McLerran Venugopalan Research Articles

Coherent γ*-nucleus scattering and coherent nuclear states

In the context of a McLerran-Venugopalan (MV) model for a large nucleus, coherent scattering of a virtual photon on that nucleus is evaluated in the A− = 0 gauge, the gauge appropriate for the target nucleus. The evaluation of the scattering in A− = 0 gauge is very intricate compared to the usual A+ = 0 gauge evaluation natural for the scattering process, but has the advantage of directly giving the scattering in terms of a partonic description of the nucleus. In the limit where a tagged forward jet puts the dipole-nucleus scattering in the saturation regime the coherent reactions are equal to the inelastic reactions. In terms of the nuclear wave function the coherent reactions come from color singlet and zero total transverse momentum quark-antiquark pairs in the wave function and in the saturation regime the nuclear wave function is a coherent state for these pairs. In the saturation region half of all quarks (or antiquarks) come from zero momentum and color charge pairs.

Multiparticle production in proton–nucleus collisions beyond eikonal accuracy

We study the effects on multigluon production at mid-rapidity in the Color Glass Condensate of the non-eikonal corrections that stem from relaxing the shockwave approximation and giving the target a finite size. We extend previous works performed in the dilute-dilute approximation suitable for proton–proton collisions, to the dilute-dense one applicable to proton–nucleus. We employ the McLerran–Venugopalan model for the projectile averages. For the target averages, we use the Golec-Biernat–Wüsthoff model and restrict to the leading contributions in overlap area that allow a factorization of ensembles of Wilson lines into products of dipoles. We make the connection with the jet quenching formalism and compare with previous results in the literature, providing a parametrization of the so-called decorated dipoles. We show that the non-eikonal effects on single inclusive particle production, contrary to what happens in jet quenching, are only sizable, of a few percent, for modest energies sqrt{s_{NN}}le 100 GeV and central rapidities. On the other hand, we find that the effects on double inclusive gluon production are larger for the same kinematics. We show that, as found previously in the dilute-dilute situation, non-eikonal corrections break the accidental symmetry in the CGC, allowing for the existence of non-vanishing odd azimuthal harmonics.

Finite volume effects in the McLerran–Venugopalan initial condition for the JIMWLK equation

We revisit the numerical construction of the initial condition for the dipole amplitude from the McLerran–Venugopalan model in the context of the JIMWLK evolution equation. We observe large finite volume effects induced by the Poisson equation formulated on a torus. We show that the situation can be partially cured by introducing an infrared regularization. We propose a procedure which has negligible finite volume corrections. The control of the finite volume and finite lattice spacings effects is crucial when considering the numerical solutions of the JIMWLK evolution equation with the collinear improvement.

Multi-particle production in proton\u2013nucleus collisions in the color glass condensate

We compute multi-gluon production in the Color Glass Condensate approach in dilute-dense collisions, hbox {p}A, extending previous calculations up to four gluons. We include the contributions that are leading in the overlap area of the collision but keep all orders in the expansion in the number of colors. We develop a diagrammatic technique to write the numerous color contractions and exploit the symmetries to group the diagrams and simplify the expressions. To proceed further, we use the McLerran–Venugopalan and Golec–Biernat–Wüsthoff models for the projectile and target averages, respectively. We use a form of the Lipatov vertices that leads to the Wigner function approach for the projectile previously employed, that we generalise to take into account quantum correlations in the projectile wave function. We provide analytic expressions for integrated and differential two gluon cumulants and show a smooth dependence on the parameters defining the projectile and target Wigner function and dipole, respectively. For four gluon correlations we find that the second order four particle cumulant is negative, so a sensible second Fourier azimuthal coefficient can be defined. The effect of correlations in the projectile on this result results qualitatively and quantitatively large.

Non-perturbative renormalization of the average color charge and multi-point correlators of color charge from a non-Gaussian small-x action

The McLerran-Venugopalan (MV) model is a Gaussian effective theory of color charge fluctuations at small-x in the limit of large valence charge density, i.e., a large nucleus made of uncorrelated color charges. In this work, we explore the effects of the first non-trivial (even C-parity) non-Gaussian correction on the color charge density to the MV model (“quartic” term) in SU(2) and SU(3) color group in the non-perturbative regime. We compare our (numerical) non-perturbative results to (analytical) perturbative ones in the limit of small or large non-Gaussian fluctuations. The couplings in the non-Gaussian action, μ¯ for the quadratic and κ4 for the quartic term, need to be renormalized in order to match the two-point function in the Gaussian theory. We investigate three different choices for the renormalization of these couplings: i) κ4 is proportional to a power of μ¯; ii) κ4 is kept constant and iii) μ¯ is kept constant. We find that the first two choices lead to a scenario where the small-x action evolves towards a theory dominated by large non-Gaussian fluctuations, regardless of the system size, while the last one allows for controlling the deviations from the MV model.

Helicity-dependent extension of the McLerran–Venugopalan model

We construct a generalization of the McLerran–Venugopalan (MV) model including helicity effects for a longitudinally polarized target (a proton or a large nucleus). The extended MV model can serve as the initial condition for the helicity generalization of the JIMWLK evolution equation constructed in our previous paper, as well as for calculation of helicity-dependent observables in the quasi-classical approximation to QCD.

Entanglement entropy, entropy production and time evolution in high energy QCD

Working in the framework of the Color Glass Condensate effective theory of high energy QCD, we revisit the momentum space entanglement entropy of the soft gluons produced in high energy dilute-dense collisions. We extend the work of [1] by considering entropy produced in a single event. This entropy arises due to decoherence of eigenstates with different energies during the time evolution after the collisions with the target. We define it rigorously as the entanglement entropy of the produced system with the experimental apparatus. We compute the time dependent single event entropy in the limit of weak projectile field. Further we compute the entropy for the ensemble of events defined by the McLerran–Venugopalan model for the projectile wave function. Interestingly the entropy of the ensemble has a much weaker time dependence than the entropy in any single event. We attribute this feature to the so called monogamy of entanglement.

Fluctuations of the gluon distribution from the small- x effective action

The computation of observables in high energy QCD involves an average over stochastic semiclassical small-x gluon fields. The weight of various configurations is determined by the effective action. We introduce a method to study fluctuations of observables, functionals of the small-x fields, which does not explicitly involve dipoles. We integrate out those fluctuations of the gluon field under which a given observable is invariant. Thereby we obtain the effective potential for that observable describing its fluctuations about the saddle point. We determine explicitly the effective potential for the covariant gauge gluon distribution both for the McLerran-Venugopalan (MV) model and for a (non-local) Gaussian approximation for the small-x effective action. This provides insight into the correlation of fluctuations of the number of hard gluons versus their typical transverse momentum. We find that the spectral shape of the fluctuations of the gluon distribution is fundamentally different in the MV model, where there is a pile-up of gluons near the saturation scale, versus the solution of the small-x JIMWLK renormalization group, which generates essentially scale invariant fluctuations above the absorptive boundary set by the saturation scale.

Broken boost invariance in the Glasma via finite nuclei thickness

We simulate the creation and evolution of non-boost-invariant Glasma in the early stages of heavy ion collisions within the color glass condensate framework. This is accomplished by extending the McLerran–Venugopalan model to include a parameter for the Lorentz-contracted but finite width of the nucleus in the beam direction. We determine the rapidity profile of the Glasma energy density, which shows deviations from the boost-invariant result. Varying the parameters both broad and narrow profiles can be produced. We compare our results to experimental data from RHIC and find surprising agreement.

Photon from the annihilation process with CGC in the pA collision

We discuss the photon production in the pA collision in a framework of the color glass condensate (CGC) with expansion in terms of the proton color source ρp. We work in a regime where the color density ρA of the nucleus is large enough to justify the CGC treatment, while soft gluons in the proton could be dominant over quark components but do not yet belong to the CGC regime, so that we can still expand the amplitude in powers of ρp. The zeroth-order contribution to the photon production is known to appear from the Bremsstrahlung process and the first-order corrections consist of the Bremsstrahlung diagrams with pair produced quarks and the annihilation diagrams of quarks involving a gluon sourced by ρp. Because the final states are different there is no interference between these two processes. In this work we elucidate calculation procedures in details focusing on the annihilation diagrams only. Using the McLerran–Venugopalan model for the color average we numerically calculate the photon production rate and discuss functional forms that fit the numerical results.

The MV model of the color glass condensate for a finite number of sources including Coulomb interactions

We modify the McLerran–Venugopalan model to include only a finite number of sources of color charge. In the effective action for such a system of a finite number of sources, there is a point-like interaction and a Coulombic interaction. The point interaction generates the standard fluctuation term in the McLerran–Venugopalan model. The Coulomb interaction generates the charge screening originating from well known evolution in x. Such a model may be useful for computing angular harmonics of flow measured in high energy hadron collisions for small systems. In this paper we provide a basic formulation of the problem on a lattice.

Pictures of Particle Production in Proton--Nucleus Collisions

This work focuses on gluon(jet) production in dilute(proton)-dense(nucleus) collisions. Depending on the frame and gauge, gluon production can be viewed as a freeing of gluons coming from either the proton wave function or from the nucleus wave function. These (apparently) very different pictures must lead to the same result and the purpose of this paper is to see how that happens. The focus is on gluons having $k_\perp\sim Q_S$ or gluons in the scaling region $k_\perp/Q_S\gg 1$. In the McLerran-Venugopalan(MV) model with $k_\perp\sim Q_S$ we are able to derive gluon production in a way that (graphically) manifestly shows $k_\perp$-factorization in terms of the number density of gluons in the nuclear wave function. We presume that this picture, and $k_\perp$-factorization, continues to hold in the presence of small-$x$ evolution although we have not been able to explicitly verify this. Our result is in agreement with usual $k_\perp$-factorization where the gluon number density of the nucleus does not appear in an explicit way.

Collinearly-improved BK evolution meets the HERA data

In a previous publication, we have established a collinearly-improved version of the Balitsky–Kovchegov (BK) equation, which resums to all orders the radiative corrections enhanced by large double transverse logarithms. Here, we study the relevance of this equation as a tool for phenomenology, by confronting it to the HERA data. To that aim, we first improve the perturbative accuracy of our resummation, by including two classes of single-logarithmic corrections: those generated by the first non-singular terms in the DGLAP splitting functions and those expressing the one-loop running of the QCD coupling. The equation thus obtained includes all the next-to-leading order corrections to the BK equation which are enhanced by (single or double) collinear logarithms. We then use numerical solutions to this equation to fit the HERA data for the electron–proton reduced cross-section at small Bjorken x. We obtain good quality fits for physically acceptable initial conditions. Our best fit, which shows a good stability up to virtualities as large as Q2=400 GeV2 for the exchanged photon, uses as an initial condition the running-coupling version of the McLerran–Venugopalan model, with the QCD coupling running according to the smallest dipole prescription.

Recent Developments at Small-x

We present a short overview of saturation physics followed by a summary of the recent progress in our understanding of nonlinear small-x evolution. Topics include McLerran–Venugopalan model, Glauber–Mueller approximation, nonlinear BK/JIMWLK evolution equations, along with the running-coupling and NLO corrections to these equations.

Transverse Flow of Gluon Fields in Heavy Ion Collision

We describe the dynamics of initial gluon fields in heavy ion collision using a formal recursive solution of the Yang Mills equations and solving for the energy momentum tensor analytically in a boost-invariant setup. We generalize the original McLerran-Venugopalan (MV) model in order to allow for realistic nuclear profiles. This leads to a transverse flow of gluon fields. This flow pattern is inherited by the quark gluon plasma fluid after thermalization. Its most interesting aspect is a rapidity-odd flow component. We show that this rapidity-odd flow does not break boost invariance and that it emerges naturally from the Yang Mills equations. It leads to directed flow of particles and introduces angular momentum to the system.

Process dependent nucleark⊥broadening effect

We study the process dependent nuclear $k_\perp$ broadening effect by employing the transverse momentum dependent(TMD) factorization approach in combination with the Mclerran-Venugopalan(MV) model. More specifically, we investigate how the parton transverse momentum distributions are affected by the process dependent gauge links in cold nuclear matter. In particular, our analysis also applies to the polarized cases including the nuclear quark Boer-Mulders function and the linearly polarized gluon distribution. Our main focus is on the nuclear TMDs at intermediate or large x.

Initial Conditions with Flow from a McLerran Venugopalan model with Transverse Dynamics

Using a recursive solution of the Yang-Mills equation, we calculate analytic expressions for the gluon fields created in ultra-relativistic heavy ion collisions at small times τ. We have worked out explicit solutions for the fields and the energy momentum tensor up to 4th order in an expansion in τ. We generalize the McLerran-Venugopalan model to allow for a systematic treatment of averaged charge densities μ2 that vary as a function of transverse coordinates. This allows us to calculate radial, elliptic and directed flow of gluon fields. Our results can serve as initial conditions for hydrodynamic simulations of nuclear collisions that include initial flow.

Initial conditions for dipole evolution beyond the McLerran–Venugopalan model

We derive the scattering amplitude N(r) for a QCD dipole on a dense target in the semi-classical approximation. We include the first subleading correction in the target thickness arising from ~\rho^4 operators in the effective action for the large-x valence charges. Our result for N(r) can be matched to a phenomenological proton fit by Albacete et al over a broad range of dipole sizes r and provides a definite prediction for the A-dependence for heavy-ion targets. We find a suppression of N(r) for finite A for dipole sizes a few times smaller than the inverse saturation scale, corresponding to a suppression of the classical bremsstrahlung tail.

Geometric scaling behavior of the scattering amplitude for DIS with nuclei

The main question, that we answer in this paper, is whether the initial condition can influence on the geometric scaling behavior of the amplitude for DIS at high energy. We re-write the non-linear Balitsky–Kovchegov equation in the form which is useful for treating the interaction with nuclei. Using the simplified BFKL kernel, we find the analytical solution to this equation with the initial condition given by the McLerran–Venugopalan formula. This solution does not show the geometric scaling behavior of the amplitude deeply in the saturation region. On the other hand, the BFKL Pomeron calculus with the initial condition at x A = 1 / m R A given by the solution to Balitsky–Kovchegov equation, leads to the geometric scaling behavior. The McLerran–Venugopalan formula is the natural initial condition for the Color Glass Condensate (CGC) approach. Therefore, our result gives a possibility to check experimentally which approach: CGC or BFKL Pomeron calculus, is more satisfactory.

Gluon spectrum in the glasma from JIMWLK evolution

The JIMWLK equation with a “daughter dipole” running coupling is solved numerically, starting from an initial condition given by the McLerran–Venugopalan model. The resulting Wilson line configurations are then used to compute the spectrum of gluons comprising the glasma initial state of a high energy heavy ion collision. The development of a geometrical scaling region makes the spectrum of produced gluons harder. Thus the ratio of the mean gluon transverse momentum to the saturation scale grows with energy. Also the total gluon multiplicity increases with energy slightly faster than the saturation scale squared.