We study integrations over light-cone momenta in the high energy effective action of QCD. After a review of the effective action, we arrive on a regularization mechanism from matching with QCD diagrams, which we apply to a re-derivation of the reggeized gluon and of the BFKL-equation. We study consequences of the proposed regularization on the analytic structure of 2->3 and 2->4 production amplitudes in the Multi-Regge-Kinematics. We derive a certain part of the 1-loop corrections to the production vertex and demonstrate that they yield the on-set of corrections demanded by the Steinmann-relations. The Reggeon-Particle-2-Reggeon vertex is determined and applied to the construction of various signature configurations of the production amplitudes. We extend the regularization method to states of three and four reggeized gluons and propose a supplement to the effective Lagrangian. We derive vertices for the 1-3 and 2-4 reggeized gluon transition for the elastic amplitude and verify that signature conservation is obeyed. Integral equations for the state of three and four reggeized gluons are formulated and shown to agree with a result by Bartels and Wusthoff. In a second part we investigate the high-energy behavior of QCD for different surface topologies of color graphs. After a brief review of the planar limit (bootstrap and gluon reggeization) and of the cylinder topology (BFKL) we investigate the 3-3 scattering in the triple Regge limit which belongs to the pair-of-pants topology. We re-derive the triple Pomeron vertex function and show that it belongs to a specific set of graphs in color space which we identify as the analog of the Mandelstam diagram. We then extend the study to the high-energy behavior of N=4 SYM where we find a new class of color graphs not present in QCD.
Within the hard pomeron approach it is shown that the triple pomeron interaction gives no contribution to the cross-section for a projectile with large enough virtuality $g^{2}\ln Q^{2}\sim 1$. For small virtualities $g^{2}\ln Q^{2}<<1$ its contribution is essential.
In the hard pomeron theory with the number of colours Nc in the infinity limit the diffractive amplitude obtained in [3] is compared with the results found for Nc=3 in [1] and in the dipole approach in [5]. It is shown that the double pomeron exchange contribution can be substituted by an equivalent triple pomeron interaction term. After such a substitution the triple pomeron vertices in [1,3,5] essentially coincide. It is demonstrated that, in any form, the triple pomeron vertex is conformal invariant. It is also shown that higher order densities in the dipole approach do not involve 1 to k p…
Hadron diffraction dissociation is considered in the dipole Pomeron model where the Pomeron is represented by a double pole in the J-plane. We find that unitarity is satisfied without decoupling of the triple Pomeron vertex. The reaction bar{p}+p --> bar{p}+X is analysed.
The triple pomeron interaction is studied in the perturbative approach of BFKL-Bartels. At finite momentum transfers $\sqrt{-t}$ the contribution factorizes in the standard manner with a triple-pomeron vertex proportional to $1/\sqrt{-t}$. At $t=0$ the contribution is finite, although it grows faster with energy than for finite $t$ and does not factorize.
Misprints and numerical coefficients corrected, a bit of phenomenology and one figure added. The case for the linear evolution of the unitarized structure functions made stronger.
We consider proton diffraction dissociation in the dipole Pomeron model, where the Pomeron is represented by a double pole in the $J-$plane, and show that unitarity can be satisfied without decoupling of the triple Pomeron vertex. Differential and total diffractive cross sections for the reaction $\bar{p}+p \to \bar{p}+X$ are analyzed and reproduced in this model.
We estimate the size of the triple Pomeron vertex in perturbative QCD and compare with the phenomenological value extracted from Regge fits to experimental data. For simplicity, the results of the QCD analysis are taken in the large-N_c limit. We find that the perturbative triple Pomeron coupling is of the same order of magnitude as the observed one. We also estimate the size of the Pomeron self energy and its contribution to the renormalization of the Pomeron intercept. The effect is very small, in agreement with previous nonperturbative estimates.
In the effective action approach the imaginary part of the triple pomeron amplitude is calculated. The found dependence on the longitudinal momentum transfer $\e$ is found to separate as a simple factor $1/|\e|$. This result is used to calculate the high-mass diffraction on a hadron and double scattering cross-section off a composite target}
The value of the important triple-Pomeron coupling is estimated by an unorthodox procedure using the known diffractive parton distribution functions. The result is g_{3P}\simeq 0.2g_N, where g_N is the Pomeron-nucleon coupling. This is in excellent agreement with an independent determination, g_{3P}\simeq 0.2g_N, previously obtained by analysing the available data in the triple-Regge region with absorptive effects taken into account.
The amplitude for 4 interacting reggeized gluons is studied in the high-colour limit. The leading order amplitude is explicitly shown to reduce to a pair of reggeons, i.e to a single BFKL pomeron. The next-to-leading order diffractive amplitude is found to split into a double pomeron exchange and triple pomeron contribution. The obtained three-pomeron vertex is different from the originally proposed in [2].
Single and double inclusive cross-sections for gluon jet production from within the triple pomeron vertex are studied in the reggeized gluon technique. It is shown that to satisfy the AGK rules the vertex has to be fully symmetric in all four reggeized gluons which form the two final pomerons. The single inclusive cross-sections are found for different cuttings of the triple pomeron vertex. They sum into the expression obtained by Yu.Kovchegov and K.Tuchin in the colour dipole picture. The found double inclusive cross-sections satisfy the AGK rules.
We will explain how it is possible to link Regge theory with DGLAP evolution using a triple-pole pomeron model. We will first show that Regge theory can be used to constrain the initial condition for DGLAP evolution. We will then spell out a method to extract Regge couplings at high Q^2 using DGLAP evolution.
Three pomerons are known to couple via the gluon loop and interactions of this kind which are responsible for high-mass diffraction are described in terms of the gluonic structure function of the QCD pomeron. We show that the triple-pomeron coupling via the light quark loop associated with the ``valence'' in the pomeron is of the same strength as the purely gluonic coupling. The large $Q^2$ behavior of this new contribution is described by the DLLA evolution from the non-perturbative $f\bar f$ valence state of the pomeron. Numerical estimates of the high-mass diffraction structure functions ba…
We show that the triple pole pomeron model \cite{CMS} provides an initial condition for a DGLAP evolution \cite{DGLAP} that produces a fit to high $Q^2$ experimental DIS data. We obtain good $χ^2$ for initial scales down to 3 GeV$^2$. Values of the initial scale smaller than 1.45 GeV$^2$ are ruled out at the 90% confidence level.
We investigate, within N=4 SYM, the high energy behavior in the triple Regge limit of a six point correlator of R-currents. Using the leading logarithmic approximation, we sum all diagrams whose color lines fit onto the surface of a sphere. We find three distinct classes of graphs, and one of them contains the triple Pomeron vertex known from QCD. We present results which, within the AdS/CFT correspondence, can be compared with scattering amplitudes in the dual string theory.
We investigate the high energy behavior of QCD for different surface topologies of color graphs. After a brief review of the planar limit (bootstrap and gluon reggeization) and of the cylinder topology (BFKL) we investigate the 3 -> 3 scattering in the triple Regge limit which belongs to the pair-of-pants topology. We re-derive the triple Pomeron vertex function and show that it belongs to a specific set of graphs in color space which we identify as the analogue of the Mandelstam diagram.
Theoretical predictions for the behavior of $A_{LL}$ asymmetry determined by the pomeron-hadron vertex are done. Strong dependence of the asymmetry on the mass creating quarks and transfer momenta is shown.
It is shown that the longitudinal double spin asymmetry $A_{ll}$ in polarized diffractive $l p \to l p+Q \bar Q$ reaction depends strongly on the spin structure of quark-pomeron vertex. The SMC spin facilities may be one of the best place to perform such experiment.
We study integrations over light-cone momenta in the gauge invariant effective action of high energy QCD. A regularization mechanism which allows for the evaluation of the longitudinal integrations is presented. After a rederivation of the reggeized gluon and the BFKL-equation from the effective action, we study the 1-3 and 2-4 reggeized gluon transition vertex of QCD Reggeon field theory and discuss their connection with the usual triple Pomeron vertex of perturbative QCD.
We study the case when the triple-Pomeron vertex is assumed to have a vectorial form, that is, the amplitude of high-mass diffractive dissociation vanishes as $V \propto \vec q_t \cdot \vec e$ as $q_t \to 0$. We find that the available data in the triple-Reggeon region may be well described in such a `weak' coupling scenario, providing that absorptive effects are taken into account. We compare this weak (vector) coupling scenario with the strong and weak (scalar) coupling scenarios. Corresponding predictions are presented for an LHC energy of 14 TeV.
The lowest order pomeron loop is calculated for the leading conformal weight with full dependence of the triple pomeron vertex on intermediate conformal weights. The loop is found to be convergent. Its contribution to the pomeron Green function begins to dominate already at rapidities 10$÷$15. The pomeron pole renormalization is found to be quite small due to a rapid fall of the triple pomeron vertex with rising conformal weights.
The triple-Pomeron coupling lies at the heart of the predictions for high energy diffractive processes. We explain why the existing determinations, which use single-particle inclusive hadronic data a+b -> a+Y, underestimate the value of the bare coupling, due to the neglect of soft rescattering which populates the rapidity gaps. We describe how data for the process gamma+p -> J/psi +Y can be used to give a much more reliable estimate of the bare coupling. We use the existing, fragmentary, J/psi data from HERA to show that the triple-Pomeron coupling is probably about 3 times larger than the pr…
The four-pomeron vertex is studied in the perturbative QCD. Its dominating terms of the leading (zeroth and first) orders in the coupling constant and subdominant in the number of colors are constructed. The vertex consists of two terms, one with a derivative in rapidity $\pd_y$ and the other with the BFKL interaction between pomerons. The corresponding part of the action and equations of motion are found. The iterative solution of the latter is possible only for rapidities smaller than 2 and quite large coupling constant $α_s$, of the order or greater than unity, when the quadruple pomeron in…
In a previous publication, we have constructed a set of non-linear evolution equations for dipole scattering amplitudes in QCD at high energy, which extends the Balitsky-JIMWLK hierarchy by including the effects of fluctuations in the gluon number in the target wavefunction. In doing so, we have relied on the color dipole picture, valid in the limit where the number of colors is large, and we have made some further approximations on the relation between scattering amplitudes and dipole densities, which amount to neglecting the non-locality of the two-gluon exchanges. In this Letter, we relax t…
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.
We present a unified direct calculation of the triple-pomeron coupling $A_{3\Pom}(Q^{2})$ for diffractive real photoproduction ($Q^{2}=0$) and deep inelastic scattering at large $Q^{2}$ in the framework of the dipole approach to the generalized BFKL pomeron. The small phenomenological value of $A_{3\Pom}(0)\approx 0.16$\,GeV$^{2}$, which was a mystery, is related to the small correlation radius $R_{c}\approx 0.3$\,fm for the perturbative gluons. We confirm the early expectations of weak $Q^{2}$ dependence of the dimensionfull coupling $A_{3\Pom}(Q^{2})$ and predict that it rises by the factor …
We compute the triple pomeron vertex from the Wilson line formalism, including both planar and non-planar contributions, and get perfect agreement with the result obtained in the Extended Generalized Logarithmic Approximation based on reggeon calculus.
Using the recent solution of the triple Pomeron coupling in the QCD dipole picture as a closed string amplitude with six legs, its analytical form in terms of hypergeometric functions and numerical value are derived.
Effective field theory of interacting BFKL pomerons is investigated and Langevin equations for the theory, which arise after the introduction of additional auxiliary field, are obtained. The Langevin equations are considered for the case of interacting BFKL pomerons with both splitting and merging vertexes and for the interaction which includes additional "toy" four pomeron interaction vertex. In the latest case an analogy with the Regge field theory in zero dimensions (RFT-0) was used in order to obtain this "toy" four pomeron interaction vertex. The comparison between the Langevin equations …
We analyse the momentum space triple Pomeron vertex in perturbative QCD. In addition to the standard form of this vertex which is used in the context of total cross-sections at high energies and in the QCD reggeon field theory, there exists an alternative form which has to be used in the study of high-mass diffraction. We review and analyse the relation between these two versions. We discuss some implications for the BK-equation. In the second part of our paper we extend this analysis to the Pomeron-Odderon-Odderon vertex.
We study properties of the momentum space Triple Pomeron Vertex in perturbative QCD. Particular attention is given to the collinear limit where transverse momenta on one side of the vertex are much larger than on the other side. We also comment on the kernels in nonlinear evolution equations.
We compute the perturbative one-to-three Pomeron vertex in the colour glass condensate using the extended generalized leading logarithmic approximation in high energy QCD. The vertex is shown to be a conformal four-point function in two-dimensional impact parameter space. Our result can be used to compare different theoretical descriptions of the colour glass condensate.
Diffractive meson production at HERA offers interesting possibilities to investigate diffractive processes and thus to learn something about the properties of the pomeron. The most succesful phenomenological description of the pomeron so far assumes it to couple like a $C = +1$ isoscalar photon to single quarks. This coupling leads, however, to problems for exclusive diffractive reactions. We propose a new phenomenological pomeron vertex, which leads to very good fits to the known data, but avoids the problems of the old vertex.
The Regge limit in perturbative QCD for processes as DIS (low x) and photon or onia-particles scattering is considered. In the first part a phenomenological approach for the inclusion of the running coupling, preserving the gluon reggeization property, is analyzed. In particular the spectrum of the hard pomeron equation and the asymptotic form of the solution are used to study scattering processes with also inclusive jet production. The singlet part of the proton structure function is evolved towards the low x region according to the full equation. In the second part the problem of unitarity i…
We study the properties of the Triple Pomeron Vertex in the perturbative QCD using the twist expansion method. Such analysis allows us to find the momenta configurations preferred by the vertex. When the momentum transfer is zero, the dominant contribution in the limit when Nc goes to infty comes from anticollinear pole. This is in agreement with result obtained without expanding, but by direct averaging of the Triple Pomeron Vertex over angles. Resulting theta functions show that the anticollinear configuration is optimal for the vertex. In the finite Nc case the collinear term also contribut…
The amplitude for the collision of two hadrons with the lowest order pomeron loop is calculated. Numerical calculations show that the loop contribution to the amplitude begins to dominate the single pomeron exchange at rapidities 8-10. Full dependence of the triple-pomeron vertex on intermediate conformal weights is taken into account.
An effective non-local quantum field theory is constructed, which describes interaction of pomerons in the high-coloured QCD. The theory includes both splitting and merging triple pomeron vertexes and diagrams with pomeronic loops. The Schwinger-Dyson equations for the 'physical' pomeron are written. Conformal invariance allows to reduce the theory to the old-fashioned Gribov pomeron theory with an infinite number of pomerons, one of which is supercritical.
Recent results by the authors on proton diffractive dissociation (single, double and central) in the low-mass resonance region with emphasis on the LHC kinematics are reviewed and updated. Based on the previous ideas that the contribution of the inelastic proton-Pomeron vertex can be described by the proton structure function, the contribution of the inelastic Pomeron-Pomeron vertex appearing in central diffraction is now described by a Pomeron structure function.
The most important consequence of Pomeron being a pole is the factorization property. However, due to Pomeron intercept being greater than 1, the extrapolated single diffraction dissociation cross section based on a classical triple-Pomeron formula is too large leading to a potential unitarity violation at Tevatron energies. It is our desire here to point out that the "flavoring" of Pomeron plays the dominant role in resolving this apparent "paradox".
We derive the full analytic expression for the QCD eikonal coupling of a quark-antiquark state to the exchanged gluon-gluon state in the BFKL formalism. The formula is valid for all conformal spin configurations of the quark-antiquark and gluon-gluon states. In particular, a new selection rule on conformal spins characterizes the non-dominant BFKL components with intercept below the Pomeron in the conformal-invariant framework.
The small-$t$ behaviour of the deep inelastic diffractive dissociation cross section in the triple Regge region is investigated, using the BFKL approximation in perturbative QCD. We show that the cross section is finite at $t=0$, but the diffusion in $\ln{k_t^2}$ leads to a large contribution of small momenta at the triple Pomeron vertex. We study the dependence upon the total energy and the invariant mass. At $t=0$, there is a decoupling of the three BFKL singularities which is a consequence of the conservation of the conformal dimension. For large invariant masses, the four gluon state in th…
Dispersive corrections to the total cross section for high-energy scattering from a heavy nucleus are calculated using a matrix model, based on the triple-Pomeron behavior of diffractive scattering from a single nucleon, for the cross section operator connecting different states of the projectile nucleon . Energy-dependent effects due to the decrease in longitudinal momentum transfers and the opening of more channels with increasing energy are included. The three leading terms in an expansion in the number of inelastic transitions are evaluated and compared to exact results for the model in th…
It is shown that the dipole pomeron model of single diffraction dissociation -- contrary to the case of the supercritical pomeron -- is compatible with the inequality $σ^{SD}\leqσ^{tot}$, imposed by unitarity, provided the triple pomeron coupling satisfies certain conditions. With the adopted approximation the model considered as a parcticular solution of the triple pomeron decoupling problem. Explicit forms of such a coupling and a qualitative comparison with the experimental data on single DD are presented. The modified factorization properties of the model are also discussed.
Elements of the pomeron phenomenology within the Regge pole exchange picture are recalled. This includes discussion of the high energy behaviour of total cross-sections, the triple pomeron limit of the diffractive dissociation and the single particle distributions in the central region. The BFKL pomeron and QCD expectations for the small $x$ behaviour of the deep inelastic scattering structure functions are discussed. The dedicated measurements of the hadronic final state in deep inelastic scattering at small $x$ probing the QCD pomeron are described. The deep inelastic diffraction is also dis…
An interaction term is added to the QCD Lagrangian involving hadron and gluon fields. An effective vertex is calculated for such interactions through exchanges of reggeized gluons. This gives rise to an effective coupling for hadron-gluon elastic scattering in the t-channel, which is used in an inclusive hadron-hadron interaction from which the Pomeron intercept alpha(0) is calculated.
Diffractive excitation is usually described by the Good--Walker formalism for low masses, and by the triple-Regge formalism for high masses. In the Good-Walker formalism the cross section is determined by the fluctuations in the interaction. By taking the fluctuations in the BFKL ladder into account, it is possible to describe both low and high mass excitation in the Good-Walker formalism. In high energy pp collisions the fluctuations are strongly suppressed by saturation, which implies that pomeron exchange does not factorise between DIS and pp collisions. The Dipole Cascade Model reproduces …
Diffractive excitation is usually described by the Good--Walker formalism for low masses, and by the triple-Regge formalism for high masses. In the Good--Walker formalism the cross section is determined by the fluctuations in the interaction. In this paper we show that by taking the fluctuations in the BFKL ladder into account, it is possible to describe both low and high mass excitation by the Good--Walker mechanism. In high energy $pp$ collisions the fluctuations are strongly suppressed by saturation, which implies that pomeron exchange does not factorise between DIS and $pp$ collisions. The…
We show that a triple-Regge parametrization of inclusive single diffraction agrees with the data in the following two domains: (a) xi > 0.03 at all t, (b) |t| > 1 GeV^2 at all xi. Since the triple-Regge parametrization fails when applied to the full xi-t range of the total single-diffractive cross section, we conclude that damping occurs only at low-xi and low-|t|. We give a (``toy'') parametrization of the damping factor, D(xi), valid at low-|t|, which describes the diffractive differential cross-section (dsig/dt) data at the ISR and roughly accounts for the observed s-dependence of diffracti…
While the main features of elastic, diffractive and total cross sections are described well by Regge theory, the measured rise of the proton-(anti)proton single diffraction dissociation cross section with energy is considerably smaller than the theoretical prediction based on factorization and a constant triple-pomeron coupling. The observed energy dependence is obtained by renormalizing the pomeron flux "carried" by a nucleon to unity. Double diffraction and double pomeron exchange cross sections are reevaluated and compared to data, and a new interpretation of hard diffraction results emerge…
Onium-onium scattering at high energy is used to illustrate a dipole picture of high energy hard scattering in the large $N_c$ limit. Single and double BFKL pomeron exchanges are calculated in the leading logarithmic approximation. An expression is given for the triple pomeron coupling when one of the pomeron's momentum is zero while the other two have momentum transfer, t. This expression is explicit and could be evaluated numerically. It has a $(-t)^{-{1\over 2}}$ singularity at t=0.
In this talk we report on recent calculation of high energy baryon scattering amplitudes in QCD. Elastic baryon impact factors for two, three and four gluons are presented and their energy evolution is described that incorporates unitarity corrections. We find that the baryon couples directly to the BFKL Pomeron, to the BKP odderon and to a new state, a three-gluon BKP Pomeron. The new state may decay into four gluons with a new 3 -> 4 transition vertex. This vertex defines the transition amplitude of the three-gluon BKP Pomeron state into two BFKL Pomerons.
We show that the latest HERA measurements of F2 are compatible with the existence of a triple pole at J=1. Such a structure also accounts for soft forward hadronic amplitudes, so that the introduction of a new singularity is not necessary to describe DIS data.
High-mass diffractive production of protons on the deuteron target is studied in the perturbative QCD in the BFKL approach. Leading order rearrangement contribution and the standard triple pomeron (the impulse approximation) are studied. In the perturbative limit $α_s\to 0$ the rearrangement contribution dominates. Numerical estimates at realistic values of $α_s$ and energies strongly depend on assumptions made about the behavior of the pomeron attached to the proton due to unitarization. They indicate that irrespective of these assumptions in the realistic situation the rearrangement and trip…
It has long been believed that the Pomeron, which has been successful in phenomenological fits to high energy scattering data, is associated with gluonic exchange. By determining the Regge-nucleon vertex in terms of previously determined glueball-quark coupling we show that the Pomeron might be related to the Regge trajectory defined by a light scalar glueball/sigma system and a tensor glueball, which involves complicated nonperturbative QCD. We predict a tensor glueball at 2.8 GeV.
We investigate consequences of the Möbius invariance of the BFKL Hamiltonian and of the triple Pomeron vertex. In particular, we show that the triple Pomeron vertex in QCD, when restricted to the large $N_c$ limit and to the space of Möbius functions, simplifies and reduces to the vertex used in the Balitsky-Kovchegov (BK) equation. As a result, the BK equation for the dipole density appears as a special case of the nonlinear evolution equation which sums the fan diagrams for BFKL Green's functions in the Möbius representation. We also calculate the corrections $O(1/(N_c^2-1)$ to the triple Po…
The Reggeon field theory with zero transverse dimensions is studied in the Hamiltonian formulation for both sub-and supercritical pomeron. Mathematical aspects of the model, in particular the scalar products in the space of quantum states, are discussed. Relation to reaction-diffusion processes is derived in absence of pomeron merging. Numerical calculations for different parameters of the models, $α(0)-1=μ$ and the triple pomeron coupling constant $λ$, show that the triple pomeron interaction always makes amplitudes fall with rapidity irrespective of the value of the intercept. The smaller th…
We give an overview of the calculation of the forward jet vertex associated to a rapidity gap (coupling of a hard pomeron to a jet) in the Balitsky-Fadin-Kuraev-Lipatov (BFKL) formalism at next-to-leading order (NLO). This result allows, together with the NLO non-forward gluon Green function, to perform NLO studies of jet production in diffractive events (Mueller-Tang dijets, as a well-known example)
High energy inclusive hadron production in the central kinematical region is analyzed within the models of unitarized pomeron. It is shown that the sum of multipomeron exchanges with intercept $α_P(0)>1$ reproduce qualitatively contribution of the triple pole (at $t=0$) pomeron to inclusive cross section. Basing on this analogy we then suggest a general form of unitarized pomeron contributions (in particular the dipole or tripole pomeron) to inclusive cross section. They lead to a parabolic form of the rapidity distribution giving $<n>\propto \ln^3s$ (tripole) or $<n>\propto \ln^2s$ (dipole). …
