dynamical regimes are also studied.
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 show that the forward-jet measurements performed at HERA allow for a detailed study of corrections due to next-to-leading logarithms (NLL) in the Balitsky-Fadin-Kuraev-Lipatov (BFKL) approach. While the description of the dσ/dx data shows small sensitivity to NLL-BFKL corrections, these can be tested by the triple differential cross section dσ/dxdk_T^2dQ^2 recently measured. These data can be successfully described using a renormalization-group improved NLL kernel while the standard next-to-leading-order QCD or leading-logarithm BFKL approaches fail to describe the same data in the whole ki…
We suggest a modified form of a unitarized BFKL equation imposing the so-called kinematic constraint on the gluon evolution in multi-Regge kinematics. The underlying nonlinear effects on the gluon evolution are investigated by solving the unitarized BFKL equation analytically. We obtain an equation of the critical boundary between dilute and dense partonic system, following a new differential geometric approach and sketch a phenomenological insight on geometrical scaling. Later we illustrate the phenomenological implication of our solution for unintegrated gluon distribution $f(x,k_T^2)$ towar…
Determination of the intercept of the BFKL pomeron is one of the pressing issues in the high energy physics. Earlier we have shown that, at the dipole size $r=r_Δ=(0.1-0.2)$f, the dipole cross section $σ(x,r)$ which is a solution of the generalized BFKL equation, exhibits a precocious asymptotic behavior $σ(x,r)\propto \left({1\over x}\right)^{Δ_{\Pom}}$. In this paper we discuss how measuring $F_{L}(x,Q^{2})$ and $\partial F_{T}(x,Q^{2})/\partial \log Q^{2}$ at $Q^{2}= (10-40)$GeV$^{2}$ and $Q^{2}=(2-10)$GeV$^{2}$, respectively, one can probe $σ(x,r_Δ)$ and directly determine the intercept $Δ…
We determine the infrared behaviour of the BFKL forward amplitude for gluon-gluon scattering. Our approach, based on the discrete pomeron solution, leads to an excellent description of the new combined inclusive HERA data at low values of x (<0.01) and at the same time determines the unintegrated gluon density inside the proton, for squared transverse momenta of the gluon less than 100 GeV^2. The phases of this amplitude are sensitive to the non-perturbative gluonic dynamics and could be sensitive to the presence of Beyond-the-Standard-Model particles at very high energies.
The asymptotic freedom is known to split the leading-$\log$ BFKL pomeron into a series of isolated poles in the complex angular momentum plane. One of our earlier findings was that the subleading hard BFKL exchanges decouple from such experimentally important observables as small-$x$ charm, $F_2^c$, and the longitudinal, $F_L$, structure functions of the proton at moderately large $Q^2$. For instance, we predicted precocious BFKL asymptotics of $F_2^c(x,Q^2)$ with intercept of the rightmost BFKL pole $α_{\Pom}(0)-1=Δ_{\Pom}\approx 0.4$. On the other hand, the small-$x$ open beauty photo- and e…
We analyse, in NLO, the physical properties of the discrete eigenvalue solution for the BFKL equation. We show that a set of eigenfunctions with positive eigenvalues, ω, together with a small contribution from a continuum of eigenfunctions with negative ω, provide an excellent description of high-precision HERA F_2 data in the region, x<0.001, Q^2 > 6 GeV^2. The phases of the eigenfunctions can be obtained from a simple parametrisation of the pomeron spectrum, which has a natural motivation within BFKL. The data analysis shows that the first eigenfunction decouples completely or almost complet…
We compare the BFKL prediction for the associated production of forward jets at HERA with fixed-order matrix element calculations taking into account the kinematical cuts imposed by experimental conditions. Comparison with H1 data of the 1993 run favours the BFKL prediction. As a further signal of BFKL dynamics, we propose to look for the azimuthal dependence of the forward jets.
Using the QCD dipole picture of the hard BFKL pomeron, we derive the general expressions of the elastic and inelastic components of the proton diffractive structure functions. We obtain a good 7 parameter fit (including a secondary reggeon contribution) to data taken at HERA by the H1 and ZEUS collaborations. The main characteristic features of the model in reproducing the data are discussed, namely the effective pomeron intercept, the scaling violations and the beta dependence. A difference obtained in the separate H1 and ZEUS fits leads us to analyse directly the differences between both mea…
We present microscopic QCD calculation of the cross section of diffractive DIS and of the partonic structure of the pomeron from the dipole approach to the generalized BFKL pomeron. We show that the pomeron can not be treated as a particle with uniquely defined structure function and flux in the proton. We find strong factorization breaking which can approximately be described by the two-component structure function of the pomeron, each component endowed with the different flux of pomerons in the proton. We predict very weak $Q^{2}$ dependence of the diffractive contribution to the proton stru…
We show that the HERA data for the inclusive structure function F_2(x,Q^2) for x < 0.01 and 0.045 < Q^2 < 45 GeV^2 can be well described within the color dipole picture, with a simple analytic expression for the dipole-proton scattering amplitude, which is an approximate solution to the non-linear evolution equations in QCD. For dipole sizes less than the inverse saturation momentum 1/Q_s(x), the scattering amplitude is the solution to the BFKL equation in the vicinity of the saturation line. It exhibits geometric scaling and scaling violations by the diffusion term. For dipole sizes larger th…
In the present note we propose a shift of the anomalous dimension function of the eigenfunctions of the BFKL equation with the NLO running coupling corrections. The calculated eigenvalue of the modified equation turns out to be conformal invariant and we discuss consequences of this result.
We resum the recently calculated second order kernel of the BFKL equation. That kernel can be viewed as the sum of a conformally invariant part and a running coupling part. The conformally invariant part leads to a corrected BFKL intercept as found earlier. The running coupling part of the kernel leads to a non-Regge term in the energy dependence of high energy hard scattering, as well as a $Q^2-$dependent intercept.
We use the BFKL equation to calculate the rate of deep inelastic scattering events containing two forward jets (adjacent to the proton remnants) at HERA. We compare the production of two forward jets with that of only one forward jet (the "Mueller" process). We obtain a stable prediction for this two to one jet ratio, which may serve as a measure of the BFKL vertex function.
We describe a formalism for solving the BFKL equation with a coupling that runs for momenta above a certain infrared cutoff. By suitably choosing matching conditions proper account is taken of the fact that the BFKL diffusion implies that the solution in the infrared (fixed coupling) regime depends upon the solution in the ultraviolet (running coupling) regime and vice versa. Expanding the BFKL kernel to a given order in the ratio of the transverse momenta allows arbitrary accuracy to be achieved.
Aspects of small x physics at HERA are discussed in honor of the work of Lev Lipatov, who influenced and motivated a whole generation of physicists to investigate this new region of phase space. The introduction of transverse momentum dependent parton densities is one of the essential aspects of small x and BFKL physics, and its extension to all flavors with application to various processes marks a new milestone.
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…
One of the well known effects of the asymptotic freedom is splitting of the leading-$\log$ BFKL pomeron into a series of isolated poles in complex angular momentum plane. Following our earlier works we explore the phenomenological consequences of the emerging BFKL-Regge factorized expansion for the small-$x$ charm ($F_2^c$) and beauty ($F_{2}^{b}$) structure functions of the proton. As we found earlier,the color dipole approach to the BFKL dynamics predicts uniquely decoupling of subleading hard BFKL exchanges from $F_2^c$ at moderately large $Q^{2}$. We predicted precocious BFKL asymptotics o…
We show that in the case of the BFKL pomeron with running coupling the diffusion pattern is strongly modified and is characterised by the sudden tunneling transition to the non-perturbative regime. We suggest that by using the b - expansion method one can suppress the non-perturbative Pomeron and isolate purely perturbative part of the gluon Green's function.
The HERA data on the proton structure function, $F_2(x,Q^2)$, at very small $x$ and $Q^2$ show the dramatic departure of the logarithmic slope, $\partial F_2/\partial\log Q^2$, from theoretical predictions based on the DGLAP evolution. We show that the running BFKL approach provides the quantitative explanation for the observed $x$ and/or $Q^2$ -dependence of $\partial F_2/\partial\log Q^2$.
Diffractive vector meson photoproduction accompanied by proton dissociation is studied for large momentum transfer. The process is described by the non-forward BFKL equation which we use to compare to data collected at the HERA collider.
This is a summary of the contributions on the next-to-leading order corrections to the BFKL equation which were presented to the `Small-x and Diffraction' working group at the 1998 Durham Workshop on HERA Physics.
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.
We analyse here in LO the physical properties of the Green function solution for the BFKL equation. We show that the solution obeys the orthonormality conditions in the physical region and fulfills the completeness requirements. The unintegrated gluon density is shown to consists of a set of few poles with parameters which could be determined by comparison with the DIS data of high precision.
The running BFKL equation gives rise to a series of moving poles in the complex j-plane. Corresponding eigenfunctions (color dipole cross sections) are the oscillating functions of the color dipole size $r$. The first nodes for all sub-leading solutions (color dipole cross sections) accumulate at $r_1\sim 0.1 fm$. Therefore the processes dominated by the dipole sizes $r\sim r_1$ are free of sub-leading BFKL corrections. A practically important example - the leptoproduction of charm. In a wide range of $Q^2$ the calculated $F_2^{cc}(x,Q^2)$ is exhausted by the leading BFKL pole and gives a perf…
The inclusive cross-section for production of a jet with a given transverse momentum off a heavy nucleus is derived in the BFKL framework with a running coupling on the basis of the bootstrap relation. The cross-section depends on the same three different coupling constants as the total cross-section unlike the cross-section for gluon production derived in the dipole approach.
We study the sea quark contribution to the BFKL kernel in the framework of Mueller's dipole model using the results of our earlier calculation. We first obtain the BFKL equation with the running coupling constant. We observe that the ``triumvirate'' structure of the running coupling found previously for non-linear evolution equations is preserved for the BFKL equation. In fact, we rederive the equation conjectured by Levin and by Braun, albeit for the unintegrated gluon distribution with a slightly unconventional normalization. We obtain the leading-N_f contribution to the NLO BFKL kernel in t…
We discuss the BFKL equation with a running gauge coupling and identify in its solutions the contributions originating from different transverse momentum scales. We show that for a running coupling constant the distribution of the gluons making up the BFKL Pomeron shifts to smaller transverse momenta so that the dominant part of Pomeron can have a nonperturbative origin. It is demonstrated how this soft physics enters into the BFKL solution through the boundary condition. We consider two kinematical regimes leading to different behaviour of the rapidity and transverse momentum dependence of th…
The $S$-matrix of diffractive scattering is diagonalized in terms of the colour dipole-dipole cross section. Recently, we have shown that the dipole cross section satisfies the generalized BFKL equation. In this paper we discuss the spectrum and solutions of our generalized BFKL equation with allowance for the finite gluon correlation radius $R_{c}$. The latter is introduced in the gauge invariant manner. We present estimates of the intercept of the pomeron and find the asymptotic form of the dipole cross section. We consider the difference between the BFKL and GLDAP evolutions and conclude th…
The running BFKL equation gives rise to a series of moving poles in the complex $j$-plane. The first nodes for all subleading solutions (color dipole cross sections) accumulate at $r_1\sim 0.1 fm$.Therefore the processes dominated by the dipole sizes $r\sim r_1 $ are free of subleading BFKL corrections. An example - the leptoproduction of charm. The calculated $F_2^{cc}$ is exhausted by the leading BFKL pole and gives a perfect description of the experimental data. The logarithmic slope of the subleading structure functions, $dF_2^{(n)}/d\log Q^2$, at small $Q^2$ is small compared to $dF_2^{(0…
I examine the form of the solution of the BFKL equation with running coupling relevant for deep inelastic scattering. The evolution of structure functions is precisely determined and well described by an effective coupling of the form 1/(beta_0(ln(Q^2/Lambda^2)+3.6(alpha_s(Q^2)ln(1/x))^1/2)) (until very small x). Corrections to the LO equation are relatively small, and the perturbative expansion is stable. Comparison to data via a global fit is very successful.
The production of forward jets of transverse momentum $p_T(j)\approx Q$ and large momentum fraction $x_{jet}\gg x$ probes the onset of BFKL dynamics at HERA. A full ${\cal O}(α_s^2)$ calculation of the inclusive forward jet cross section is presented and compared to the expected BFKL cross section. The kinematical region populated by these events and the scale dependence of the fixed order perturbative QCD cross sections are discussed.
To study diffractive photon production at HERA, we compute the projection of the virtual photon-photon impact-factor on the BFKL leading-order eigenfunctions E(n,nu) for non-zero transfer. This calculation supplements former ones performed for n=0. We provide an expression for n=+/-2 and check that all the other components are zero.
Diffractive heavy vector meson photoproduction accompanied by proton dissociation is studied for arbitrary momentum transfer. The process is described by the non-forward BFKL equation, for which a complete analytical solution is found, giving the scattering amplitude. The impact of non-leading corrections to the BFKL equation is also analysed. Results are compared to the HERA data on J/psi production.
We report an evaluation of subleading eigenvalues and eigenfunctions of the BFKL equation in the color dipole representation with a running gauge coupling. We present an expansion of the small-$x$ proton structure function $F_{2p}(x,Q^2)$ in terms of the rightmost BFKL-singularities. The BFKL-Regge phenomenology of DIS structure functions is developed which is shown to provide remarkably good description of the data on $F_{2p}(x,Q^2)$ from E665 to HERA.
Experimental results from HERA on diffractive vector meson production and their theoretical interpretation within microscopic QCD are reviewed with an emphasis on the BFKL color dipole and kt-factorization approaches.
Diffractive vector meson photoproduction accompanied by proton dissociation is studied for large momentum transfer. The process is described by the non-forward BFKL equation, for which an analytical solution is found for all conformal spins, giving the scattering amplitude. Results are compared to HERA data on rho production.
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 $(m_{V…
In this talk, I will argue that the HERA experimental data show that the typical parameter ($κ$) responsible for the value of the shadowing corrections (SC) in DIS is so large that the BFKL Pomeron is hidden under SC. The SC turn out to be large enough but mostly for the gluon structure function which is not well determined by the available experimental data and by the current theoretical procedure.
We discuss forward jet production data recently published by the H1 and ZEUS collaborations at HERA. We review how several Monte-Carlo models compare to the data. QCD calculations based on the BFKL formalism and on fixed NLO perturbation theory with and without resolved virtual photons are described.
An iterative solution best suited for a Monte Carlo implementation is presented for the non-forward BFKL equation in a generic color representation. We introduce running coupling effects compatible with bootstrap to all orders in perturbation theory. A numerical analysis is given showing a smooth transition from a hard to a soft pomeron when accounting for running effects.
I calculate the anomalous dimension governing the Q^2 evolution of the gluon (and structure functions) coming from the running coupling BFKL equation. This may be expressed in an exact analytic form, up to a small ultraviolet renormalon contribution, and hence the corresponding splitting function may be determined precisely. Rather surprisingly it is most efficient to expand the gluon distribution in powers of alpha_s(Q^2) rather than use the traditional expansion where all orders of alpha_s\ln(1/x) are kept on an equal footing. The anomalous dimension is very different from that obtained from…
The production of forward jets of transverse momentum $p_T(j)\approx Q$ and large momentum fraction $x_{jet}\gg x$ probes the onset of BFKL dynamics at HERA. A full ${\cal O}(α_s^2)$ calculation of the inclusive forward jet cross section is presented and compared to the expected BFKL cross section.
Processes of heavy quark production at HERA, TEVATRON and THERA energies are considered using the semihard ($k_T$ factorization) QCD approach with emphasis on the BFKL dynamics of gluon distributions.
We present approximations of varying degree of sophistication to the integral equations for the (gluon) structure functions of a hadron (``the partonic flux factor'') in a model valid in the Leading Log Approximation with a running coupling constant. The results are all of the BFKL-type, i.e. a power in the Bjorken variable x_B^{-λ} with the parameter λdetermined from the size α_0 of the ``effective'' running coupling \barα\equiv 3α_s/π= α_0/\log(k_{\perp}^2) and varying depending upon the treatment of the transverse momentum pole. We also consider the implications for the transverse momentum …
We introduce a coupled pair of evolution equations for the unintegrated gluon distribution and the sea quark distribution which incorporate both the resummed leading $\ln (1/x)$ BFKL contributions and the resummed leading $\ln (Q^2)$ DGLAP contributions. We solve these unified equations in the perturbative QCD domain. With only two physically motivated parameters we obtain an excellent description of the HERA $F_2$ data.
The proton structure function F_2(x,Q^2) for x < 0.01 and 0.045< Q^2 < 45 GeV^2, measured in the deep inelastic scattering at HERA, can be well described within the framework of the Color Glass Condensate.
