Running Coupling Corrections Research Articles

Running coupling and non-perturbative corrections for O(N) free energy and for disk capacitor

We reconsider the complete solution of the linear TBA equation describing the energy density of finite density states in the O(N) nonlinear sigma models by the Wiener-Hopf method. We keep all perturbative and non-perturbative contributions and introduce a running coupling in terms of which all asymptotic series appearing in the problem can be represented as pure power series without logs. We work out the first non-perturbative contribution in the O(3) case and show that (presumably because of the instanton corrections) resurgence theory fails in this example. Using the relation of the O(3) problem to the coaxial disks capacitor problem we work out the leading non-perturbative terms for the latter and show that (at least to this order) resurgence theory, in particular the median resummation prescription, gives the correct answer. We demonstrate this by comparing the Wiener-Hopf results to the high precision numerical solution of the original integral equation.

Helicity evolution at small x: the single-logarithmic contribution

We calculate single-logarithmic corrections to the small-x flavor-singlet helicity evolution equations derived recently [1–3] in the double-logarithmic approximation. The new single-logarithmic part of the evolution kernel sums up powers of αs ln(1/x), which are an important correction to the dominant powers of αs ln2(1/x) summed up by the double-logarithmic kernel from [1–3] at small values of Bjorken x and with αs the strong coupling constant. The single-logarithmic terms arise separately from either the longitudinal or transverse momentum integrals. Consequently, the evolution equations we derive employing the light-cone perturbation theory simultaneously include the small-x evolution kernel and the leading-order polarized DGLAP splitting functions. We further enhance the equations by calculating the running coupling corrections to the kernel.

Extended collinearly-improved Balitsky-Kovchegov evolution equation in target rapidity

An extended collinearly improved Balitsky-Kovchegov evolution equation in the target rapidity representation is derived by including the running coupling corrections during the expansion of the ``real'' $S$-matrix. We find that the running coupling brings important modifications to the evolution equation, as one can see that there are extra contributions to the evolution kernel once the running coupling is included. To identify the significance of the modifications, we numerically solve the evolution equation with and without the running coupling contributions during the $S$-matrix expansion. The numerical results show that the scattering amplitude is largely suppressed by the running coupling corrections, which indicate that one needs to consider the running coupling contributions during the derivation of the nonlinear evolution equation in the target rapidity representation.

HERA data and collinearly-improved BK dynamics

Within the framework of the dipole factorisation, we use a recent collinearly-improved version of the Balitsky-Kovchegov equation to fit the HERA data for inclusive deep inelastic scattering at small Bjorken x. The equation includes an all-order resummation of double and single transverse logarithms and running coupling corrections. Compared to similar equations previously proposed in the literature, this work makes a direct use of Bjorken x as the rapidity scale for the evolution variable. We obtain excellent fits for reasonable values for the four fit parameters. We find that the fit quality improves when including resummation effects and a physically-motivated initial condition. In particular, the resummation of the DGLAP-like single transverse logarithms has a sizeable impact and allows one to extend the fit up to relatively large photon virtuality Q2.

Forward dihadron back-to-back correlations in pA collisions

We study the disappearance of the away-side peak of the di-hadron correlation function in p+A vs p+p collisions at forward rapidities, when the scaterring process presents a manifest dilute-dense asymmetry. We improve the state-of-the-art description of this phenomenon in the framework of the Color Glass Condensate (CGC), for hadrons produced nearly back-to-back. In that case, the gluon content of the saturated nuclear target can be described with transverse-momentum-dependent gluon distributions, whose small-$x$ evolution we calculate numerically by solving the Balitsky-Kovchegov equation with running coupling corrections. We first show that our formalism provides a good description of the disappearance of the away-side azimuthal correlations in d+Au collisions observed at BNL Relativistic Heavy Ion Collider (RHIC) energies. Then, we predict the away-side peak of upcoming p+Au data at $~\sqrt[]{s}=200$ GeV to be suppressed by about a factor 2 with respect to p+p collisions, and we propose to study the rapidity dependence of that suppression as a complementary strong evidence of gluon saturation in experimental data.

Particle multiplicities in the central region of high-energy collisions from kT-factorization with running coupling corrections

Horowitz and Kovchegov have derived a kT-factorization formula for particle production at small x which includes running coupling corrections. We perform a first numerical analysis to confront the theory with data on the energy and centrality dependence of particle multiplicities at midrapidity in high-energy p+A (and A+A) collisions. Moreover, we point out a strikingly different dependence of the multiplicity per participant on Npart in p+Pb vs. Pb+Pb collisions at LHC energies, and argue that the observed behavior follows rather naturally from the convolution of the gluon distributions of an asymmetric vs. symmetric projectile and target.

Testing the running couplingkT-factorization formula for the inclusive gluon production

The inclusive gluon production at midrapidities is described in the color glass condensate formalism using the ${k}_{T}$-factorization formula, which was derived at fixed coupling constant considering the scattering of a dilute system of partons with a dense one. Recent analysis demonstrated that this approach provides a satisfactory description of the experimental data for the inclusive hadron production in $pp/pA/AA$ collisions. However, these studies are based on the fixed coupling ${k}_{T}$-factorization formula. This formula does not take into account the running coupling corrections, which are important to set the scales present in the cross section. In this paper we consider the running coupling corrected ${k}_{T}$-factorization formula conjectured some years ago and investigate the impact of the running coupling corrections on the observables. In particular, the pseudorapidity distributions and charged hadrons' multiplicity are calculated considering $pp$, $dAu/pPb$, and $AuAu/PbPb$ collisions at RHIC and LHC energies. We compare the corrected running coupling predictions with those obtained using the original ${k}_{T}$-factorization assuming a fixed coupling or a prescription for the inclusion of the running of the coupling. Considering the Kharzeev-Levin-Nardi unintegrated gluon distribution and a simplified model for the nuclear geometry, we demonstrate that the distinct predictions are similar for the pseudorapidity distributions in $pp/pA/AA$ collisions and for the charged hadrons' multiplicity in $pp/pA$ collisions. On the other hand, the running coupling corrected ${k}_{T}$-factorization formula predicts a smoother energy dependence for $dN/d\ensuremath{\eta}$ in $AA$ collisions.

Collinearly improved JIMWLK evolution in Langevin form

The high-energy evolution of Wilson line operators, which at leading order is described by the Balitsky-JIMWLK equations, receives large radiative corrections enhanced by single and double collinear logarithms at next-to-leading order and beyond. We propose a method for resumming such logarithmic corrections to all orders, at the level of the Langevin formulation of the JIMWLK equation. The ensuing, collinearly-improved Langevin equation features generalized Wilson line operators, which depend not only upon rapidity (the logarithm of the longitudinal momentum), but also upon the transverse size of the color neutral projectile to which the Wilson lines belong. This additional scale dependence is built up during the evolution, via the condition that the successive emissions of soft gluons be ordered in time. The presence of this transverse scale in the Langevin equation furthermore allows for the resummation of the one-loop running coupling corrections.

Rapidity evolution of Wilson lines at the next-to-leading order: Balitsky–JIMWLK equation at NLO

Abstract Scattering amplitudes of proton–Nucleus or Nucleus–Nucleus collisions at high energy are described by matrix elements of Wilson line operators — infinite gauge factors ordered along the straight line trajectories of the fast moving particles. The energy dependence of such amplitudes is described by the evolution equation of Wilson lines with respect to the rapidity parameter — the Balitsky–JIMWLK evolution equation. Most of the current phenomenology of high-energy and high-density QCD is based on the leading-order evolution equation with only running coupling corrections. I will present the derivation of the Balitsky–JIMWLK evolution equation at the next-to-leading order.

Energy evolution of the moments of the hadron distribution in QCD jets including NNLL resummation and NLO running-coupling corrections

The moments of the single inclusive momentum distribution of hadrons in QCD jets, are studied in the next-to-modified-leading-log approximation (NMLLA) including next-to-leading-order (NLO) corrections to the alpha_s strong coupling. The evolution equations are solved using a distorted Gaussian parametrisation, which successfully reproduces the spectrum of charged hadrons of jets measured in e+e- collisions. The energy dependencies of the maximum peak, multiplicity, width, kurtosis and skewness of the jet hadron distribution are computed analytically. Comparisons of all the existing jet data measured in e+e- collisions in the range sqrt(s)~2-200 GeV to the NMLLA+NLO* predictions allow one to extract a value of the QCD parameter Lambda_QCD, and associated two-loop coupling constant at the Z resonance alpha_s(m_Z^2)= 0.1195 +/- 0.0022, in excellent numerical agreement with the current world average obtained using other methods.

Heavy quark pair production in high-energy pA collisions: Quarkonium

Quarkonium production in high-energy proton (deuteron)–nucleus collisions is investigated in the color glass condensate framework. We employ the color evaporation model assuming that the quark pair produced from dense small-x gluons in the nuclear target bounds into a quarkonium outside the target. The unintegrated gluon distribution at small Bjorken x in the nuclear target is treated with the Balitsky–Kovchegov equation with running coupling corrections. For the gluons in the proton, we examine two possible descriptions, unintegrated gluon distribution and ordinary collinear gluon distribution. We present the transverse momentum spectrum and nuclear modification factor for J/ψ production at RHIC and LHC energies, and those for ϒ(1S) at LHC energy, and discuss the nuclear modification factor and the momentum broadening by changing the rapidity and the initial saturation scale.

Running coupling corrections to non-linear evolution for diffractive dissociation

We determine running coupling corrections to the kernel of the non-linear evolution equation for the cross section of single diffractive dissociation in high energy DIS. The running coupling kernel for diffractive evolution is found to be exactly the same as the kernel of the rcBK evolution equation.

Running coupling corrections to inclusive gluon production

We calculate running coupling corrections for the lowest-order gluon production cross section in high energy hadronic and nuclear scattering using the BLM scale-setting prescription. At leading order, there are three powers of fixed coupling; in our final answer, these three couplings are replaced by seven factors of running coupling: five in the numerator and two in the denominator, forming a ‘septumvirate’ of running couplings, analogous to the ‘triumvirate’ of running couplings found earlier for the small-x BFKL/BK/JIMWLK evolution equations. It is interesting to note that the two running couplings in the denominator of the ‘septumvirate’ run with complex-valued momentum scales, which are complex conjugates of each other, such that the production cross section is indeed real. We use our lowest-order result to conjecture how running coupling corrections may enter the full fixed-coupling kT-factorization formula for gluon production which includes nonlinear small-x evolution.

Exclusive processes in electron–ion collisions

The exclusive processes in electron–ion (eA) interactions are an important tool to investigate the QCD dynamics at high energies as they are in general driven by the gluon content of the target which is strongly subject to parton saturation effects. In this Letter we compute the cross sections for the exclusive vector meson production as well as the deeply virtual Compton scattering (DVCS) relying on the color dipole approach and considering the numerical solution of the Balitsky–Kovchegov equation including running coupling corrections (rcBK). The production cross sections obtained with the rcBK solution and bCGC parametrization are very similar, the former being slightly larger.

Running coupling corrections to high energy inclusive gluon production

We calculate running coupling corrections for the lowest-order gluon production cross section in high energy hadronic and nuclear scattering using the BLM scale-setting prescription. In the final answer for the cross section the three powers of fixed coupling are replaced by seven factors of running coupling, five in the numerator and two in the denominator, forming a ‘septumvirate’ of running couplings, analogous to the ‘triumvirate’ of running couplings found earlier for the small- x BFKL/BK/JIMWLK evolution equations. It is interesting to note that the two running couplings in the denominator of the ‘septumvirate’ run with complex-valued momentum scales, which are complex conjugates of each other, such that the production cross section is indeed real. We use our lowest-order result to conjecture how running coupling corrections may enter the full fixed-coupling k T -factorization formula for gluon production which includes nonlinear small- x evolution.

Non-linear QCD dynamics and exclusive production in ep collisions

The exclusive processes in electron-proton ($ep$) interactions are an important tool to investigate the QCD dynamics at high energies as they are in general driven by the gluon content of proton which is strongly subject to parton saturation effects. In this paper we compute the cross sections for the exclusive vector meson production as well as the deeply virtual Compton scattering (DVCS) relying on the color dipole approach and considering the numerical solution of the Balitsky-Kovchegov equation including running coupling corrections. We show that the small-$x$ evolution given by this evolution equation is able to describe the DESY-HERA data and is relevant for the physics of the exclusive observables in future electron-proton colliders and in photoproduction processes to be measured in coherent interactions at the LHC.

Testing non-linear evolution with running coupling corrections in ep and pp collisions

Perturbative QCD predicts that the growth of the gluon density at small x (high energies) should saturate, forming a Color Glass Condensate (CGC), which is described in mean field approximation by the Balitsky–Kovchegov (BK) equation. Recently, the next-to-leading order corrections for the BK equation were derived and a global fit of the inclusive ep HERA data was performed, resulting in a parameterization for the forward scattering amplitude. In this paper we compare this parameterization with the predictions of other phenomenological models and investigate the saturation physics in diffractive deep-inelastic electron–proton scattering and in the forward hadron production in pp collisions. Our results demonstrate that the running coupling BK solution is able to describe these observables.

Nonlinear QCD meets data: A global analysis of lepton-proton scattering with running coupling Balitsky-Kovchegov evolution

We perform a global fit to the structure function ${F}_{2}$ measured in lepton-proton experiments at small values of Bjorken-$x$, $x\ensuremath{\le}0.01$, for all experimentally available values of ${Q}^{2}$, $0.045\text{ }\text{ }{\mathrm{GeV}}^{2}\ensuremath{\le}{Q}^{2}\ensuremath{\le}800\text{ }\text{ }{\mathrm{GeV}}^{2}$. We show that the recent improvements resulting from the inclusion of running coupling corrections allow for a description of data in terms of nonlinear QCD evolution equations. In this approach ${F}_{2}$ is calculated within the dipole model with all Bjorken-$x$ dependence described by the running coupling Balitsky-Kovchegov equation. Two different initial conditions for the evolution are used, both yielding good fits to data with ${\ensuremath{\chi}}^{2}/\mathrm{d}.\mathrm{o}.\mathrm{f}.<1.1$. The proton longitudinal structure function ${F}_{L}$, not included in the fits, is also well described. Our analysis allows to perform a first principle extrapolation of the proton-dipole scattering amplitude once the initial condition has been fitted to presently available data. We provide predictions for ${F}_{2}$ and ${F}_{L}$ in the kinematical regions of interest for future colliders and ultra-high energy cosmic rays. A numerical implementation of our results down to $x={10}^{\ensuremath{-}12}$ is released as a computer code for public use.

High-energy scattering in the saturation regime including running coupling and rare fluctuation effects

The analytic form of the asymptotic behavior of the S matrix in the saturation regime including the running coupling is obtained. To get this result, we solve the Balitsky and Kovchegov-Weigert evolution equations in the saturation regime, which include running coupling corrections. We study also the effect of rare fluctuations on top of the running coupling. We find that the rare fluctuations are less important in the running coupling case as compared to the fixed coupling case.

Collinear singularities and running coupling corrections to gluon production in CGC

We analyze the structure of running coupling corrections to the gluon production cross section in the projectile-nucleus collisions calculated in the Color Glass Condensate (CGC) framework. We argue that for the gluon production cross section (and for gluon transverse momentum spectra and multiplicity) the inclusion of running coupling corrections brings in collinear singularities due to final state splittings completely unaffected by CGC resummations. Hence, despite the saturation/CGC dynamics, the gluon production cross section is not infrared-safe. As usual, regularizing the singularities requires an infrared cutoff Λ coll that defines a resolution scale for gluons. We specifically show that the cutoff enters the gluon production cross section in the argument of the strong coupling constant α s ( Λ coll 2 ) . We argue that for hadron production calculations one should be able to absorb the collinear divergence into a fragmentation function. The singular collinear terms in the gluon production cross section are shown not to contribute to the energy density ϵ of the produced matter, which is indeed an infrared-finite quantity.