Low-x QCD with CMS at the LHC

The physics of gluon saturation and non-linear evolution at small values of parton momentum fraction x in the proton and nucleus is discussed in the context of experimental results at HERA and RHIC. The rich physics potential of low-x QCD studies at the LHC is discussed and some measurements in pp, pA and AA collisions accessible with the Compact Muon Solenoid (CMS) experiment are presented.


Parton structure and evolution
Deep inelastic scattering (DIS) electron-proton ep (and electron-nucleus, eA) collisions provide a precise means to study the partonic structure of the proton (and nucleus). The inclusive DIS hadron cross section, d 2 σ/dx dQ 2 , is a function of the virtuality Q 2 of the exchanged gauge boson (i.e. its "resolving power"), and the Bjorken-x fraction of the total nucleon momentum carried by the struck parton. The differential cross section for the neutral-current (γ, Z exchange) process is written in terms of the target structure functions as where Y ± = 1 ± (1 − ) 2 is related to the inelasticity of the collision, and the structure functions F 2,3,L (x, Q 2 ) describe the density of quarks and gluons in the hadron 1 . F 2 is the dominant contribution to the cross section over most of phase space. One of the most significant discoveries at HERA is the strong growth of the inclusive DIS cross section for decreasing Bjorken-x at fixed Q 2 as well as for increasing Q 2 at fixed x (Fig. 1). The strong scaling violations evident at small x in Fig. 1 are indicative of the increasing gluon radiation. At small x, F 2 is sensitive to the sea quark distribution, driven by the gluon splitting, and since the gluon density xg(x, Q 2 ) can be thus determined (Fig. 2).
Once measured at an input scale Q 2 0 2 GeV 2 , the parton distribution functions (PDFs) at any other Q 2 are given by the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) evolution equations which govern the probability of parton branchings (gluon splitting, q, g−strahlung) in QCD [2]. The DGLAP parton evolution, however, only takes into account the Q 2 -dependence of the PDFs, resumming over single logarithms in α s ln(Q 2 ), "leading twist", but neglecting the 1/x terms. At large energies (small x), the probability of emitting an extra gluon increases as ∝ α s ln(1/x). In this regime, the evolution of parton densities proceeds over a large rapidity region, ∆y ∼ ln(1/x), and the finite transverse momenta of the partons become increasingly important. Thus, their appropriate description is in terms of k T -unintegrated PDFs, xg(x, k T ), described by the Balitski-Fadin-Kuraev-Lipatov (BFKL) equation which governs parton evolution in x at fixed Q 2 [3]. Hints of extra BFKL radiation have been recently found in the enhanced production of forward jets at HERA compared to DGLAP expectations [4,5].

Parton saturation and non-linear evolution at low x
As shown in Figures 1 and 2, the gluon density rises very fast for decreasing x. For x < 0.01, the growth in F 2 is well described by F 2 (x, Q 2 ) ∝ x −λ(Q 2 ) with λ ≈ 0.1 -0.3 logarithmically rising with Q 2 [6]. Eventually, at some small enough value of x one expects to enter a regime where the gluon density becomes so large that non-linear (gg fusion) effects become important, taming the growth of the parton densities. In such a high-gluon density regime three things are expected to occur: (i) the standard DGLAP and BFKL linear equations should no longer be applicable since they only account for single parton branchings (1 →2 processes) but not for non-linear (2 →1) gluon recombination; (ii) pQCD (collinear and k T ) factorization should break due to its (now invalid) assumption of incoherent parton scattering; and, as a result, (iii) standard pQCD calculations lead to a violation of unitarity even for Q 2 ≫ Λ 2 . Figure 3 schematically depicts the different domains of the parton density as a function of y = ln(1/x) and Q 2 . The transition to the regime of saturated PDFs is expected for small x values below an energy-dependent "saturation momentum", Q s , intrinsic to the (size of the) hadron. Since xg(x, Q 2 ) can be interpreted as the number of gluons with transverse area r 2 ∼ 1/Q 2 in the hadron wavefunction, an increase of Q 2 effectively diminishes the 'size' of each parton, partially compensating for the growth in their number (i.e. the higher Q 2 is, the smaller the x at which saturation sets in). Saturation effects are, thus, expected to occur when the size occupied by the partons becomes similar to the size of the hadron, πR 2 . This provides a definition for the saturation scale of an arbitrary hadron with A nucleons (i.e. with gluon density xG = A · xg): with λ ≈ 0.25 [8]. Equation (3) indicates that Q s grows with the number of nucleons, A, of the target, and the energy of the collision, √ s, or equivalently, the rapidity of the gluon y = ln(1/x). The mass number, A, dependence implies that, at equivalent energies, saturation effects will be enhanced by factors as large as A 1/3 ≈ 6 in heavy nuclear targets (A = 208 for Pb) compared to protons. In the last fifteen years, an effective field theory of QCD in the high-energy (high density, small x) limit has been developed -the Colour Glass Condensate (CGC) [9] -which describes the hadrons in terms of classical fields (saturated gluon wavefunctions) below the saturation scale Q s . The saturation momentum Q s introduces a (semi-)hard scale, Q s ≫ Λ, which not only serves as an infrared cut-off to unitarize the cross sections but allows weak-coupling perturbative calculations (α s (Q s ) ≪1) in a strong F µν colour field background. Hadronic and nuclear collisions are seen as collisions of classical wavefunctions which "resum" all gluon recombinations and multiple scatterings. The quantum evolution in the CGC approach is given by the JIMWLK [10] non-linear equations (or by their mean-field limit for N c → ∞, the Balitsky-Kovchegov equation [11]) which reduce to the standard BFKL kernel at higher x values.

Parton saturation: Experimental studies
The main source of information on the quark densities is obtained from measurements of (i) the structure functions F 2,3 in lepton-hadron scattering, and (ii) lepton pair (Drell-Yan) production in hadron-hadron collisions. The gluon densities, xG enter at LO directly in hadron-hadron scattering processes with (i) prompt photons and (ii) jets in the final state, as well as in the (difficult) measurement of (iii) the longitudinal DIS structure function F L (and also indirectly in F 2 through the derivative in Eq. (2)). In addition, (iv) heavy vector mesons (J/ψ, ϒ) from diffractive photoproduction processes 2 are a valuable probe of the gluon density since their cross sections are proportional to the square of xG [12,13]: with In hadronic collisions, one commonly measures (real and virtual) photons and jets at central rapidities (y = 0) where M the characteristic scale of the hard scattering. However, one can probe smaller x 2 values in the target by measuring the corresponding cross sections in the forward direction. Indeed, for a 2 → 2 parton scattering the minimum momentum fraction probed in a process with a particle of momentum p T produced at pseudo-rapidity η is [14] x min 2 = i.e. x min 2 decreases by a factor of ∼10 every 2 units of rapidity. Though Eq. (6) is a lower limit at the end of phase-space (in practice the x 2 values in parton-parton scatterings are at least 10 larger than x min 2 [14]), it provides the right estimate of the typical x 2 = (p T / √ s) e η values reached in non-linear 2 → 1 processes (in which the momentum is balanced by the gluon "medium") as described in parton saturation models [15]. Figure 4 summarizes the range of experimental processes sensitive to the gluon density and their approximate x coverage. Figure 5 shows the kinematical map in (x, Q 2 ) of the DIS, DY, direct γ and jet data used in the PDF fits. Results from HERA and the Tevatron cover a substantial range of the proton structure (10 −4 x 0.8, 1 Q 2 10 5 GeV 2 ) but the available measurements are much rarer in the case of nuclear targets (basically limited to fixed-target studies, 10 −2 x 0.8 and 1 Q 2 10 2 GeV 2 ). As a matter of fact, the nuclear parton distributions are basically unknown at low x (x < 0.01) where the only available measurements are fixed-target data in the state -in which the p (A) remains intact (or in a low excited state) and separated by a rapidity gap from the rest of final-state particles.
non-perturbative range (Q 2 < 1 GeV 2 ) dominated by Regge dynamics rather than quark/gluon degrees of freedom. An example of the current lack of knowledge of the nuclear densities at low x is presented in Fig. 6 where different available parametrizations of the ratio of Pb to proton gluon distributions, consistent with the available nDIS data at higher x, show differences as large as a factor of three [19].  Available measurements in the (x, Q 2 ) plane used for the determination of the proton [16] (top) and nuclear [17] (bottom) PDFs.

HERA results
Though the large majority of ep DIS data collected during the HERA-I phase are consistent with standard DGLAP predictions, more detailed and advanced experimental and theoretical results in the recent years have pointed to interesting hints of non-linear QCD effects in the data. Arguably, the strongest manifestation of such effects is given by the socalled "geometric scaling" property observed in inclusive σ DIS for x < 0.01 [20] as well as in various diffractive cross sections [21,22]. For inclusive DIS events, this feature manifests itself in a total cross section at small x (x < 0.01) which is only a function of τ = Q 2 /Q 2 s (x), instead of being a function of x and Q 2 /Q 2 s separately (Fig. 7). The saturation momentum follows Q s (x) = Q 0 (x/x 0 ) λ with parameters λ ∼ 0.3, Q 0 = 1 GeV, and x 0 ∼ 3·10 −4 . Interestingly, the scaling is valid up to very large values of τ, well above the saturation scale, in an "extended scaling" region with Q 2 [23,7]. The saturation formulation is suitable to describe not only   inclusive DIS, but also inclusive diffraction γ ⋆ p → X p. The very similar energy dependence of the inclusive diffractive and the total cross section in γ ⋆ p collisions at a given Q 2 is easily explained in the Golec-Biernat Wüsthoff model [20] but not in standard collinear factorization. Furthermore, geometric scaling has been also found in different diffractive DIS cross sections (inclusive, vector mesons, deeply-virtual Compton scattering DVCS) [21,22]. All these results suggest that the observed scalings are indeed manifestations of the saturation regime of QCD. Unfortunately, the value of Q s ∼ 1 GeV at HERA lies in the transition region between the soft and hard sectors and, therefore, non-perturbative effects obscure the obtention of clearcut experimental signatures.

RHIC results
The expectation, based on Eq. (3), of enhanced parton saturation effects in the nuclear wave functions accelerated at ultrarelativistic energies has been one of the primary physics motivations for the heavy-ion programme at RHIC 3 [9]. Further, the properties of the high-density matter produced in the final-state of AA interactions cannot be properly interpreted without having determined the influence of initial state modifications of the nuclear PDFs. In this context, after five years of operation, two main experimental observations at RHIC have been found consistent with CGC predictions: (i) the modest hadron multiplicities measured in AuAu reactions, and (ii) the suppressed hadron yield at forward rapidities in dAu collisions. The bulk, dN ch /dη| η=0 ≈ 700, multiplicities measured at mid-rapidity in central AuAu at √ s NN = 200 GeV are comparatively lower than the dN ch /dη| η=0 ≈ 1000 predictions [24] of "minijet" scenarios, soft Regge models, or extrapolations from an incoherent sum of proton-proton collisions, but they can be reproduced by approaches based on gluon saturation [8,25] which take into account a reduced parton flux in the nuclear targets, i.e. f a/A (x, Q 2 ) < A · f a/N (x, Q 2 ). In the CGC calculations, the final hadron multiplicities are assumed to be simply related to the initial number of released partons (local parton-hadron duality) which are depleted in the initial state compared to pp collisions due to non-linear gluon-gluon recombinations [8]. Simple assumptions related to the dependence of the saturation scale on energy and overlapping area of the colliding nuclei, describe the centrality and center-of-mass (c.m.) energy dependences of the bulk AA hadron production (Fig. 8).  [26] vs the predictions of the saturation approach [25].
The second manifestation of saturation-like effects in the RHIC data is the BRAHMS observation [27] of suppressed yields of moderately high-p T hadrons (p T ≈ 2 − 4 GeV/c) in dAu relative to pp collisions at η ≈ 3.2. Hadron production at  [27] versus DGLAP shadowing [15] and CGC [28] predictions. Fig. adapted from [15]. such small angles is sensitive to partons in the Au nucleus with x ≈ O(10 −3 ). The observed nuclear modification factor, R dAu ≈ 0.8, cannot be reproduced by pQCD calculations that include standard leading-twist shadowing of the nuclear PDFs [15,14] but can be described by CGC approaches [28] that parametrize the Au nucleus as a saturated gluon wavefunction. In addition, a recent analysis of the nuclear DIS F 2 data also confirms the existence of "geometrical scaling" for x <0.017 [25].

Low-x QCD at the LHC
The Large Hadron Collider (LHC) at CERN will provide pp, pA and AA collisions at √ s NN = 14, 8.8 and 5.5 TeV respectively with luminosities L ∼ 10 34 , 10 29 and 5·10 26 cm −2 s −1 . Such large c.m. energies and luminosities will allow detailed QCD studies at unprecedented low x values thanks to the copious production of hard probes (jets, quarkonia, prompt γ, Drell-Yan pairs, etc.). The expected advance in the study of low-x QCD phenomena will be specially substantial for nuclear systems since the saturation momentum, Eq. (3), Q 2 s ≈ 5 -10 GeV 2 , will be in the perturbative range [8], and the relevant x values, Eq. (6), will be 30-70 times lower than AA and pA reactions at RHIC: x ≈ 10 −3 (10 −5 ) at central (forward) rapidities for processes with Q 2 ∼10 GeV 2 (Fig. 10).

The CMS experiment
The CMS experiment is one of the two large general-purpose detectors being installed at the LHC. Its experimental capabilities are extremely well adapted for the study of low-x phenomena with proton and ion beams featuring: (i) Very large acceptance at midrapidity (|η| < 2.5, full φ) for charged and neutral hadrons as well as µ ± , e ± , and γ over a wide range of p T (the 4 T magnetic field results in the best track momentum resolution at LHC). (ii) Excellent muon reconstruction leading to the best mass resolution for J/ψ, and ϒ measurements at the LHC. (iii) Complete electromagnetic (EM) and hadronic (HAD) calorimetry for full jet reconstruction over |η| < 3 and ∆φ = 2π with a large statistical significance for single jet and jet+X (X = jet, γ, Z) channels. (iv) Unparalleled forward physics capabilities thanks to the forward hadronic calorimeter (HF, 3< |η| <5), TOTEM T1 (3.1 < |η| < 4.7) and T2 (5.5< |η| <6.6) trackers, and CASTOR (5.3< |η| <6.7) and zero-degree (ZDC, |η| >8.1 for neutrals) calorimeters. The combination of HF, TOTEM, CASTOR and ZDC (Fig. 11) makes of CMS the largest acceptance detector ever built at a hadron collider. The HF [29], located 11.2 m away on both sides of the interaction point (IP), is a steel plus quartz-fiberČerenkov calorimeter with 1200 channels (∆η × ∆φ ∼0.18×0.18, 1.65 m absorber corresponding to 10.3λ I ) sensitive to the deposited EM and HAD energy, allowing jet reconstruction at very forward rapidities. The T1 and T2 telescopes are part of the the TOTEM experiment [30] which shares the same IP as CMS and are mainly designed to measure charged tracks from diffractive dissociation processes. CASTOR [31] is an azimuthally symmetric electromagnetic/hadronic calorimeter situated at 14.37 m from the interaction point covering the same acceptance as T2. The calorime-ter is aČerenkov-light device, consisting of successive layers of tungsten absorber and fused silica (quartz) plates as active medium arranged in 2 EM (10 HAD) sections of about 22X 0 (10.3λ I ) radiation (interaction) lengths. The ZDC [32] is also a tungsten+quartz samplingČerenkov calorimeter with 5 EM (19X 0 , divided in x) and 4 HAD (5.6λ I , divided z) sections. It is located at 140 m from the CMS vertex at the end of the straight sections of the two LHC pipes containing the countercirculating beams. The purpose of the ZDC is to measure very forward going neutrons and photons with ∼10% (2 mm) energy (position) resolution.

Low-x QCD measurements in CMS
The following three measurements in pp, pA and AA collisions are being considered in CMS to look for signatures of high gluon density effects at low x.

Forward jets (pp, pA, AA)
The cross section for dijet production in the forward direction, "Müller-Navelet" jets [33], is a particularly sensitive measure of the small x parton dynamics in hadronic collisions [5]. The two HF calorimeters (3< |η| <5), specifically designed to measure energetic forward jets 4 , have an energy resolution of ∼20% for typical jets with E T ∼ 40 GeV (i.e. E = E T cosh η ≈ 1 TeV at η= 4). In the presence of low-x saturation effects, the forward-backward dijet production cross section (separated by ∆η ∼ 9 and, thus, measurable in each of the HFs) is expected to be suppressed by a factor of ∼2 in pp at 14 TeV (Fig. 12). A study is underway to determine the feasibility of such measurements in CMS.  4 The HF plays a prominent role in forward jet tagging for the vector-boson-fusion (qq → qqH) Higgs production channel.

QQ photoproduction (electromagnetic AA collisions)
High-energy diffractive production of heavy vector mesons (J/ψ, ϒ) proceeds through colourless two-gluon exchange (which couples to γ → QQ) and is thus a sensitive probe of the low-x gluon densities, see Eq. (4). Ultra-peripheral interactions (UPCs) of high-energy heavy ions generate strong electromagnetic fields which help constrain the low-x behaviour of xG via quarkonia produced in γ-nucleus collisions [34]. Lead beams at 2.75 TeV have Lorentz factors γ = 2930 leading to maximum (equivalent) photon energies ω max ≈ γ/R ∼ 100 GeV, and c.m. energies W max γ γ ≈ 160 GeV and W max γ A ≈ 1 TeV. From Eq. (5), the x values probed in γ A → J/ψ A processes at y = 2 can be as low as x ∼ 10 −5 . The CMS experiment can measure ϒ → µ + µ − produced in electromagnetic PbPb collisions tagged with neutrons detected in the ZDCs (as done at RHIC [34]). Figure 13 shows the expected dimuon invariant mass distributions predicted by STARLIGHT [35] within the CMS acceptance for an integrated PbPb luminosity of 0.5 nb −1 . An ϒ peak with ∼1200 counts is clearly seen on top of the µ + µ − continuum.  Fig. 13. Expected µ + µ − invariant mass from γ Pb → ϒ Pb ⋆ → µ + µ − Pb ⋆ and γ γ → µ + µ − as given by Starlight [35] for UPC PbPb collisions at √ s NN = 5.5 TeV in the CMS acceptance.

Forward Drell-Yan pairs (pp, pA, AA)
High-mass Drell-Yan pair production at the very forward rapidities covered by CASTOR and T2 (|η| ∼ 5 − 6) can probe the parton densities down to x ∼ 10 −6 . A study is underway in CMS to combine the CASTOR electromagnetic energy measurement together with the good position resolution of T2 for charged tracks, to trigger on and reconstruct the e + e − invariant mass in pp collisions at 14 TeV, and perform a two-dimensional study of xg in the M 2 and x plane.

Conclusion
We have reviewed the physics of non-linear QCD and high gluon densities at small fractional momenta x with emphasis on the existing data at HERA (proton) and RHIC (nucleus) which support the existence of a parton saturation regime (also known as colour-glass-condensate). The future perspectives at the LHC have been presented, including the promising capabilities of the forward CMS detectors to study the parton densities down to x ∼ 10 −6 with various hard probes (jets, quarkonia, Drell-Yan). The programme of investigating the dynamics of low-x QCD is not only appealing in its own right but it is an essential prerequisite for predicting a large variety of hadron-, photon-and neutrino-scattering cross sections at very high energies.