Saturation of gluon distribution is a consequence of the non-linear evolution equations of QCD. Saturation implies the existence of so called saturation momentum which is defined as a gluon density per unit rapidity per transverse area. At large energies for certain kinematical domains saturation momentum is the only scale for physical processes. As a consequence different observables exhibit geometrical scaling (GS). We discuss a number of examples of GS in different reactions.
We present a model based on the concept of saturation for small $Q^2$ and small $x$. With only three parameters we achieve a good description of all Deep Inelastic Scattering data below $x=0.01$. This includes a consistent treatment of charm and a successful extrapolation into the photoproduction regime. The same model leads to a roughly constant ratio of diffractive and inclusive cross section.
We propose a new description of inclusive diffraction in deep inelastic scattering (DIS). The diffractive structure functions are expressed in the dipole picture and contain heavy-quark contributions. The dipole scattering amplitude, a saturation model fitted on inclusive DIS data, features a saturation scale Q_s(x) larger than 1 GeV for x=10^{-5}. The q\bar{q}g contribution to the diffractive final state is modeled in such a way that both the large-Q^2 and small-beta limits are implemented. In the regime xpom<0.01 in which saturation is expected to be relevant, we obtain a parameter-free desc…
We successfully describe the HERA-data on diffractive deep inelastic scattering using a saturation model which has been applied in our earlier analysis of the inclusive $ep$-scattering data. No further parameters are needed. Saturation already turned out to be essential in describing the transition from large to small values of $Q^2$ in inclusive scattering. It is even more important for diffractive processes and naturally leads to a constant ratio of the diffractive versus inclusive cross sections. We present an extensive discussion of our results as well as detailed comparison with data.
We analyse the dissociation of a photon in diffractive deep inelastic scattering in the kinematic regime where the diffractive mass is much bigger than the photon virtuality. We consider the dominant q\bar{q}g component keeping track of the transverse momentum of the gluon which can be measured as a final-state jet. We show that the diffractive gluon-jet production cross-section is strongly sensitive to unitarity constraints. In particular, in a model with parton saturation, this cross-section is sensitive to the scale at which unitarity effects become important, the saturation scale. We argue…
In this paper we investigate the saturation physics in diffractive deep inelastic electron-ion scattering. We estimate the energy and nuclear dependence of the ratio $σ^{diff}/σ^{tot}$ and predict the $x_{\pom}$ and $β$ behavior of the nuclear diffractive structure function $F_{2,A}^{D(3)}(Q^2, β, x_{IP})$. Moreover, we analyze the ratio $R^{diff}_{A1,A2}(Q^2, β, x_{IP}) = F_{2,A1} ^{D(3)}/F_{2,A2} ^{D(3)}$, which probes the nuclear dependence of the structure of the Pomeron. We show that saturation physics predicts that approximately 37 % of the events observed at eRHIC should be diffractive.
We discuss emergence of geometrical scaling as a consequence of the nonlinear evolution equations of QCD, which generate a new dynamical scale, known as the saturation momentum: Qs. In the kinematical region where no other energy scales exist, particle spectra exhibit geometrical scaling (GS), i.e. they depend on the ratio pT=Qs, and the energy dependence enters solely through the energy dependence of the saturation momentum. We confront the hypothesis of GS in different systems with experimental data.
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.
The inelasticity in nucleus-nucleus collisions at high energies is calculated in the framework of geometrical multichain model. The very fast increase of the inelasticity is found as a result of a second-stage cascading process. The same behaviour is expected for all the models using the wounded nucleon idea. The simple formula for the number of wounded nucleons inside colliding nuclei does not need to be exact.
Analyzing recent ALICE data on inelastic pp scattering at the LHC energies we show that charged particle distributions exhibit geometrical scaling (GS). We show also that the inelastic cross-section is scaling as well and that in this case the quality of GS is better than for multiplicities. Moreover, exponent $λ$ characterizing the saturation scale is for the cross-section scaling compatible with the one found in deep inelastic ep scattering at HERA. Next, by parametrizing charged particles distributions by the Tsallis-like formula, we find a somewhat unexpected solution that still exhibits G…
Starting from a short range expansion of the inelastic overlap function, capable of describing quite well the elastic pp and $\bar{p}p$ scattering data, we obtain extensions to the inelastic channel, through unitarity and an impact parameter approach. Based on geometrical arguments we infer some characteristics of the elementary hadronic process and this allows an excellent description of the inclusive multiplicity distributions in $pp$ and $\bar{p}p$ collisions. With this approach we quantitatively correlate the violations of both geometrical and KNO scaling in an analytical way. The physical…
We investigate the role of unitarization effects in virtual photon-proton (γ^*p) interactions at small x. The q\bar{q}-fluctuation of the initial photon is separated into a small distance and a large distance component and a model for the unitarization of each component is proposed. The Born approximation for the small size component is calculated using QCD perturbation theory. Reggeon diagram technique is used in order to obtain a self-consistent scheme for both total γ^*p cross section and diffractive production. The model gives a good description of HERA data in the small-x region, with a s…
Consistency of the previously suggested color-dipole representation of deep-inelastic scattering (DIS_) and vector-meson production at low x with DGLAP evolution allows one to predict the exponent of the W^2 dependence of the saturation scale Lambda_sat^2(W^2) ~ (W^2)^C-2. One finds C_2^theory = 0.27 in agreement with the model-independent analysis of the experimental data from HERA on deep-inelastic electron scattering.
The solution to the BFKL equation grows like a power of center of mass energy, s, violating unitarity conditions at high energies. The growth of the cross section can be tamed by taking into account multiple pomeron exchanges. This is known as saturation and it is expressed in the Balitsky-Kovchegov equation. Conservation of energy should also slow down the growth of the cross section, and our aim in this work is to study the effects of enforcing energy conservation in DIS-events. Using the dipole picture, onium-onium collisions can be viewed, in the large N_c and the leading logarithmic limit…
Considering the high-energy limit of the QCD gluon distribution inside a nucleus, we calculate the proton-nucleus total inelastic cross section using a simplified dipole model. We show that, if gluon saturation occurs in the nuclear surface region, the total cross section of proton-nucleus collisions increases more rapidly as a function of the incident energy compared to that of a Glauber-type estimate. We discuss the implications of this with respect to recent ultra-high-energy cosmic ray experiments.
Gluon distributions of colliding hadrons saturate as a result of the non-linear evolution equations of QCD. As a consequence there exists the so called saturation momentum, which is related to the gluon density per unit rapidity per transverse area. When saturation momentum is the only scale for physical processes, different observables exhibit geometrical scaling (GS). We show a number of examples of GS and its violation in different reactions.
Saturation and geometrical scaling (GS) of gluon distributions are a consequence of the non-linear evolution equations of QCD. We argue that in pp GS holds for the inelastic cross-section rather than for the multiplicity distributions. We also discuss possible fluctuations of the proton saturation scale in pA collisions at the LHC.
We show how the AGBS model, originally developed for deep inelastic scattering applied to HERA data on the proton structure function, can also describe the RHIC data on single inclusive hadron yield for $d+Au$ and $p+p$ collisions through a new simultaneous fit. The single inclusive hadron production is modeled through the color glass condensate, which uses the quark(and gluon)--condensate amplitudes in momentum space. The AGBS model is also a momentum space model based on the asymptotic solutions of the BK equation, although a different definition of the Fourier transform is used. This aspect…
The geometric scaling property observed in the HERA data at small x, that the deep inelastic scattering (DIS) total cross-section is a function of only the variable Q^2*x^λ, has been argued to be a manifestation of the saturation regime of QCD and of the saturation scale Q^2_s(x) x^{-λ}. We show that this implies a similar scaling in the context of diffractive DIS and we observe, for several diffractive observables, that the experimental data from HERA confirm the expectations of this scaling.
We analyze geometrical scaling in deep inelastic scattering using experimental data from HERA ep collider. Parallel analyses are performed for energy and Bjorken-x data binnings. In particular, value of parameter λwhich governs x-dependence of saturation scale is found for both binnings: λ_En = 0.352 \pm 0.008 and λ_Bj = 0.302 \pm 0.004. We use the following method: λ_En is found as a value for which ratios σ^{W1}_{γ*p}(τ)/σ^{W2}_{γ*p}(τ) are closest to 1 (σ^{Wi}_{γ*p} is a photon-proton cross section with definite energy W_i and τ= Q^2 x^λ is a scaling variable); λ_Bj is found analogously usi…
Deep-inelastic scattering at low Bjorken x and elastic vector meson electroproduction are analyzed on the basis of the off-shell s-channel unitarity. We discuss behavior of the total cross-section of virtual photon-proton scattering and obtain, in particular, that the exponent in the power-like dependence of $σ^{tot}_{γ^* p}$ is related to the interaction radius of a constituent quark. The explicit mass dependence of the exponent in the power energy behavior of the vector meson electroproduction has been obtained. Angular distributions at large momentum transfers are considered. The energy dep…
Progresses towards the calculation and the understanding of NLO/NLL contributions to Deep Inelastic Scattering at low x with gluon saturation are being reviewed.
For gauge theories which admit a dual string description, we analyze deep inelastic scattering at strong 't Hooft coupling and high energy, in the vicinity of the unitarity limit. We discuss the onset of unitarity corrections and determine the saturation line which separates weak scattering from strong scattering in the parameter space of rapidity and photon virtuality. We discover that the approach towards unitarity proceeds through two different mechanisms, depending upon the photon virtuality Q^2 : single Pomeron exchange at relatively low Q^2 and, respectively, multiple graviton exchanges …
Deep-inelastic scattering at low x and elastic vector meson production are considered on the basis of the off-shell extension of the s-channel unitarity. We discuss behavior of the structure function F_2(x,Q^2) at low x and the total cross-section of virtual photon-proton scattering and obtain, in particular, the dependence sigma^{tot}_{gamma^* p}\sim (W^2)^{lambda(Q^2)} where exponent lambda(Q^2) is related to the interaction radius of a constituent quark. The energy dependence of the total cross-section of gamma^*gamma^*-interactions is calculated. The explicit mass dependence of the exponen…
We propose a method to assess the quality of geometrical scaling in Deep Inelastic Scattering and apply it to the combined HERA data on $γ^{\ast}p$ cross-section. Using two different approaches based on Bjorken $x$ binning and binning in $γ^{\ast}p$ scattering energy $W$, we show that geometrical scaling in variable $τ\sim Q^{2} x^λ$ works well up to Bjorken $x$'s 0.1. The corresponding value of exponent $λ$ is 0.32 -- 0.34.
The saturation model with DGLAP evolution is shown to give good description of the production of the charm and beauty quarks in deep inelastic scattering. The modifications of saturation properties caused by the presence of heavy quarks are also discussed.
We propose a new criterium for saturation of the density of partons both in nucleons and nuclei. It is applicable to any multiple scattering model which would be used to compute the number of strings exchanged in $ep$ and $eA$ collisions. The criterium is based on percolation of strings, and the onset of percolation is estimated from expectations coming from the study of heavy ion collisions at high energies. We interpret this onset as an indication of saturation of the density of partons in the wave function of the hadron. In order to produce quantitative results, a particular model fitted to…
We study the semi-inclusive hadron production in deep inelastic scattering at small $x$. A transverse momentum dependent factorization is found consistent with the results calculated in the small-$x$ approaches, such as the color-dipole framework and the color glass condensate, in the appropriate kinematic region at the lowest order. The transverse momentum dependent quark distribution can be studied in this process as a probe for the small-$x$ saturation physics. Especially, the ratio of quark distributions as a function of transverse momentum at different $x$ demonstrates strong dependence o…
Deep inelastic scattering at low x can be described by essentially only two fitted parameters. The interpretation of J/Psi photoproduction in terms of the gluon structure function is elaborated upon.
We review the recent finding of the two-plateau momentum distribution of sea quarks in deep inelastic scattering off nuclei in the saturation regime. The diffractive plateau which dominates for small $\bp$ measures precisely the momentum distribution of quarks in the beam photon, the rôle of the nucleus is simply to provide an opacity. The plateau for truly inelastic DIS exhibits a substantial nuclear broadening of the momentum distribution. Despite this nuclear broadening, the observed final state and initial state sea quark densities do coincide exactly. We emphasize how the saturated sea is…
At low $x \cong Q^2/W^2 << 1$, in deep inelastic scattering, the photon fluctuates into a $q \bar q$ vector state that interacts via two gluons with the proton. The energy dependence is determined by the saturation scale, in our approach given by $Λ^2_{sat} (W^2) \sim (W^2)^{C_2}$. Imposing DGLAP evolution, we find $C_2^{theory} = 0.27$ in agreement with the model-independent analysis of the HERA data. Different values of the exponent $C_2$ are correlated with different ratios of the longitudinal to transverse structure function. This stresses the need for experiments to separate longitudinal …
We analyze deep inelastic scattering at small Bjorken x, using the approximate conformal invariance of QCD at high energies. Hard pomeron exchanges are resummed eikonally, restoring unitarity at large values of the phase shift in the dual AdS geometry. At weak coupling this phase is imaginary, corresponding to a black disk in AdS. In this saturated regime, cross sections exhibit geometric scaling and have a simple universal form, which we test against available experimental data for the proton structure function F_2(x,Q^2). We predict, in particular, the dependence of the cross section on the …
We use recently proposed method of ratios to assess the quality of geometrical scaling in deep inelastic scattering for different forms of the saturation scale. We consider original form of geometrical scaling (motivated by the Balitski-Kovchegov (BK) equation with fixed coupling) studied in more detail in our previous paper, and four new hypotheses: phenomenologically motivated case with $Q^2$ dependent exponent $λ$ that governs small $x$ dependence of the saturation scale, two versions of scaling (running coupling 1 and 2) that follow from the BK equation with running coupling, and diffusive…
We derive the two-plateau momentum distribution of final state (FS) quarks produced in deep inelastic scattering (DIS) off nuclei in the saturation regime. The diffractive plateau which dominates for small transverse momentum p measures precisely the momentum distribution of quarks in the beam photon, the role of the nucleus is simply to provide an opacity. The plateau for truly inelastic DIS exhibits a substantial nuclear broadening of the FS momentum distribution. We discuss the relationship between the FS quark densities and the properly defined initial state (IS) nuclear quark densities.Th…
The Lund dipole model DIPSY is based on BFKL evolution and saturation. It can be applied to collisions between electrons, protons, and nuclei. In this talk I present some recent results for exclusive final states in inelastic collisions, a method to generate final states in diffractive excitation, and some results for collisions with nuclei.
We review QCD based descriptions of diffractive deep inelastic scattering emphasizing the role of models with parton saturation. These models provide natural explanation of such experimentally observed facts as the constant ratio of diffractive and total DIS cross sections as a function of the Bjorken variable x and Regge factorization of diffractive parton distributions.
Our approach to low-x deep inelastic scattering based on the ell-perp factorization of perturbative QCD (the color-dipole picture) yields parameter-free absolute predictions for J/Psi production. The connection of J/Psi production to the low-x saturation scale and to the gluon structure function is clarified.
A modification of the saturation model of deep inelastic scattering at small x which includes the Altarelli-Parisi (DGLAP) evolution is presented. Significant improvement of the description of the structure function F_2 at large Q^2 is achieved and a good description of diffractive data is preserved.
I discuss the saturation regime of QCD: the weak-coupling regime that describes large parton densities inside hadrons and the resulting high-energy behavior of scattering amplitudes. I briefly review past successes in the context of deep inelastic scattering at HERA and forward particle production at RHIC, and I present some recent progresses that significantly improved our understanding of high-energy scattering in QCD.
Recent developments in theory and phenomenology relevant to deep inelastic lepton scattering are reviewed, concentrating on the following topics: Predicted behaviour of non-singlet and polarized structure functions at small $x$; Theoretical studies of saturation and unitarity effects at small $x$ in quarkonium scattering; Renormalons and higher twist contributions; Next-to-leading-order calculations of jet cross sections; Forward jet production as a probe of small-$x$ dynamics.
In large-mass diffraction in deep inelastic scattering, we study the gluon jet close to the rapidity gap and show that its transverse momentum spectrum is peaked around a value which caracterizes the onset of S-matrix unitarity. We argue that such a measurement of diffractive jets could help understanding if at the present energies, unitarity comes as a consequence of saturation.
In this paper we discuss the example of one rapidity gap process: the inclusive cross sections of the leading neutrons in deep inelastic scattering with protons (DIS). The equations for this process are proposed and solved, giving the example of theoretical calculation of the survival probability for one rapidity gap processes. It turns out that the value of the survival probability is small and it decreases with energy
In Part I (in this journal) we argued that the structure function $F_2^{γp}(x,Q^2)$ in deep inelastic $ep$ scattering, regarded as a cross section for virtual $γ^*p$ scattering, has a saturated Froissart-bounded form behaving as $\ln^2 (1/x)$ at small $x$. This form provides an excellent fit to the low $x$ HERA data, including the very low $Q^2$ regions, and can be extrapolated reliably to small $x$ using the natural variable $\ln(1/x)$. We used our fit to derive quark distributions for values of $x$ down to $x=10^{-14}$. We use those distributions here to evaluate ultra-high energy (UHE) cros…
Following recent progresses in the understanding of high-energy scattering in QCD, we derive the first phenomenological consequences of Pomeron loops, in the context of both inclusive and diffractive deep inelastic scattering. In particular, we discuss diffusive scaling, a new scaling law that emerges for sufficiently high energies and up to very large values of $Q^2,$ well above the proton saturation momentum.
We use data on the deep inelastic structure function F_2 in order to constrain the cross-section for scattering a colour dipole off a proton. The data seem to prefer parameterisations which include saturation effects. That is they indicate that the strong rise with energy of the dipole cross-section, which holds for small dipoles, pertains only for r < r_s(x) where r_s(x) decreases monotonically as x decreases. Subsequent predicitions for the diffractive structure function F_2^{D(3)} also hint at saturation, although the data are not really sufficiently accurate.
The charm and beauty quark production in deep inelastic scattering at low values of the Bjorken variable x is considered in the DGLAP improved saturation model. After fitting parameters of the model to the structure function F_2, the heavy quark contributions Fc_2 and Fb_2 are predicted. A good description of the data is found. Predictions for the longitudinal structure function F_L and the diffractive structure function FD_2 are also presented.
Nuclear shadowing corrections to the structure functions of deep inelastic scattering on intermediate-mass nuclei are calculated at very small values of Bjorken x and small values of Q^2 (Q^2<5 Gev^2). The two-component approach developed in previous works of authors for a description of the nucleon structure functions of deep inelastic scattering is used. It is shown that the hard component of the nucleon structure functions that arises, in terms of the colour dipole model, from qq-pairs with a high transverse momentum, is almost not shadowed. It is shown that a change of the slope of the sha…
In the context of both inclusive and diffractive deep inelastic scattering, we derive the first phenomenological consequences of the inclusion of Pomeron loops in the QCD evolution equations towards high energy. We discuss the transition between the well-known geometric scaling regime and the new diffusive scaling, that emerges for sufficiently high energies and up to very large values of Q^2, well above the proton saturation momentum.
Saturation effects in diffractive deep inelastic scattering
University of California, Los Angeles（加州大学洛杉矶分校）；Mani L. Bhaumik Institute for Theoretical Physics, University of California, Los Angeles（加州大学洛杉矶分校马尼·L·巴乌米克理论物理研究所）；The University of Kansas（堪萨斯大学）；The University of Helsinki（赫尔辛基大学）
After introducing the concept of geometrical scaling (GS) on the example of deep inelastic ep scattering, we show that GS is also present in the p_T spectra measured by the LHC. We discuss simple phenomenological signatures of GS and its applications.
This is a short presentation of our talks given at eRHIC Workshop at the BNL. We give here a status report of our attempts to understand how gluon saturation will manifest itself in deep inelastic scattering with nuclei. This summary reflects our current understanding and shows directions of our research rather then a final answer to the question. Nevertheless, we are able to share with our reader our tentative answer to the question:``Why do we need to measure DIS with nuclei and why these data will be complementary to the information obtained from proton DIS".
We argue that the deep inelastic structure function $F_2^{γp}(x, Q^2)$, regarded as a cross section for virtual $γ^*p$ scattering, is hadronic in nature. This implies that its growth is limited by the Froissart bound at high hadronic energies, giving a $\ln^2 (1/x)$ bound on $F_2^{γp}$ as Bjorken $x\rightarrow 0$. The same bound holds for the individual quark distributions. In earlier work, we obtained a very accurate global fit to the combined HERA data on $F_2^{γp}$ using a fit function which respects the Froissart bound at small $x$, and is equivalent in its $x$ dependence to the function u…
We briefly report on the contribution of QCD-instantons to the phenomenon of saturation in deep-inelastic scattering (DIS) at small Bjorken-x. The explicitly known instanton gauge field serves as a concrete realization of an underlying non-perturbative saturation mechanism associated with strong classical fields. Two independent strategies were pursued, with consistent results: On the one hand, an approach starting from instanton-perturbation theory was considered and on the other hand, the scattering of a Wilson loop in an instanton gauge field background. In both cases, the conspicuous, intr…
Following recent predictions that the geometric scaling properties of deep inelastic scattering data in inclusive gamma*-p collisions are expected also in exclusive diffractive processes, we investigate the diffractive production of vector mesons. Using analytic results in the framework of the BK equation at non-zero momentum transfer, we extend to the non-forward amplitude a QCD-inspired forward saturation model including charm, following the theoretical predictions for the momentum transfer dependence of the saturation scale. We obtain a good fit to the available HERA data and make predictio…
This talk discusses some recent studies of gluon saturation in nuclei. We stress the connection between the initial condition in heavy ion collisions and observables in deep inelastic scattering (DIS). The dominant degree of freedom in the small x nuclear wavefunction is a nonperturbatively strong classical gluon field, which determines the initial condition for the glasma fields in the initial stages of a heavy ion collision. A correlator of Wilson lines from the same classical fields, known as the dipole cross section, can be used to compute many inclusive and exclusive observables in DIS.
A new geometrical scaling method with gluon saturation rapidity limit is proposed to study the gluon saturation feature of the central rapidity region of relativistic nuclear collisions. The net-baryon number is essentially transported by valence quarks that probe the saturation regime in the target by multiple scattering. We take advantage of the gluon saturation model with geometric scaling of the rapidity limit to investigate the net baryon distributions, nuclear stopping power and gluon saturation features in the SPS and RHIC energy regions. Predictions for net-baryon rapidity distribution…
We study the impact parameter dependence of inelasticity in the framework of an updated geometrical model for multiplicity distribution. A formula in which the inelasticity is related to the eikonal is obtained. This framework permits a calculation of the multiplicity distributions as well as the inelasticity once the eikonal function is given. Adopting a QCD inspired parametrization for the eikonal, in which the gluon-gluon contribution dominates at high energy and determines the asymptotic behavior of the cross sections, we find that the inelasticity decreases as collision energy is increase…
This paper investigates the configurational entropic content of hadron-nucleus collisions. Hadron multiplicities and Au nuclei are employed to compute the critical points of the configurational entropy as a function of the saturation scale in deep inelastic scatterings, in QCD. The results match phenomenological data to the precision of 0.39%.
I review the problem of parton saturation and its implications through three in principal different approaches, but somewhat related: saturation in a geometrical approach, QCD saturation through the Color Glass Condensate and shadowing in a Pomeron approach.
