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Register a new userA matrix formulation for small-x singlet evolution
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We propose a matrix evolution equation in (x,kt)-space for flavour singlet, unintegrated quark and gluon densities, which generalizes DGLAP and BFKL equations in the relevant limits. The matrix evolution kernel is constructed so as to satisfy renormalization group constraints in both the ordered and antiordered regions of exchanged momenta kt, and incorporates the known NLO anomalous dimensions in the MSbar scheme as well as the NLx BFKL kernel. We provide a hard Pomeron exponent and effective eigenvalue functions that include the n_f-dependence, and give also the matrix of resummed DGLAP splitting functions. The results connect smoothly with those of the single-channel approach. The novel P_{qa} splitting functions show resummation effects delayed down to x=0.0001, while both P_{ga} entries show a shallow dip around x=0.001, similarly to the gluon-gluon single-channel results. We remark that the matrix formulation poses further constraints on the consistency of a BFKL framework with the MSbar scheme, which are satisfied at NLO, but marginally violated by small n_f/N_c^2-suppressed terms at NNLO.
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We rederive the small-$x$ evolution equations governing quark helicity distribution in a proton using solely an operator-based approach. In our previous works on the subject, the evolution equations were derived using a mix of diagrammatic and operator-based methods. In this work, we re-derive the double-logarithmic small-$x$ evolution equations for quark helicity in terms of the polarized Wilson lines, the operators consisting of light-cone Wilson lines with one or two non-eikonal local operator insertions which bring in helicity dependence. For the first time we give explicit and complete expressions for the quark and gluon polarized Wilson line operators, including insertions of both the gluon and quark sub-eikonal operators. We show that the double-logarithmic small-$x$ evolution of the polarized dipole amplitude operators, made out of regular light-cone Wilson lines along with the polarized ones constructed here, reproduces the equations derived in our earlier works. The method we present here can be used as a template for determining the small-$x$ asymptotics of any transverse momentum-dependent (TMD) quark (or gluon) parton distribution functions (PDFs), and is not limited to helicity.
It is well understood that the leading logarithmic approximation for the amplitudes of high energy processes is insufficient and that the next-to-leading logarithmic effects are very large and lead to instability of the solution. The resummation at low $x$, which includes kinematical constraints and other corrections leads to stable result. Using previously established resummation procedure we study in detail the preasymptotic effects which occur in the solution to the resummed BFKL equation when the energy is not very large. We find that in addition to the well known reduction of the intercept, which governs the energy dependence of the gluon Greens function, resummation leads to the delay of the onset of its small $x$ growth. Moreover the gluon Greens function develops a dip or a plateau in wide range of rapidities, which increases for large scales. The preasymptotic region in the gluon Greens function extends to about $8$ units in rapidity for the transverse scales of the order of $30-100 ; {rm GeV} $. To visualize the expected behavior of physical processes with two equal hard scales we calculate the cross section of the process $gamma^{*}+gamma^{*}to X$ to be probed at future very high-energy electron-positron colliders. We find that at $gamma^*gamma^*$ energies below $100 ; rm GeV$ the BFKL Pomeron leads to smaller value of the cross section than the Born approximation, and only starts to dominate at energies about $100 ; rm GeV$. This pattern is significantly different from the one which we find using LL approximation. We also analyze the transverse momentum contributions to the cross section for different virtualities of the photons and find that the dominant contributions to the integral over the transverse momenta comes from lower values than the the external scales in the process under consideration.
We present a method to include colour-suppressed effects in the Mueller dipole picture. The model consistently includes saturation effects both in the evolution of dipoles and in the interactions of dipoles with a target in a frame-independent way. When implemented in a Monte Carlo simulation together with our previous model of energy--momentum conservation and a simple dipole description of initial state protons and virtual photons, the model is able to reproduce to a satisfactory degree both the gamma*-p cross sections as measured at HERA as well as the total p-p cross section all the way from ISR energies to the Tevatron and beyond.
In this paper we present the results of numerical studies of the JIMWLK and BK equations with a particular emphasis on the universal scaling properties and phase space structure involved. The results are valid for near zero impact parameter in DIS. We demonstrate IR safety due to the occurrence of a rapidity dependent saturation scale Q_s(tau). Within the set of initial conditions chosen both JIMWLK and BK equations show remarkable agreement. We point out the crucial importance of running coupling corrections to obtain consistency in the UV. Despite the scale breaking induced by the running coupling we find that evolution drives correlators towards an asymptotic form with near scaling properties. We discuss asymptotic features of the evolution, such as the tau- and A-dependence of Q_s away from the initial condition.
We present a global fit to the structure function F_2 measured in lepton-proton experiments at small values of Bjorken-x, x< 0.01, for all experimentally available values of Q^2, 0.045< Q^2 < 800 GeV^2, using the Balitsky -Kovchegov equation including running coupling corrections. Using our fits to F_2, we reproduce available data for F_L and perform predictions, parameter-free and completely driven by small-x evolution, to the kinematic range relevant for the LHeC.
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