In this contribution we describe in some detail different aspects of the construction of BFKL cross sections. We focus on several effects which are relevant at next-to-leading order. In particular, we describe QCD coherence in DIS final states, improvements of the collinear region in multi-Regge kinematics, inclusive jet production at next-to-leading order, and azimuthal angle decorrelations of Mueller-Navelet jets at hadron colliders.
The total $γ^*γ^*$ cross-section is derived in the Leading Order QCD dipole picture of BFKL dynamics, and compared with the one from 2-gluon exchange. The Double Leading Logarithm approximation of the DGLAP cross-section is found to be small in the phase space studied. Cross sections are calculated for realistic data samples at the $e^+e^-$ collider LEP and a future high energy linear collider. Next to Leading order corrections to the BFKL evolution have been determined phenomenologically, and are found to give very large corrections to the BFKL cross-section, leading to a reduced sensitivity …
We calculate inclusive dijet production cross section in high energy hadron collisions within the BFKL resummation formalism for the QCD Pomeron. We take into account the Pomerons which are adjacent to the hadrons. With these adjacent Pomerons we define a new object - the BFKL structure function of hadron - which enables one to calculate the inclusive dijet production for any rapidity intervals. We present predictions for the dijet K-factor and azimuthal angle decorrelation. Estimations for some NLO BFKL corrections are also given. (To appear in the Proceedings of the Workshop on Particle Theo…
Inclusive cross-sections for gluon production in collision of two mucleons with two nucleons are studied in the BFKL approach. Various contributions include emission from the pomerons attached to the participants, from the BFKL interactions in between these pomerons and from the intermediate BKP state. The last contribution may be observable provided the growth with energy of the pomeron contribution is tamed in accordance with unitarity.
We perform a numerical study of various improvements of the LO BFKL prediction for the gamma^* gamma^* total cross section in the kinematical region of the L3 detector at LEP. In particular, we study the effects of a massive charm quark and of different photon polarizations. The variation of the BFKL prediction under changes of alpha_s is investigated.
We obtain a simple analytic expression for the high energy $γ^* γ^*$ scattering cross section at the next-to-leading order in the logarithms-of-energy power counting. To this end we employ the eigenfunctions of the NLO BFKL equation constructed in our previous paper. We also construct the eigenfunctions of the NNLO BFKL kernel and obtain a general form of the solution for the NNLO BFKL equation, which confirms the ansatz proposed in our previous paper.
We calculate inclusive dijet production cross section in high energy hadron collisions within the BFKL resummation formalism for the QCD Pomeron. Unlike the previous calculations with the Pomeron developing only between tagging jets, we include also the Pomerons which are adjacent to the hadrons. With these adjacent Pomerons we define a new object --- the BFKL structure function of hadron --- which enables one to calculate the inclusive dijet production for any rapidity intervals. We present predictions for the K-factor and the azimuthal angle decorrelation in the inclusive dijet production fo…
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.
We study in the BFKL approach the total hadronic cross section for the collision of two virtual photons for energies in the range of LEP2 and in the range of future linear colliders. The BFKL resummation is done at the next-to-leading order in the BFKL Green's function; photon impact factors are taken instead at the leading order, but with the inclusion of the subleading terms required by invariance under changes of the renormalization scale and of the BFKL scale s_0. We compare our results with previous estimates based on a similar kind of approximation.
In the framework of the dipole picture of the BFKL pomeron we discuss a few possibilities of calculating the total $γ^{\star}γ^{\star}$ cross section of the virtual photons. We argue that the existing successful fits of the dipole picture formulae to the measured structure functions $F_2(Q^2,x)$ favor one of the possible extensions of the dipole picture of the BFKL pomeron formulated recently by one of us.
We present a numerical estimate of the $γ^* γ^*$ total cross section at the designed 500 GeV $e^+e^-$ Linear Collider, based upon the BFKL Pomeron. We find that the event rate is substantial provided electrons scattered under small angles can be detected, and a measurement of this cross section provides an excellent test of the BFKL Pomeron.
We present a numerical estimate of the $γ^* γ^*$ total cross section at LEP and at the designed $e^+e^-$ Next Linear Collider (NLC), based upon the BFKL Pomeron. We find for the linear collider that the event rate is substantial provided electrons scattered under small angles can be detected, and a measurement of this cross section provides an excellent test of the BFKL Pomeron. For LEP, although the number of events is substantially smaller, an initial study of this process is feasible.
In this work the double vector meson production in ultraperipheral heavy ion collisions is investigated within the BFKL approach. The integrated cross sections and event rates for the processes AA --> V1 V2 AA (V_i = rho, omega, phi, J/Psi, Upsilon) are computed and theoretical estimates for scattering on both light and heavy nuclei are given for energies of RHIC and LHC.
Double inclusive cross-sections for gluon production in collision of two nucleons with two nucleons are studied in the BFKL approach. Various contributions include emission from the pomerons attached to the participants, from the BFKL interactions between these pomerons and from the intermediate BKP state. The last contribution may be observable provided the growth with energy of the pomeron contribution is tamed in accordance with unitarity. Possibility of long-range azimuthal correlations due to the BKP state are discussed.
The behaviour of the pomeron propagator in the external nuclear field is studied in the (2+1)-dimensional QCD. It is shown that in the physically interesting case when the field does not vanish at large rapidities the propagator in the field vanishes much faster than in the vacuum, in agreement with the results found in the local Regge-Gribov model. However if the nuclear field vanishes at high rapidities the field does not change the behaviour of the pomeron propagator.
The DGLAP, BFKL, modified DGLAP and modified BFKL equations are constructed in a unified partonic framework. The antishadowing effect in the recombination process is emphasized, which leads to two different small $x$ behaviors of gluon distribution. In the meantime, the BFKL equation and modified DGLAP equation are viewed as the corrections of the initial gluon correlations to the evolution dynamics at twist-2 and twist-4, respectively. A partonic explanation of Regge theory for the BFKL dynamics and the relation of the BFKL dynamics with the helicity configuration of the initial gluons are pr…
The BFKL dynamics can be successfully tested at the future high energy $e^+e^-$ linear collider. The total $γ^*γ^*$ cross-section is calculated in the Leading Log QCD dipole picture of BFKL dynamics, and compared with the one from 2-gluon exchange. The rapidity dependence of the cross-section remains a powerful tool to increase the ratio between the BFKL and the 2-gluon cross-sections and is more sensitive to BFKL effects, even in the presence of higher order corrections. Other potential signals like diffractive $J/ Ψ$ and 'forward' jet productions will also be discussed.
We study the $γ^* γ^*$ total cross section in the NLA BFKL approach. We have extracted the NLO corrections to the photon impact factor from two recent papers of Balitsky and Chirilli and Chirilli and Kovchegov and used them to build several representations of the total cross section, equivalent within the NLA. We have combined these different representations with two among the most common methods for the optimization of a perturbative series, namely PMS and BLM, and compared their behavior with the energy with the only available experimental data, those from the LEP2 collider.
The dependence of the subprocess cross section for dijet production at fixed transverse momentum on the (large) rapidity separation Delta y of the dijets can be used to test for `BFKL physics', i.e. the presence of higher--order (alphas log(Delta y)^n contributions. Unfortunately in practice these subtle effects are masked by the additional, stronger dependence arising from the parton distributions. We propose a simple ratio test using two different collider energies n which the parton distribution dependence largely cancels. Such a ratio does distinguish between the `asymptotic' analytic BFKL…
We study the total cross section for the collision of two highly-virtual photons at large energies, taking into account the BFKL resummation of energy logarithms with full next-to-leading accuracy. A necessary ingredient of the calculation, the next-to-leading order impact factor for the photon to photon transition, has been calculated by Balitsky and Chirilli using an approach based on the operator expansion in Wilson lines. We extracted the result for the photon impact factor in the original BFKL calculation scheme comparing the expression for the photon-photon total cross section obtained i…
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 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 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…
The QCD dipole picture allows to build an unified theoretical description -based on BFKL dynamics- of the total and diffractive nucleon structure functions measured at HERA. We use a four parameter fit to describe the 1994 H1 proton structure function $F_{2}$ data in the low x, moderate $Q^{2}$ range. The diffractive dissociation processes are discussed within the same framework, and a 6 parameter fit of the 1994 H1 diffractive structure function data is performed. The BFKL dynamics can also be successfully tested at the $e^+e^-$ collider LEP and a future high energy linear collider. The total…
As noticed by Fadin, Kuraev and Lipatov in 1975 incorporation of asymptotic freedom into the BFKL equation splits the QCD pomeron into a series of isolated poles in complex angular momentum plane. Following our earlier work on the proton structure function we explore the phenomenological consequences of the emerging BFKL-Regge factorized expansion for the small-$x$ structure function of the pion $F_{2π}(x,Q^2)$. We calculate $F_{2π}$ in a small-$x$ region and find good agreement with the recent H1 determination of $F_{2π}(x,Q^2)$. We also present the BFKL-Regge factorization based evaluation o…
We investigate whether the Balitsky-Fadin-Kuraev-Lipatov (BFKL) low x evolution equation is capable to describe the energy dependence of the exclusive photo-production cross-section of vector mesons J/Psi and Upsilon on protons. Such cross-sections have been measured by both HERA experiments H1 and ZEUS in electron-proton collisions and by LHC experiments ALICE, CMS and LHCb in ultra-peripheral proton-proton and ultra-peripheral proton-lead collisions. Our approach provides a perturbative description of the rise with energy and relies only on a fit of the initial transverse momentum profile of…
In this paper we continue to pursue a goal of finding the effective theory for high energy interaction in QCD based on the colour dipole approach, for which the BFKL Pomeron Calculus gives a low energy limit. The two key problems, that we consider are the following: the probabilistic interpretation of the BFKL Pomeron Calculus and the possible scenario for the asymptotic behaviour of the scattering amplitude at high energy in QCD. In this paper we show that the generating functional approach is equivalent to the BFKL Pomeron Calculus and leads to a clear interpretation of this calculus as an a…
We investigate whether NLO BFKL evolution is capable to describe the energy dependence of the exclusive photo-production cross-section of vector mesons $J/Ψ$ and $Υ$ on protons. Our description is based on available NLO BFKL fits of the proton impact factor in inclusive DIS, which allow us to construct the necessary scattering amplitude at zero momentum transfer $t=0$. Assuming an exponential drop-off with $t$, this result allows us to calculate the exclusive photoproduction cross-section. Comparing our results with both HERA data (measured by H1 and ZEUS collaborations in $ep$ collision) and …
A study of the inclusive production of a pair of hadrons (a "dihadron" system), having high transverse momenta and separated by a large interval of rapidity, is presented. This process has much in common with the widely discussed Mueller--Navelet jet production and can be also used to access the BFKL dynamics at proton colliders. Large contributions enhanced by logarithms of energy can be resummed in perturbation theory within the BFKL formalism in the next-to-leading logarithmic accuracy. The experimental study of dihadron production would provide with an additional clear channel to test the …
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…
We present results on dijet and W+dijet production at hadron colliders obtained by supplementing the leading log BFKL resummation with energy and momentum conservation. For pure dijet production, the inclusion of the BFKL radiation in the energy conservation leads to a decrease in the parton flux sufficient to counter-act the expected exponential increase in the cross section obtained for the partonic cross section. Other BFKL signatures such as the dijet azimuthal angle decorrelation do still survive.
Starting from the multi-Regge effective action for high-energy scattering in QCD a $t$-channel approach can be developed which is similar to the approach by White based on general Regge arguments. The BFKL kernel of reggeized gluon interaction, contributions to the $2 \rightarrow 4 $ reggeized gluon vertex function and the one-loop correction to the BFKL kernel are considered. The conditions are discussed under which this approach can provide a simple estimante of the next-to-leading logarithmic corrections to the BFKL perturbative pomeron intercept.
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.
We study next-to-leading corrections to the integral kernel of the BFKL equation for high energy cross-sections in QCD and in supersymmetric gauge theories. The eigenvalue of the BFKL kernel is calculated in an analytic form as a function of the anomalous dimension γof the local gauge-invariant operators and their conformal spin n. For the case of an extended N=4 SUSY the kernel is significantly simplified. In particular, the terms non-analytic in n are canceled. We discuss the relation between the DGLAP and BFKL equations in the N=4 model.
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.
The Balitsky-Fadin-Kuraev-Lipatov (BFKL) approach for the cross sections at high energy $\sqrt s$ in perturbative QCD is briefly reviewed. The role of gluon Reggeization in the derivation of the BFKL equation and its compatibility with $s$-channel unitarity (``bootstrap'') are discussed.
The production in hadron-hadron collisions of jet pairs with large rapidity separation and comparable modest transverse momentum is, in principle, described by the perturbative QCD BFKL equation. The measurement of such jet pairs appears well suited to the LHC detectors, with their ability to detect forward jets. We present predictions for dijet cross sections and correlations obtained using a BFKL Monte Carlo which allows kinematic and other subleading effects to be incorporated. The enhanced phase space for gluon emission at the LHC makes the BFKL behavior somewhat easier to observe than at …
It is shown that the next-to-leading order (NLO) corrections to the QCD Pomeron intercept obtained from the BFKL equation, when evaluated in non-Abelian physical renormalization schemes with BLM optimal scale setting do not exhibit the serious problems encountered in the $\overline{\mbox{MS}}$-scheme. A striking feature of the NLO BFKL Pomeron intercept in the BLM approach is its rather weak dependence on the virtuality of the reggeized gluon. This remarkable property yields an important approximate conformal invariance. The results obtained provide an opportunity for applications of NLO BFKL …
Equation for the sum of BFKL pomeron fan diagrams is rederived by direct summation and solved numerically for rapidities $y\leq 50$. At high rapidities y>20 the resulting cross-sections for the scattering of a longitudinally polarized $q\bar q$ pair on the nucleus cease to depend on its transverse dimension and tend to a constant limit 0.1768 $R_A^2$, which corresponds to scattering of a colour dipole on a black disk. Thus the unitarity is restored and the singularity in the j plane is reduced to a simple pole at j=1.The nuclear structure function at small x behaves as $Q^2\ln(1/x)$. The found…
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…
We find one-loop correction to the integral kernel of the BFKL equation for the total cross section of the high energy scattering in QCD and calculate the next-to-leading contribution to anomalous dimensions of twist-2 operators near $j=1$.
In this letter we show that the behaviour of $F_{2}$, at very small $x_B$, agrees with the behaviour expected from the BFKL evolution equation, when screening corrections are included. We obtain a description which is consistent with the data, however, we require the screening corrections to be relatively large (about a quarter of the total DIS cross section). The relation between the screening corrections and the diffractive DIS cross section is discussed.
We investigate the high-energy scattering in the spontaneously broken Yang - Mills gauge theory in 2+1 space--time dimensions and present the exact solution of the leading $\ln s$ BFKL equation. The solution is constructed in terms of special functions using the earlier results of two of us (L.N.L. and L.S.). The analytic properties of the $t$-channel partial wave as functions of the angular momentum and momentum transfer have been studied. We find in the angular momentum plane: (i) a Regge pole whose trajectory has an intercept larger than 1 and (ii) a fixed cut with the rightmost singularity…
The Balitsky-Fadin-Kuraev-Lipatov (BFKL) evolution equation is known to be ``unstable'' with respect to fluctuations in gluon virtuality, transverse momentum and energy requiring to go beyond the leading order BFKL. Still, these instabilities point to fruitful improvements of our deep understanding of QCD. Recent applications to next-leading order and to saturation problems are outlined.
Using the solution of the BFKL equation including the leading and subleading conformal spin components, we show how the conformal invariance underlying the leading log 1/x expansion of perturbative QCD leads to elastic amplitudes described by two effective Pomeron singularities. One Pomeron is the well-known "hard" BFKL leading singularity while the new one appears from a shift of the higher conformal spin BFKL singularities from subleading to leading position. This new effective singularity is compatible with the "soft" Pomeron and thus, together with the "hard" Pomeron, meets at large Q^2 th…
A truncated BFKL series is studied and applied to hadronic processes. The proton-(anti)proton cross sections are described with good agreement with data and in a way consistent with the unitarity bound. The elastic scattering amplitude is calculated at momentum transfer $t$ different from zero, introducing two distinct ansatze for the proton impact factor. The elastic differential cross section is obtained at small $t$ approximation and compared with the data.
The high energy limit of scattering processes in QCD is, at least on the purely theoretical level, described by the BFKL equation. However, many phenomenological studies of BFKL fail miserably when confronted with data. In this talk we will briefly review the application of (LL) BFKL in phenomenology, and critically examine the application of LL eigenfunctions in the study of the NLL BFKL kernel. We then introduce a recently proposed iterative solution of the NLL BFKL equation that allows for a detailed study of physical properties of the BFKL evolution.
The colour dipole cross section is the principal quantity in the lightcone $s$-channel description of the diffractive scattering. Recently we have shown that the dipole cross section satisfies the generalized BFKL equation. In this paper we discuss properties and solutions of our generalized BFKL equation with allowance for the finite gluon correlation radius $R_{c}$. The latter is introduced in a gauge invariant manner. We present estimates of the intercept of the pomeron and find the asymptotic form of the dipole cross section.
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 …
This is the talk given at ``Gribov-75 Memorial Workshop on Quarks, Hadrons, and Strong Interactions", May 22-24, 2005, Budapest. Hungary. In this talk we discuss the BFKL Pomeron Calculus and its interrelation with thecolour dipole approach. The two key problems we consider are the probabilistic interpretation of the BFKL Pomeron Calculus and the possiblescenario for the asymptotic behaviour of the scattering amplitude at high energyin QCD.
The differences between two standard Monte Carlo models, Lepto and Ariadne, for deep inelastic scattering at small-x is analysed in detail. It is shown that the difference arises from a `unorthodox' suppression factor used in Ariadne which replaces the normal ratio of parton densities. This gives rise to a factor that qualitatively is similar to what one would expect from BFKL dynamics for some observables like the energy flow and forward jets but not for the 2+1 jet cross-section. It is also discussed how one could use the 2+1 jet cross-section as a probe for BFKL dynamics.
We discuss how the multi-Regge factorisation of QCD amplitudes can be used in the study of multi-jet processes at colliders. We describe how the next-to-leading logarithmic (NLL) BFKL evolution can be combined with energy and momentum conservation. By recalculating the quark contribution to the next-to-leading logarithmic corrections to the BFKL kernel we can study several properties of the NLL corrections. We demonstrate that in the standard analysis, the NLL corrections to a single gluon emission includes contributions from significantly more energetic quark--anti-quark configurations, somet…
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.
We discuss virtual photon scattering in the region dominated by BFKL exchange and report results for the cross sections at present and future e+ e- colliders.
We investigate the QCD resummation for the Higgs boson plus a high $P_T$ jet production with large rapidity separations in proton-proton collisions at the LHC. The relevant Balitsky-Fadin-Kuraev-Lipatov (BFKL) and Sudakov logs are identified and resummed. In particular, we apply recent developments of the transverse momentum dependent factorization formalism in the impact factors, which provides a systematic framework to incorporate both the BFKL and Sudakov resummations.
In this paper the exact analytical solution is found for the BFKL Pomeron calculus in QCD, in which all Pomeron loops have been included. This solution manifests the geometrical scaling behaviour and matches with the solution to the linear evolution equation on the line $r^2= 1/Q^2_s(x)$, where $r$ is the dipole size and $Q_s$ is the saturation momentum.
We demonstrate that the diffraction slope of the generalized BFKL pomeron amplitude has the conventional Regge growth $B(s) = B(0) + 2α_{\Pom}'\log(s)$. This proves that the generalized BFKL pomeron is described by the moving $j$-plane singularity. We give an estimate for the slope $α_{\Pom}'$ in terms of the correlation radius for the perturbative gluons.
