Nuclear Parton Distribution Functions After the First Decade of LHC Data

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Nuclear Parton Distribution Functions After the First Decade of LHC Data © 2024 by the author(s) Published version Klasen, Michael; Paukkunen, Hannu Klasen, M., & Paukkunen, H. (2024). Nuclear Parton Distribution Functions After the First Decade of LHC Data. Annual Review of Nuclear and Particle Science, 74, 49-87. https://doi.org/10.1146/annurev-nucl-102122-022747 2024 Downloaded from www.annualreviews.org. Guest (guest) IP: 130.234.241.32 On: Mon, 21 Oct 2024 05:55:46 NS74_Art03_Klasen ARjats.cls August 26, 2024 9:52 Annual Review of Nuclear and Particle Science Nuclear Parton Distribution Functions After the First Decade of LHC Data Michael Klasen1and Hannu Paukkunen2,3 1Institute for Theoretical Physics, University of Münster, Münster, Germany; email: [email protected] 2Department of Physics, University of Jyväskylä, Jyväskylä, Finland; email: [email protected] 3Helsinki Institute of Physics, Helsinki, Finland Annu. Rev. Nucl. Part. Sci. 2024. 74:49–87 First published as a Review in Advance on April 26, 2024 The Annual Review of Nuclear and Particle Science is online at nucl.annualreviews.org https://doi.org/10.1146/annurev-nucl-102122- 022747 Copyright © 2024 by the author(s). This work is licensed under a Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. See credit lines of images or other third-party material in this article for license information. Keywords quantum chromodynamics, nuclear structure, parton distribution functions, collider physics, future developments Abstract We present a review of the conceptual basis, current knowledge, and recent progress regarding global analysis of nuclear parton distribution functions (PDFs). After introducing the theoretical foundations and methodological approaches for the extraction of nuclear PDFs from experimental data, we discuss how different measurements in fixed-target and collider experiments provide increasingly precise constraints on various aspects of nuclear PDFs, including shadowing, antishadowing, the EMC effect, Fermi motion, flavor separation,deuteron binding,and target-mass and other higher-twist effects. Particularemphasis isgiven tomeasurements carried out in proton–lead collisions at the Large Hadron Collider, which have revolutionized the global analysis during the past decade. These measurements include electroweak boson, jet, light hadron, and heavy flavor observables. Finally, we outline the expected impact of the future Electron Ion Collider and discuss the role and interplay of nuclear PDFs with other branches of nuclear, particle, and astroparticle physics. 49 Downloaded from www.annualreviews.org. Guest (guest) IP: 130.234.241.32 On: Mon, 21 Oct 2024 05:55:46 NS74_Art03_Klasen ARjats.cls August 26, 2024 9:52 Contents 1. INTRODUCTION........................................................... 50 2. THEORETICALFOUNDATIONS........................................... 53 2.1. Factorization in the Operator Product Expansion and the Quantum Chromodynamics–Improved Parton Model. ... ... ... ... ... ... ... .. . .. . .. . .. 53 2.2. Target-MassCorrections.................................................. 55 2.3. Higher-TwistCorrections................................................. 56 3. GLOBAL ANALYSES OF NUCLEAR PARTON DISTRIBUTIONFUNCTIONS.............................................. 56 3.1. ThenCTEQFramework.................................................. 57 3.2. TheEPPSFramework.................................................... 58 3.3. ThenNNPDFFramework................................................ 59 3.4. Next-to-Next-to-Leading-Order and Model-Dependent Approaches .. ... ... 60 3.5. Overview of Global Nuclear Parton Distribution Function Analyses .. ... .. . . 60 4. FIXED-TARGETDATA....................................................... 63 4.1. Early Deep Inelastic Scattering Data and Constraints on Quarks... ... ... ... . 63 4.2. Deep Inelastic Scattering at High xand Nuclear Effects in the Deuteron.. . .. 65 4.3. The Drell–Yan Process and Constraints on Antiquarks . . .. . .. . .. . .. . .. . .. . .. 66 4.4. Neutrino Deep Inelastic Scattering Data and Flavor Separation. .. . .. . .. . .. . . 67 5. COLLIDERDATA............................................................ 69 5.1. ElectroweakBosons....................................................... 69 5.2. Photons.................................................................. 71 5.3. LightHadrons............................................................ 71 5.4. Jets....................................................................... 72 5.5. HeavyQuarksandQuarkonia.............................................. 73 5.6. Exclusive and Inclusive Observables in Ultraperipheral Collisions... . .. . .. . .. 75 6. OTHERDEVELOPMENTS.................................................. 76 6.1. ElectronIonCollider...................................................... 76 6.2. LatticeQuantumChromodynamics........................................ 77 6.3. RelationstoOtherPhenomena............................................ 78 7. CONCLUSION............................................................... 79 1. INTRODUCTION Nuclear structure at high energies is an important current research topic; it is relevant not only to our understanding of the fundamental quark and gluon dynamics in protons and neutrons bound in nuclei but also to elucidating the formation, properties, and evolution of a deconfined state of hadronicmatter thatexisted inthe early Universe—the so-called quark–gluon plasma (QGP).Nuclear parton distribution functions (PDFs) encode cold binding effects in nuclei (1, 2), determine the preequilibrium phase and initial-state phase transition to the QGP (3),and are correlated with the precise extraction of important QGP properties, such as its temperature (4) and the final-state phase transition during chemical freeze-out (5). While the evolution of PDFs with the scale Q2, at which they are probed, can be computed in perturbative quantum chromodynamics (QCD) (6–9), their dependence on the longitudinal parton momentum fraction inside the hadron is nonperturbative and must be fitted to experimental data. Traditionally, deep inelastic scattering (DIS) of charged leptons or neutrinos and Drell–Yan (DY) dilepton production with fixed targets have 50 Klasen •Paukkunen Downloaded from www.annualreviews.org. Guest (guest) IP: 130.234.241.32 On: Mon, 21 Oct 2024 05:55:46 NS74_Art03_Klasen ARjats.cls August 26, 2024 9:52 provided the bulk of the data.However, over the last decade,collider data from the Large Hadron Collider (LHC) at CERN and the Relativistic Heavy Ion Collider (RHIC) at BNL have led to significant improvements in our knowledge of collinear, unpolarized nuclear PDFs, which are reviewed here. In the naive parton model, the double-differential charged lepton DIS cross section per nucleon, d2σℓA dxdy=2πα2 Q4s[1+(1 −y)2]FA 2(x), 1. is directly related to the nuclear structure function FA 2(x)=x∑qe2 qfA q(x)and thus the quark PDFs fA q(x) in the nucleus A,while the gluon PDF and Q2dependence enter only in the QCD-improved parton model (see Section 2).Here, αis the electromagnetic fine-structure constant,eqis the fractional charge of quark q,Q2is the virtuality of the exchanged photon, s=Q2/(xy) is the hadronic center-of-mass energy, yis the lepton inelasticity, and xis the Bjorken variable. The nuclear structurefunctionsdifferfrom free-nucleon structurefunctionsnotonly due toanadmixtureof protons and neutrons but also in various other ways depending on the region in x. At small x, a depletion (called shadowing) is observed in FA 2(x) at x≲0.05,followed with increasing xby an enhancement (called antishadowing) at 0.05 ≲x≲0.3, then another depletion—the famous EMC effect—at 0.3 ≲x≲0.7, and ultimately an enhancement due to Fermi motion at 0.7 ≲x. The ratios of large-Aisoscalar structure functions to the structure function of the deuteron (D)—a loosely bound isoscalar state of a proton (p) and a neutron (n)—can be parameterized and fitted, albeit with considerable uncertainties,to SLAC (10,11) and NMC (12) data in the following form (13): R(x)=1.10 −0.36x−0.28e−21.9x+2.77x14.4, 2. that is, without a dependence on the nuclear mass number Aor the hard scale Q2. However, a logarithmic decrease in Ahas been observed in the EMC region in the 1984 SLAC data, and a logarithmic increase in Q2has been observed in the shadowing region in the more precise NMC data (14). To first approximation, this is also how quark PDFs in a bound nucleon Ndiffer from the free-nucleon PDFs. This review focuses on model-independent global fits of nuclear PDFs and the progress due to LHC data over the last decade. To obtain a first impression of the nuclear dynamics, it is nonetheless interesting to discuss the main historical experimental measurements and theoretical interpretations in the different xregions. Shadowing had been known to be present in real and virtual photon scattering on nuclei since the 1970s (15,16).At the hadron level,it can be interpreted by assuming that the photon fluctuates from its pointlike state into a superposition of vector mesons (ρ,ω,ϕ),which then interact strongly with the nucleons on the surface of the target nucleus (vector meson dominance). Such nucleons absorb most of the incoming “hadron” flux and therefore cast a shadow onto the inner ones (17). At the parton level and in the nuclear rest frame, the photon can be seen to split into a quark– antiquark dipole with lifetime τ, which scatters coherently from multiple partons in the nucleus if τ≥RA∼A1/3fm or x≤1/(2MNRA)∼0.1A−1/3, resulting again in a reduced nuclear cross section (18–20). The relative motion of nucleons inside the nucleus was considered first in the 1970s for the deuteron(21)andalsointhe1980sforheaviernucleineartheFermisurface(22–24).Thestructure function of a nucleon in a nucleus can then be expressed as a convolution, FA 2(xN)=∫A xN dy fA(y)FN 2(xN y),where fA(y)∼1 √2π1A exp{−[y−(1 −δA)]2 212 A}, 3. www.annualreviews.org •Nuclear PDFs After the First Decade of LHC Data 51 Downloaded from www.annualreviews.org. Guest (guest) IP: 130.234.241.32 On: Mon, 21 Oct 2024 05:55:46 NS74_Art03_Klasen ARjats.cls August 26, 2024 9:52 of the nucleon structure function FN 2(xN) with the nucleon momentum distribution fA(y). Its peak is shifted away from unity due to soft nuclear interactions by an amount δA∼0.04, which corresponds roughly to the ratio of nucleon separation energy over its mass. The width 1Ais determined by a fraction of the Fermi momentum kF∼250 MeV divided by the nucleon mass MN∼1 GeV and is thus small. Equation 3 can therefore be approximated by a simple rescaling, FA 2(xN)=FN 2(xN/(1 −δA)). The net result of this rescaling is to deplete the partons in the intermediate-xNregion, implying FA 2/FB 2<1 for A>B, and to enrich the large-xN∼1 region with FA 2/FB 2>1. The region xN>1 can be modeled by modifying the Gaussian ansatz for fA(y) in Equation 3 (e.g., with a power law tail; 25) and also in deconfinement or cluster models (2), but this region is usually neglected (26). The discovery of a suppression of FFe 2/FD 2at xN=0.65 of ∼0.89 in 1983 by the EMC Collaboration in muon DIS (27) and its confirmation in reanalyzed iron and aluminum SLAC data from the early 1970s (28, 29) came as a big surprise, since Fermi motion models predicted an enhancement of ∼1.25 at this value of xNand a suppression only for xN<0.5. It triggered many theoretical explanations at both the nuclear and partonic levels,and a consensus has yet to emerge (30).Models withnucleons as theonly degrees offreedom inthenucleus mustbe incomplete,since the convolution in Equation 3 violates baryon number and momentum sum rules. The missing momentum could be carried by pions, whose exchanges lead to an intermediate-range nuclear attraction of 300 to 500 MeV, which is canceled by short-distance vector exchanges of almost equal size. The net effect is an average binding energy of 8 MeV per nucleon as observed (31). However, one would then expect an enhancement of antiquarks and therefore of the DY process, which has not been seen (32). The failure of nucleon-only and nucleon–pion models indicates that the nucleon structure itself is modified by the medium. The parton model interpretation of the EMC effect is that the medium reduces the number of high-momentum quarks. This momentum reduction leads, via the uncertainty principle, to the notion that quarks in nuclei are confined in a larger volume than that of a free nucleon. There are two proposals to realize this simple idea: Either scalar and vector mean-field effects cause bound nucleons to be larger than free ones, or short-range correlations (SRCs) cause the nucleon structure to be modified by including either NN ∗configurations or deconfined six-quark configurations that are orthogonal to the two-nucleon wave functions.Interestingly,two-nucleon SRCs might explain the observed linear correlation between the magnitude of the EMC effect at 0.3 ≤xN≤0.7 and the size of the plateau observed in quasi-elastic scattering at 1.5 ≤xN≤2, which would solve the single-nucleon sum rule problem (33, 34). A similar compensation mechanism could be at work in the antishadowing region, which is imposed by shadowing through the momentum sum rule. In the Breit frame, small-momentum quarks and gluons, because of the uncertainty principle, spread over a distance comparable to the nucleon–nucleonseparation.Quarks andgluons from different nucleons canthen overlapspatially and fuse, thus increasing the density of high-momentum partons (antishadowing) at the expense of that of lower-momentum ones (shadowing) (35). In perturbative QCD, this process is flavor dependent,and q¯ q→gfusion results,for instance,in shadowing for antiquarks and antishadowing for gluons. The fact that there is no clear evidence of antishadowing in the DY process can be interpreted either with an important role of valence (v) quarks or as a consequence of the evolution in Q2(1, 2). The nonperturbative nature of nuclear interactions, the need for phenomenological models, and the incongruous nuclear and partonic interpretations of the effects described above are strong motivations to parameterize and fit nuclear PDFs to the available data in a model-independent way. Improving on Equation 2, Eskola parameterized the ratio of heavier nuclear structure functions over deuterons separately in each region at the starting scale Q2 0, matched it at the transition 52 Klasen •Paukkunen Downloaded from www.annualreviews.org. Guest (guest) IP: 130.234.241.32 On: Mon, 21 Oct 2024 05:55:46 NS74_Art03_Klasen ARjats.cls August 26, 2024 9:52 points (whose definition depended on A), and evolved it in Q2(36). As observed experimentally (16) and predicted theoretically (37), shadowing then vanished only very slowly toward larger Q2, in particular for quarks and antiquarks. When the parameterized ratio was fitted to DIS and pA DY data, while imposing baryon number and momentum conservation, the nuclear data could be described fairly independently of the underlying proton PDFs (38). A rigorous statistical analysis of DIS data initially led to rather large values of χ2per degree of freedom (df) of 1.82 to 1.93 (39),which could,however, be reduced to 1.35 in leading-order (LO) and 1.21 in next-to-leading- order (NLO) QCD using more precise DIS and pA DY data (40, 41). Collider (RHIC) data on π0 production from 2006 introduced sensitivity to the gluon density beyond scaling violations with a resulting χ2/df =0.79 at LO and NLO (42). An equally good value of χ2/df =0.83 was obtained at NLO with a similar dataset but with a direct A-dependent parameterization of nuclear PDFs at the starting scale Q2 0(43). The inclusion of neutrino DIS proved to be more difficult (44), triggering a discussion about the universality of nuclear effects in charged lepton and neutrino scattering (45–49). In the last decade, a wealth of LHC data on electroweak boson, photon, light and heavy hadron, and jet production has become available for proton–lead (p+Pb) collisions, which has already had a significant impact on the determination of nuclear PDFs (50–52). These modern developments are reviewed thoroughly in this article, whereas other recent reviews have mostly focused on proton PDFs (53–55) or sketched a larger multidimensional picture of the nucleus (56). The remainder of this article is organized as follows. In Section 2, we briefly review the theoretical foundations of nuclear DIS and its factorization. In Section 3, we describe the different methodological approaches to global fits of nuclear PDFs. In Sections 4 and 5, we discuss the impact of the different experimental data in roughly chronological order; our main focus is on LHC data. The impact of the future Electron Ion Collider (EIC) and connections to other fields in nuclear, particle, and astroparticle physics (lattice QCD, the search for gluon saturation, the QGP, and astrophysical phenomena) are briefly addressed in Section 6 before we conclude the review in Section 7. 2. THEORETICAL FOUNDATIONS Westart ourdiscussionofnuclear PDFs byreviewingtheir theoretical foundationsbothwithinthe operator product expansion (OPE) and the QCD-improved parton model, including target-mass and other higher-twist effects. 2.1. Factorization in the Operator Product Expansion and the Quantum Chromodynamics–Improved Parton Model High-energy lepton DIS is the key process for studying the hadronic structure of nucleons N or nuclei Awith mass MNor MA=A MNand four-momentum PNor PAin terms of their partonic (quark and gluon) degrees of freedom (see Figure 1) (57, 58). The charged lepton ℓand (anti)neutrino νhave incoming four-momentum kand outgoing four-momentum k′, the squared center-of-mass energy is sN,A=(k+PN,A)2,and Xrepresents all final-state hadrons with total fourmomentum PXand squared mass W2 N,A=(q+PN,A)2. In the laboratory frame, the lepton energy loss is ν=q·PN,A/MN,A=E−E′, and the exchanged vector boson Vhas squared momentum transfer Q2= −q2>0. The inclusive differential cross section d˜σA∼Lµν ˜ WA µν can be written as a combination of a pointlike leptonic tensor Lµν and the hadronic tensor ˜ WA µν (PA,q)=1 4π∫d4z eiq·z⟨A|J† µ(z)Jν(0)|A⟩ = 1 4πdisc.˜ Tµν (PA,q), 4. www.annualreviews.org •Nuclear PDFs After the First Decade of LHC Data 53 Downloaded from www.annualreviews.org. Guest (guest) IP: 130.234.241.32 On: Mon, 21 Oct 2024 05:55:46 NS74_Art03_Klasen ARjats.cls August 26, 2024 9:52 a b c γ, Z (q) ℓ' (k')µ µ ℓ (k)ν ν W (q) c µ ν W (q) W s/d s/dq q'q q N, A (PN,A)N, A (PN,A)N, A (PN,A) X (PX)X (PX)X (PX) Figure 1 Leading-order diagrams for (a) neutral-current deep inelastic scattering with charged leptons, (b) charged-current deep inelastic scattering with neutrinos or antineutrinos, and (c) charm dimuon production. The charm quark is understood to hadronize before the semileptonic decay. where the latter is given in terms of a product of hadronic currents and can be related to the discontinuity of the virtual forward Compton scattering amplitude: ˜ TA µν (PA,q)=∫d4z eiq·z⟨A|TJ† µ(z)Jν(0)|A⟩.5. The OPE then allows one to expand the hadronic matrix element of the forward scattering amplitude in a complete set of local operators (59): ˜ TA µν (PA,q)= −2i∑ j,τ,n cj,µ1···µn τ,µν ⟨A|Oj,τ µ1···µn|A⟩ = −2i∑ j,k 22k Q4kC2k jA2k˜ 5j,k µν +O(τ > 2), 6. where cj,µ1···µn τ,µν denotes the hard scattering, τdenotes the twist of the operator O(defined as its mass dimension minus its spin),and jdenotes different operators with the same twist.Up to power corrections, one can identify the product of the perturbative Wilson coefficientsC2k iand reduced hadronic matrix elements A2kas integer Mellin moments of structure functions ∫1 0dy y2k−1˜ FA i(y,Q2)=C2k iA2k+O(τ > 2), 7. with y2k−1→y2k−2for i=2. The Lorentz structure in terms of metric tensors and momenta is encoded in ˜ 5j,k µν . In the QCD-improved parton model, the nuclear structure functions ˜ FA i(xA,Q2)=∑ j=q,g∫1 xA dyA yA Ci,j˜ fA j(yA,Q2)+O(τ > 2) 8. depend on the Bjorken scaling variable xN,A=Q2/(2q·PN,A)=Q2/(2νMN,A) with xN[0, A] (xA=xN/A[0, 1]) with logarithmic scaling violation in Q2(see below). They are given as convolutions of target-independent short-distance Wilson coefficients Ciand Cjwith universal nuclear PDFs ˜ fA j. Inspection of Equations 7 and 8 shows that nuclear PDFs can be understood as moments of matrix elements of local twist-two operators composed of quark and gluon fields (60). The nuclear PDFs ˜ fA i(xA,Q2) above are related to the more familiar average-nucleon nuclear PDFs fA i(xN,Q2) through fA i(xN,Q2)=˜ fA i(xA,Q2)/A. This rescaling is a key step that allows us to compare structure functions across different nuclei, including the free nucleon. The evolution of the PDFs with the scale Q2is perturbatively calculable and given by the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) evolution equations (6–9): dfA i(xN,Q2) dlnQ2=αs(Q2) 2π∫A xN dyN yN Pi j (xN yN)fA j(yN,Q2), 9. 54 Klasen •Paukkunen Downloaded from www.annualreviews.org. Guest (guest) IP: 130.234.241.32 On: Mon, 21 Oct 2024 05:55:46 NS74_Art03_Klasen ARjats.cls August 26, 2024 9:52 where αsis the QCD coupling and Pij denotes the partonic splitting functions. Furthermore, the PDFs satisfy sum rules due to charge, baryon number, and momentum conservation, ∫A 0dxN{fA uv,fA dv}(xN,Q2)= {2Z+N,Z+2N},∫A 0dxNxN∑ i fA i(xN,Q2)=A, 10. where Zis the electric charge of the nucleus with baryon number A=Z+N. It is therefore common (but not necessary) to decompose the nuclear PDFs as fA i(xN,Q2)=Z Afp/A i(xN,Q2)+A−Z Afn/A i(xN,Q2), 11. where the bound-neutron PDFs fn/A i(xN,Q2) are commonly obtained from those of the bound-proton fp/A i(xN,Q2) by assuming isospin symmetry: fn/A u,¯ u(xN,Q2)=fp/A d,¯ d(xN,Q2), fn/A d,¯ d(xN,Q2)=fp/A u,¯ u(xN,Q2).12. In principle, the above integrations extend to A, although the dominant support of the PDFs is expected to be in the region xA≤1/A, or xN≤1. One therefore usually assumes fA i(xN,Q2)=0 for xN>1, which has the advantage that the same evolution equations can be used for all nuclei in the interval x[0, 1]. Accounting for the heavy quark masses is essential in an accurate description of the freeproton data (61). Standard methods to handle the quark masses include general-mass (GM) variable-flavor-number schemes (VFNSs) (62) such as the simplified Aivazis–Collins–Olness– Tung (SACOT) schemes (63–65) or the fixed-order plus next-to-leading logarithms (FONLL) schemes (66, 67), which provide systematic ways to interpolate between the fixed-flavor-number scheme (FFNS), in which heavy quarks are not considered as partons, and the zero-mass (ZM) VFNS, in which heavy quarks are treated as massless partons. 2.2. Target-Mass Corrections Target-mass corrections (TMCs) can be discussed in terms of collinear factorization (68) or can be obtained from the OPE by inverting moments of structure functions (59) (see Equation 7). They can be written in the following general form (69): FA,TMC i(xN,Q2)=∑ j Aj iFA j(ξN,Q2)+Bj ihA j(ξN,Q2)+CigA 2(ξN,Q2), 13. where ξN=2xN/(1 +rN) is the Nachtmann variable (70) with rN=√1+4x2 NM2 N/Q2, and hA j(ξN,Q2)∼∫A ξN dξ′ N FA j(ξ′ N,Q2) ξ′ N and gA 2(ξN,Q2)=∫A ξN dξ′ NhA 2(ξ′ N,Q2) 14. are auxiliary functions with 1/ξ′ N→1/ξ′ N 2for j=2. Specifically, the proportionality factor in hA 2 is unity, and we have the following: FA,TMC 2(xN,Q2)=(x2 N ξ2 Nr3 N)FA 2(ξN,Q2)+(6M2 Nx3 N Q2r4 N)hA 2(ξN,Q2)+(12M4 Nx4 N Q4r5 N)gA 2(ξN,Q2).15. Quark masses modify ξNby ξN→RijξN, where the factor Rij depends on the incoming and outgoing quark masses miand mj. In the case of mi=0, one obtains the slow-rescaling limit Rij =1+(nmj)2/Q2with n=1 for charged-current (CC) DIS (71) and n=2 for neutral-current (NC) DIS (61). www.annualreviews.org •Nuclear PDFs After the First Decade of LHC Data 55 Downloaded from www.annualreviews.org. Guest (guest) IP: 130.234.241.32 On: Mon, 21 Oct 2024 05:55:46 NS74_Art03_Klasen ARjats.cls August 26, 2024 9:52 a b c q xp q xp q xp Jet A A A Jet Jet Figure 2 Classification of multiple parton scattering in a nuclear medium: (a) interactions internal to the nucleus, (b) initial-state interactions, and (c) final-state interactions. Figure adapted from Reference 72. 2.3. Higher-Twist Corrections Interactions internal to the nucleus as in Figure 2achange the nuclear PDFs with respect to those of the free nucleon. However, since only a single parton participates in the hard scattering, the structure functions can still be factorized as in Equation 8, and the leading-twist nuclear PDFs can be parameterized at an initial scale Q0, evolved with (in principle A-dependent) evolution equations and fitted to experimental data or modeled theoretically. Next-to-leading power corrections to Equation 8 of O(r2 T∼1/p2 T), O(m2 J/p2 T), and O(αs(Q2)32/Q2) arise from the transverse size rTof the initial nucleus, the nonvanishing invariant mass mJof the final jet, and initial-state and final-state interactions involving more than one parton as shown in Figure 2b,c. In hadron–nucleus collisions, both are enhanced by A1/3due to the large density of soft partons in the nucleus. For initial-state interactions, 32∼0.01 GeV2 is the squared scale of the twist-four correlation function of single parton pairs and proportional to the transverse field strength. For final-state interactions, long-range soft parton interactions must also be considered. It can be shown that the A1/3enhancement can be factorized to all powers in hadron–nucleus but not nucleus–nucleus collisions and then involves correlation functions of multiple parton pairs. In general, however, even the hadron–nucleus DY cross section cannot be factorized beyond next-to-leading power (72). 3. GLOBAL ANALYSES OF NUCLEAR PARTON DISTRIBUTION FUNCTIONS The inverse problem of extracting nuclear PDFs from experimental data is approached in a similar way as the determination of (free) proton PDFs (55); that is, all global analyses are based on optimizing the correspondence between theoretical calculations and experimental measurements by minimizing a figure-of-merit function that is typically of the following form: χ2=∑ i,j (Di−Ti)C−1 i j (Dj−Tj).16. Here, Didenotes the experimental values for observables in the fit, and Tidenotes the corresponding theoretical values, which depend on the PDFs. The covariance matrix is defined as Ci j =σ2 iδi j +∑α¯σiα¯σjα, where σiis the total uncorrelated uncertainty added in quadrature and ¯σiαis the correlated systematic uncertainty from source α. How to exactly assign values for σiand ¯σiαvaries from one analysis to another and depends on whether uncertainties are multiplicative or additive (73,74).In the case of fitting nuclear PDFs,only a few datasets provide complete information on the correlated systematic uncertainties, and in most cases only the overall normalization uncertainty is given. An essential part of global PDF analyses is the propagation of experimental uncertainties into the PDFs.From the practical point of view,the two principal methods are the Hessian (75,76) and Monte Carlo methods (77, 78). The Hessian uncertainty analysis (75, 76) is based on expanding 56 Klasen •Paukkunen Downloaded from www.annualreviews.org. Guest (guest) IP: 130.234.241.32 On: Mon, 21 Oct 2024 05:55:46 NS74_Art03_Klasen ARjats.cls August 26, 2024 9:52 1.8 1.4 1.0 0.6 0.2 10–5 10–4 10–3 10–2 10–1 1 10–5 10–4 10–3 10–2 10–1 1 10–5 10–4 10–3 10–2 10–1 1 1.8 1.4 1.0 0.6 0.2 1.8 1.4 1.0 0.6 0.2 1.8 1.4 1.0 0.6 0.2 1.8 1.4 1.0 0.6 0.2 1.8 1.4 1.0 0.6 0.2 x x x EPPS21 nCTEQ15HQ nNNPDF3.0 RPb (x, Q2 = 10 GeV2) u RPb (x, Q2 = 10 GeV2) d RPb (x, Q2 = 10 GeV2) u RPb (x, Q2 = 10 GeV2) d RPb (x, Q2 = 10 GeV2) s RPb (x, Q2 = 10 GeV2) g Figure 4 Comparison of the 208Pb nuclear modifications resulting from the EPPS21 (solid blue lines) (51), nCTEQ15HQ (dashed purple lines) (50), and nNNPDF3.0 (dotted-dashed green lines) (52) global analyses of nuclear PDFs—that is, the PDFs of lead divided by the summed PDFs of 82 free protons and 126 free neutrons. Uncertainty bands (shaded areas) correspond to 90% confidence levels. Abbreviation: PDF, parton distribution function. 4. FIXED-TARGET DATA We now turn to a detailed discussion of the available experimental data and their impact on global nuclear PDF analyses in roughly chronological order. We first focus on the early fixed-target NC and CC DIS and DY data, but we also review the more recent JLab DIS data. 4.1. Early Deep Inelastic Scattering Data and Constraints on Quarks Measurements of fixed-target electron and muon NC DIS on various nuclei from SLAC, EMC, and NMC,as well as from the BCDMS,FNAL,and HERMES experiments,form the backbone of globalnuclearPDF determinations(2).Most ofthese datasets canbefitted withanexcellent χ2/df, but a few of them, such as the 1988 EMC measurement of FSn 2/FD 2(114), are difficult to describe (41, 50, 52, 80, 81, 115). In contrast, the NMC FSn 2/FC 2data (14, 116) can be fitted well (50–52, 80, 81). Other outliers include the E665 data (117) on FC 2/FD 2,FCa 2/FD 2, and FPb 2/FD 2, though, for instance, the ratio (FPb 2/FD 2)/(FC 2/FD 2)=FPb 2/FC 2is consistent with the NMC data (116). The kinematic reach of these fixed-target data, x≳5×10−3and Q2≲140 GeV2, is naturally more limited than that of the HERA experiments H1 and ZEUS, which provide the bulk of the datainfree-proton analyses(54,55).Inaddition,cutsare often appliedtolimit theeffectsof TMCs and other higher-twist corrections,which could be larger in nuclear reactions (72). At low Q2, NC DIS is governed by virtual photon exchange. In the kinematic region of fixed-target experiments, the cross section d2σℓA dxdQ2=4πα2 Q4[FA 2(x,Q2)(y2 2+1−y−xyM2 s−M2)−xy2FA L(x,Q2)]32. www.annualreviews.org •Nuclear PDFs After the First Decade of LHC Data 63 Downloaded from www.annualreviews.org. Guest (guest) IP: 130.234.241.32 On: Mon, 21 Oct 2024 05:55:46 NS74_Art03_Klasen ARjats.cls August 26, 2024 9:52 is dominated by the structure function FA 2(x,Q2), which at LO is sensitive only to the squared charge-weighted sum of quarks and antiquarks (see Section 1). As the ℓADIS data that enter the global fits are given in terms of ratios, dσℓA1 dσℓA2≈FA1 2 FA2 2 , 33. where A2is typically D or carbon (C), the DIS data can mainly directly constrain the overall nuclear modification of valence and sea quarks. The contributions of gluons enter the cross section only at order αs, and the direct constraints for the gluon densities are therefore weak. However, the gluons drive the Q2dependence of FA 2(x,Q2) at small values of x(118): dFA 2(x,Q2) dlogQ2≈10αs(Q2) 27πx f A g(2x,Q2), x→0.34. Through this relation, it was understood early on that the Q2dependence of the ratios FSn 2/FC 2 measured by the NMC Collaboration (14) around x≈0.01. . .0.02 and Q2≈1. . .10 GeV2can constrain the Adependence of the gluon nuclear modifications (119). In particular, a very strong Adependence of gluons would contradict the measured positive Q2slopes of FSn 2/FC 2. There are alsosimilar HERMES datafor FKr 2/FD 2(120),but theQ2leverarm isnotas longinthe perturbative regime.In principle,the longitudinal structure function FA Lcarries a direct sensitivity to the gluon (121), but data are scarce (122). Likewise, the cross sections for charm production would provide more direct information on the gluons—modulo a possible intrinsic charm PDF (123)—but not many data are available (124, 125). Since FA 2probes predominantly the squared charge-weighted sum of quark PDFs, the flavor decomposition is also difficult to pin down. To understand this, let us write, for instance, the valence quark distributions as follows: fA uv=RA v(Z Afp uv+A−Z Afp dv)+δRA v(2Z A−1)fp uvfp dv fp uv+fp dv , 35. fA dv=RA v(Z Afp dv+A−Z Afp uv)−δRA v(2Z A−1)fp uvfp dv fp uv+fp dv , 36. where RA v≡(Rp/A uvfp uv+Rp/A dvdp dv)/(fp uv+fp dv) is an average nuclear modification of the valence quarks, and the difference is δRA v≡Rp/A uv−Rp/A dv. The first terms dominate, and thus the large-x data constrain very tightly the average modification RA v. Having data for several different combinations of Zand A, combined with the fact that the nuclear effects are expected to scale with A, will give also constraints on δRA vand thereby to the mutual differences of nuclear effects in up and down valence quarks. The same reasoning naturally applies for the up and down sea quarks. Equations 35 and 36 also clearly demonstrate the anticorrelation of the nuclear effects between up and down quarks. For an isoscalar nucleus Z=A/2, the up and down quark distributions are always equal. To facilitate the interpretation of nuclear effects, many early NC DIS experiments corrected their data for isospin effects—that is, for the unequal numbers of protons and neutrons in heavy nuclei compared with the deuteron.These data were long taken at face value (and are still done so, e.g., in nNNPDF3.0) and fitted by setting Z=N=A/2. However, this is not necessary in global fits and has in the past even caused some confusion about the nuclear valence quark modification (97). Comparing FA 2=[ZF p/A 2+(A−Z)Fn/A 2]/A(see Equation 11) with the isoscalar expression 64 Klasen •Paukkunen Downloaded from www.annualreviews.org. Guest (guest) IP: 130.234.241.32 On: Mon, 21 Oct 2024 05:55:46 NS74_Art03_Klasen ARjats.cls August 26, 2024 9:52 ˆ FA 2=[Fp/A 2+Fn/A 2]/2 leads to ˆ FA 2=βFA 2with β=A 2(1+Fn/A 2 Fp/A 2)/(Z+(A−Z)Fn/A 2 Fp/A 2).37. The experiments then assumed Fn/A 2/Fp/A 2=Fn 2/Fp 2and parameterized this ratio from DIS data on protons and deuterons with, for instance, 1 −0.8x(11) or 0.92 −0.86x(126). These functions then allow one to calculate βand either apply it to the theoretical calculations (105) or remove the isoscalar correction from the data and fit the true nuclei (26, 97). 4.2. Deep Inelastic Scattering at High xand Nuclear Effects in the Deuteron Recent precise JLab measurements taken with 6- to 10-GeV electron beams on various nuclear targets (127–130) at low to intermediate Q2and W2can considerably reduce the nuclear PDF uncertainties in the high-xregion (26, 131). However, this region requires good control over potential TMCs (see Section 2.2) and other higher-twist corrections (see Section 2.3), hadronic resonances, and (as with the other fixed-target DIS data) nuclear effects in the deuteron. TMCs scale with powers of M2 N/Q2and are thus suppressed even for heavy nuclei. Their leading effect is a shift in the probed momentum fraction xNto the Nachtmann variable ξN(69). When the kinematic cuts are relaxed from Q2>4 GeV2and W2>12.25 GeV2to Q2>1.69 GeV2and W2>2.89 GeV2, the subleading target-mass effects provide a uniform shift of less than 1% for all nuclei, leaving the ratios FA 2/FD 2unaffected (26). Some global analyses therefore include the JLab data with lower kinematic cuts (26, 51, 81, 131). The compatibility of the JLab data with global fits can be taken as evidence that other higher-twist effects can be neglected (51).They can, however, also be parameterized as FA 2(x,Q)→FA 2(x,Q)[1+A1/3h0xh1(1 +h2x) Q2], 38. where the values {h0,h1,h2}={−3.3 GeV2, 1.9, −2.1} come from the CJ15 proton PDF analysis (132) and A1/3scaling is assumed (see Section 2.3). This form of higher-twist corrections leads to a slight reduction of FA 2at intermediate x∼0.3 and a substantial enhancement at high xand Q2<16 GeV2, and it improves the global χ2by around 3% within the nCTEQ15HIX global analysis (26). In the resonance region at very low W2[1.21; 2.89] GeV2, the nuclear effects at large xare surprisingly similar to those in DIS, which may signal the applicability of quark hadron duality due to the averaging over nuclear resonances (133). As described in Section 1, Fermi motion can be accounted for by a convolution of the nucleon structure function with the nucleon momentum distribution or effectively by a rescaling of the variable x. The rise of FA 2/FD 2at large xcan be well described by the following parameterization (26): x′=x−εxκlog10 A.39. As the PDFs decrease with x, the negative shift ensures that the transformed function is larger than the unmodified one and nonvanishing as x→1. The overall size of the rescaling effect is controlled by ε,κ > 0 ensures that only the large-xregion is modified, and the log10Aterm implies an increasing modification across the full range of nuclear Avalues from the proton (A=1) to lead (A=208). A good description of the JLab data is obtained with κ=10 and ϵ∼0.03. While this suggests that it may be possible to expand the kinematic reach to W2< 2.89 GeV2, the resonance region is currently avoided in all global fits (see Table 1). Given that the nuclear DIS data are usually presented as ratios of FA 2/FD 2and that most global fits of proton PDFs use deuteron data as well,good control over the nuclear effects in the deuteron www.annualreviews.org •Nuclear PDFs After the First Decade of LHC Data 65 Downloaded from www.annualreviews.org. Guest (guest) IP: 130.234.241.32 On: Mon, 21 Oct 2024 05:55:46 NS74_Art03_Klasen ARjats.cls August 26, 2024 9:52 is required. The deuteron is much more loosely bound than heavier nuclei and is therefore often approximated as an isoscalar combination of a free proton and neutron. However, its structure at large xis still modified by Fermi motion, nuclear binding, and off-shell effects, while at small x rescattering still induces some shadowing. These nuclear effects are of the order of a few percent in the available DIS data and below 1% in the available DY data. They can be accounted for in different ways; one option is to use a free-proton baseline fitted to deuteron DIS data without nuclear corrections (51). In this case the deuteron nuclear effects are, to some extent, fitted into the uversus dquark flavor separation. Then no additional nuclear correction should be applied, although this also has been done, and in such a case the effect of double counting should be quantified (81).Alternatively,one can rescale the fitted FA 2/FD 2data by a ratio of FD 2/Fp 2(26) as modeled, for instance, within the CJ15 global proton analysis (132; see also 134). If this is done with a free-proton PDF that already includes deuteron data (26), the same deuteron correction should be applied in both cases (86). A third possibility is to fit the free proton without deuteron data and then the deuteron in the same way as the other nuclei (80). The theoretical treatment of the deuteron affects the description of all NC DIS data and thus also the question of the compatibility of CC DIS data with NC DIS and electroweak boson production at the LHC (see Sections 4.4 and 5.1). 4.3. The Drell–Yan Process and Constraints on Antiquarks The DY process—the inclusive production of electroweak gauge bosons in hadron collisions, followed by a leptonic decay of the gauge boson—has been of enormous historical importance for the quark flavor separation in protons (53). For heavier nuclei, fixed-target measurements have been made in the FNAL E605 (135), E772 (32), and E866 (136) experiments in pA collisions covering several nuclei from carbon to tungsten in the kinematic range x>10−2and dilepton mass Mℓℓ <15 GeV. At such low values of Mℓℓ, well below the Zboson peak, the DY process is dominated by an off-shell intermediate photon with the cross section differential in the lepton pair rapidity yℓℓ and Mℓℓ given by d2σpA DY dMℓℓdyℓℓ ∼∑ q e2 q[fp q(x1)fA ¯ q(x2)+fp ¯ q(x1)fA q(x2)]with x1,2 =Mℓℓe±yℓℓ √s, 40. which tests a squared charge-weighted combination of quarks and antiquarks. Fixed-target experiments generally have larger acceptance in the x1≫x2region, where x1is defined with respect to the proton beam, and in this case the first term in Equation 40 is the dominant one, where fp q(x1) is mostly determined by the valence quark content of the proton. For isoscalar nuclei, fA ¯ u=fA ¯ d, and the cross section ratios between pA and pD collisions, measured in the E772 and E866 experiments, become dσpA DY dσpD DY isoscalar A≈fA ¯ u(x2) fD ¯ u(x2)=fA ¯ d(x2) fD ¯ d(x2).41. As a result, the measured DY ratios are sensitive to the nuclear modifications of sea quark distributions at x2∼0.03. . .0.3 (137, 138). In principle, the dependence on Mℓℓ should also retain sensitivity to the gluon through the DGLAP evolution.Combining the photon-mediated DIS and DY measurements historically provided the first chance to disentangle the nuclear effects in valence and sea quarks,leading to the conclusion that there was not such a clear antishadowing effect in sea quarks as there was for valence quarks.The E605 data (135) for p+Cu collisions are given in 66 Klasen •Paukkunen Downloaded from www.annualreviews.org. Guest (guest) IP: 130.234.241.32 On: Mon, 21 Oct 2024 05:55:46 NS74_Art03_Klasen ARjats.cls August 26, 2024 9:52 termsofabsolute cross sectionsandare oftenusedinfits ofprotonPDFs.In thefuture,fixed-target pA data from the FNAL E906/SeaQuest experiment (139) are expected to improve the precision of the available data.The renewed facilities at RHIC should also enable new measurements of the DY process (140), and similar measurements are planned at the LHCb experiment at the LHC (141). In principle, the pion–nucleus DY process has the potential to constrain the flavor decomposition of the valence quarks (142). It depends on the pion PDFs, but this dependence cancels largely in ratios of nuclear cross sections. Unfortunately, the precision of the πADY data from the CERN NA3 (143), NA10 (144), and FNAL E615 (145) experiments is not high enough to provide significant discrimination power on top of the DIS data. However, the new CERN-based facility AMBER (146) may be able to improve upon the current precision. 4.4. Neutrino Deep Inelastic Scattering Data and Flavor Separation Because of the weak nature of neutrino interactions,heavy nuclear targets such as iron or lead have traditionally been used to obtain CC DIS data with sufficient statistics.Despite the nuclear targets, these data have routinely been included in global analyses of proton PDFs,with or without nuclear corrections (147), and have formed the principal constraint for a possible sversus ¯ sasymmetry (148). In addition, determinations of the weak mixing angle in neutrino DIS (149) have relied on a sufficient understanding of the nuclear structure (150–152). In the case of CC neutrino DIS, the differential cross section is d2σν,¯νA dxdy=G2 FM4 W 2πxyQ2(Q2 Q2+M2 W)2[(1−y−x2y2M2 N Q2)Fν,¯ νA 2+y2xFν,¯ νA 1±(y−y2 2)xFν,¯νA 3], 42. where GFis the Fermi constant, MWis the Wboson mass, and the plus–minus sign is taken for incoming neutrinos (+) and antineutrinos (−). At LO and high Q2, d2σνA∝(fA d+fA s+fA b)+(1 −y)2(fA ¯ u+fA ¯ c), 43. d2σ¯νA∝(fA ¯ d+fA ¯ s+fA ¯ b)+(1 −y)2(fA u+fA c).44. Due to the suppressing factor (1 −y)2, there is an increased sensitivity to the strange quark distribution in comparison to NC charged lepton DIS, particularly for antineutrinos. In addition, the up and down quarks enter the cross sections with different weights than in the case of NC charged lepton DIS. Adding the neutrino data thus helps in constraining the differences between nuclear effects in up and down quarks (see Equations 35 and 36). However, in a global fit these data are also sensitive to the assumed proton PDFs. Data on inclusive neutrino DIS have been taken by, for instance, the CDHSW (153), CCFR (154), and NuTeV (13) experiments on iron and by the CHORUS experiment on lead (155). Charm production has been measured as well through muonic decays of produced charmed hadrons (see Figure 1) (156, 157). In principle, all of these data should be relevant for nuclear PDFs, but they are only partially included in global analyses due to concerns about possible mutual tensions between the neutrino datasets, tensions with the charged lepton DIS data, and the fact that some of these neutrino data are in some cases already used in the proton PDF fits that are used as baselines in the fits of nuclear PDFs. Another difficulty is that there are no references from νpor νD scattering, so the data are reported as absolute cross sections. Figure 5 shows neutrino (Figure 5a) and antineutrino (Figure 5b) cross sections divided by the theoretical NLO predictions with CT18A proton PDFs in the SACOT-χscheme, including www.annualreviews.org •Nuclear PDFs After the First Decade of LHC Data 67 Downloaded from www.annualreviews.org. Guest (guest) IP: 130.234.241.32 On: Mon, 21 Oct 2024 05:55:46 NS74_Art03_Klasen ARjats.cls August 26, 2024 9:52 Average σdata/σNLO (CT18A) EPPS21 ν+Fe nCTEQ15HQ ν+Fe nNNPDF3.0 ν+Fe SLAC/NMC parameterization CHORUS ν+Pb NuTeV ν+Fe CDHSW ν+Fe EPPS21 ν+Fe nCTEQ15HQ ν+Fe nNNPDF3.0 ν+Fe SLAC/NMC parameterization CHORUS ν+Pb NuTeV ν+Fe CDHSW ν+Fe 1.2 1.1 1.0 0.9 0.8 Average σdata/σNLO (CT18A) 1.2 1.1 1.0 0.9 0.8 10–2 10–1 1 10–2 10–1 1 a b x x Figure 5 Average ratios of (a) neutrino cross sections and (b) antineutrino cross sections, as measured in the CHORUS (155), NuTeV (13), and CDHSW (153) experiments, to a theoretical next-to-leading-order prediction with CT18A parton distribution functions within the kinematic range of Q2>4 GeV2and W2>12.25 GeV2. The data are compared with EPPS21 (51), nCTEQ15HQ (158), and nNNPDF3.0 (52) predictions. The SLAC/NMC neutral-current deep inelastic scattering parameterization (see Equation 2) is also shown as a reference. approximate TMCs and electroweak corrections as used in Reference 45.The ratios are evaluated as weighted averages over Q2>4 GeV2and W2>12.25 GeV2as in Reference 158.The obtained ratios are compared with predictions for NuTeV data using EPPS21 (51), nCTEQ15HQ (158), andnNNPDF3.0(52).Also,the SLAC/NMCNC DISparameterization(Equation2) isshownfor comparison. Tensions between different datasets, nuclear PDFs, and data and theory are clearly visible. The largest differences between nuclear PDFs occur in the case of antineutrino DIS at x≳0.2, where the nNNPDF3.0 values are significantly above those from EPPS21 and nCTEQ15HQ. This can be explained by the large enhancement of ¯ dand sdensities in nNNPDF3.0 compared with those in EPPS21 and nCTEQ15HQ (see Supplemental Figure 1). From the datasets, the NuTeV neutrino data in particular stand out from the others, but larger deviations between the CHORUS and CDHSW data can also be observed if less restrictive kinematic cuts are imposed and electroweak corrections are neglected (158). To some degree, the observed tensions can be alleviated by normalizing the data by the cross sections integrated over xand y(45, 47), by neglecting the NuTeV systematic error correlations, and by introducing additional cuts in x(158). The current consensus seems to be that at least the CHORUS data can be included in global analyses without significant tensions. In addition, the charm dimuon data are used in nCTEQ15ν(158) and nNNPDF3.0 (52), and the CDHSW data are used in TUJU21 (80) and KSASG20 (81). There have been speculations about differences in nuclear shadowing in CC and NC processes (20), at least at low Q2(19), but W±and Zproduction at the LHC probing nuclear PDFs at significantly higher Q2can be fitted well in global analyses. In the future, novel neutrino–nucleus DIS data may become available through dedicated experiments measuring neutrinos produced in high-luminosity pp collisions at the LHC. Indeed, the first observations of such collider neutrinos have already been made by the FASER (159) and SND@LHC (160) Collaborations.The impact of such future measurements on nuclear PDFs has been considered recently in Reference 161. 68 Klasen •Paukkunen Downloaded from www.annualreviews.org. Guest (guest) IP: 130.234.241.32 On: Mon, 21 Oct 2024 05:55:46 NS74_Art03_Klasen ARjats.cls August 26, 2024 9:52 5. COLLIDER DATA We now turn to a discussion of the LHC (p+Pb) as well as the RHIC (D+Au) measurements used in global fits of nuclear PDFs. Within collinear factorization, the pA (or DA) cross sections dσ(pA →O+X)=∑ i,j[,k] fp i⊗fA j⊗dˆσ(i j →O,[k]+X) [⊗DO k] 45. for the observable Oinvolve convolutions of (nuclear) PDFs fp,A i,jwith perturbative partonic cross sections dˆσand, in the case of inclusive light or heavy flavor hadron (h) production, nonperturbative FFs Dh k. These are numerically costly and in many cases must be evaluated with precomputed grids (162).Since one of the colliding objects is a proton and the nuclear PDFs also depend on the proton, absolute LHC cross sections depend directly and indirectly on the proton PDFs. To reduce this dependence and also cancel other theoretical and experimental uncertainties,the nuclear modification ratio, RpA ≡dσ(pA →O+X)/dσ(pp →O+X), 46. and the forward-to-backward ratio, RFB ≡dσ(pA →O+X)y>0/dσ(pA →O+X)y<0, 47. are often introduced. Here yrefers to the rapidity of the observable O. In fits of collider data, normalization uncertainties originating from the luminosity measurements play a special role. For example, the measured and calculated Rp+Pb values for hadron production in the y≫0 region (small xN) at the LHC are often rather flat, and changes in the nuclear PDFs can be compensated by treating the normalization uncertainty as a correlated systematic uncertainty (see, e.g., 158). If the luminosity uncertainty is common for y>0 and y<0, the RFB is free from this additional freedom. Today, all global fits of nuclear PDFs account for the systematic normalization uncertainties. 5.1. Electroweak Bosons The first global analysis of nuclear PDFs to include Wand Zboson data from p+Pb collisions was EPPS16 (97). However, the impact of the Run 1 ATLAS (163) and CMS (164, 165) data was still rather limited due to low statistics.Since then,measurements for these electroweak processes have been published from all four LHC experiments at √s=5.02 TeV (Run 1) and from the ALICE, CMS, and LHCb experiments at √s=8.16 TeV (Run 2). They are now, in different combinations, used in all recent global analyses (50–52) and also in the NNLO and model-dependent fits TUJU21 (80) and KP16 (113). The available data are summarized in Table 2. At the moment, the most stringent constraints come from the Run 2 CMS Wboson data (170). Figure 6 compares the nuclear modification ratio Rp+Pb constructed from the CMS Run 2 (170,171) measurements with NLO calculations using the EPPS21 (51),nCTEQ15HQ (50),and nNNPDF3.0 (52) nuclear PDFs. While the EPPS21 analysis included the Rp+Pb data shown in the figure, the nCTEQ15HQ and nNNPDF3.0 analyses fitted absolute p+Pb cross sections. As one can see, the spread between the different predictions is still rather significant. In comparison to a calculation with no nuclear effects—that is, 82 free protons and 126 free neutrons—the data indicate a clear sign of shadowing at forward rapidities or x≪1.The relative ordering of EPPS21, nCTEQ15HQ, and nNNPDF3.0 values follows that of the corresponding gluon shadowing in Figure 4. Also, as can be seen from Figure 4, even after inclusion of these electroweak data, the overall variation in the strange quark PDF is still quite significant, which indicates that the constraints for the nuclear strange quark PDFs are still not very strong. www.annualreviews.org •Nuclear PDFs After the First Decade of LHC Data 69 Downloaded from www.annualreviews.org. Guest (guest) IP: 130.234.241.32 On: Mon, 21 Oct 2024 05:55:46 NS74_Art03_Klasen ARjats.cls August 26, 2024 9:52 Table 2 Summary of Z,W±, and low-invariant-mass Z/γ∗rapidity distributions available from p+Pb collisions at the Large Hadron Collider Dataset nCTEQ15HQ (50) EPPS21 (51) nNNPDF3.0 (52) TUJU21 (80) KP16 (113) Run 1 ATLAS Z(163) ✓ ✓ ✓ ✓ ✓ CMS Z(164) ✓ ✓ ✓ ✓ ✓ ALICE Z(166) — — ✓a— — LHCb Z(167) ✓—✓a— — ATLAS W±(168)b✓— — — ✓ CMS W±(165) ✓ ✓ ✓ — — ALICE W±(166) ✓—✓a— — Run 2 CMS Z(172) — — ✓a— — CMS Z/γ ∗(172) — — ✓a— — ALICE Z(346) — — ✓a— — LHCb Z(169) — — — — — CMS W±(170, 171) ✓ ✓c✓ ✓ — ALICE W±(347) — — — — — aAdded in nNNPDF3.0 (52). bPreliminary data. cAdded in EPPS21 (51). The CMS Run 2 measurement for Zboson production (172) reports similarly small uncertainties. However, it is not possible to obtain a good quantitative description of these data with any nuclear PDFs because of the large fluctuations of the data around midrapidity (yℓℓ =0), which lead,for instance, to an RFB that does not tend to unity toward yℓℓ →0 as one would expect.Along with the on-shell Zproduction, the CMS experiment also measured low-mass cross sections in Lepton rapidity (c.m. frame) Lepton rapidity (c.m. frame) NLO QCD EPPS21 nCTEQ15HQ nNNPDF3.0 Isospin only NLO QCD EPPS21 nCTEQ15HQ nNNPDF3.0 Isospin only Rp+Pb Rp+Pb 1.2 1.3 1.1 1.0 0.9 0.8 0.7 0.6 –2 –1 1 20 –2 –1 1 2 0 1.2 1.3 1.4 1.1 1.0 0.9 0.8 0.7 CMS W+,p+Pb, s = 8.16 TeV CMS W–,p+Pb, s = 8.16 TeV a b Figure 6 Nuclear modification ratios for (a)W+bosons and (b)W−bosons at CMS Run 2 (170, 171) compared with EPPS21 (51), nCTEQ15HQ (50), nNNPDF3.0 (52), and a calculation with 82 free protons and 126 free neutrons. Abbreviations: c.m., center of mass; NLO, next-to-leading order; QCD, quantum chromodynamics. 70 Klasen •Paukkunen Downloaded from www.annualreviews.org. Guest (guest) IP: 130.234.241.32 On: Mon, 21 Oct 2024 05:55:46 NS74_Art03_Klasen ARjats.cls August 26, 2024 9:52 the window 15 GeV <Mℓℓ <60 GeV. Within the TUJU21 analysis (80),it was noticed that to simultaneously reproduce the normalization of the CMS low-mass and Zcross sections,the NNLO QCD corrections appear to be necessary.This is the first time the necessity of NNLO corrections has been seen in the case of p+Pb collisions. Dielectron pairs have also been measured in the ALICE experiment at Run 1 in the low-mass region Mℓℓ <3 GeV and with 0 <pT,ℓℓ <8 GeV (173), which is in principle very sensitive to the gluon density and avoids the fragmentation contribution present for real photons (174–176). Currently, the data are unfortunately still dominated by the heavy flavor (c,b) decay background, but the statistics should be improved in Run 2 and the background should be reducible with heavy flavor tagging, in particular in the LHCb experiment (141). 5.2. Photons Another electroweak probe of nuclear PDFs is the prompt production of real photons with finite pT(177–180).It proceeds directly through quark–antiquark annihilation (q¯ q→gγ) and the QCD Compton process (qg →qγ),which dominates at large pT(181).The radiation of massless photons from quarks in pure QCD processes gives rise to a photon fragmentation contribution (182–184), which is important at small pT. Isolating the photon and thus reducing the surrounding hadronic energy (e.g., in a cone) suppresses the fragmentation component and also the nonprompt photon background from hadronic (in particular pion) decays (185). Direct photon production in proton (and pion) nucleus collisions was first measured in fixed-target mode at the Fermilab E706 experiment (186). PHENIX (187) and STAR (188) measurements in D+Au collisions at RHIC are also available. At the LHC, high-pTisolated photons in p+Pb (189) collisions have been measured by the ATLAS Collaboration. Ratios of these ATLAS cross section measurements to the corresponding pp data (190) have a reduced sensitivity to missing higher-order effects, FFs, and proton PDFs. They can be reasonably described at NLO QCD and are therefore included in the nNNPDF3.0 (52) analysis. The absolute pp and p+Pb cross sections are, however, underestimated by NLO QCD by up to 30% at the lowest values of pT∼20 GeV.This could indicate the need to include NNLO corrections (191).The impact of the ATLAS prompt photon data in the global fit is small compared with dijet and heavy flavor production due to larger uncertainties. Looking toward the future, the ALICE Collaboration has proposed building a new forward calorimeter (FoCal) for LHC Run 4 (192, 193), which would be optimized for direct photons in the rapidity region 3.2 < η < 5.8. 5.3. Light Hadrons As discussed in Section 4, before the beginning of the LHC era, not much was known about the nuclear gluon PDFs. The first direct evidence for the presence of shadowing, antishadowing, and the EMC effect in gluons came from inclusive hadron production in DA collisions at RHIC. This process involves a gluon contribution already at LO and is therefore a candidate to constrain the nuclear gluons in the perturbative region—that is, when the transverse momentum of the hadron is sufficiently large.This possibility was first discussed in References 40,95, and 194 in the light of early RHIC data (195–198). The first global analysis to fit this type of data was EPS08 (96), which includednegatively charged hadrondata from theBRAHMS experiment (196)as well asPHENIX (197, 199) and STAR (200) pion data. It was, however, noticed that the rapidity dependence of the BRAHMS data at low values of pT≳2 GeV was too strong to be optimally reproduced within a globalfit,inducing tensions withthe NMCdata forFSn 2/FC 2(14).These negativelycharged hadron BRAHMS data were eventually dropped from the EPS09 analysis (42) when it was noticed that it was difficult to reproduce even the pp reference data. The later EP(P)S analyses (51, 97) have www.annualreviews.org •Nuclear PDFs After the First Decade of LHC Data 71 Downloaded from www.annualreviews.org. Guest (guest) IP: 130.234.241.32 On: Mon, 21 Oct 2024 05:55:46 NS74_Art03_Klasen ARjats.cls August 26, 2024 9:52 retained only PHENIX π0data (199), while nCTEQ15 (43) included these and updated STAR π0data (188). Inclusive hadron production is sensitive not only to PDFs but also to the final-state hadronization encoded in the parton-to-hadron FFs.In the EP(P)S and nCTEQ fits,the FFs are taken from global fits of hadron production in e+e−,eN, and pp collisions (201–205). The sensitivity to FFs was recently studied within the nCTEQ15WZ+SIH (88) analysis, which also propagated the FF uncertainties,when available,into the fit.Combining the latest RHIC D+Au data on pions,kaons, and ηmesons (188,199,200,206) with the corresponding ALICE p+Pb measurements (207–209) led to a consistent description of the data and to a considerable reduction in the nuclear gluon uncertainty. The fact that consistent global fits down to pT>3 GeV are possible can also be taken as an indication that higher-twist final-state rescattering (see Section 2.3) is indeed a subleading effect. The inclusive hadron data have therefore been retained as well in the latest nCTEQ15HQ analysis (50).Alternatively, the RHIC data in D+Au collisions have also been interpreted in terms of nuclear-modified FFs (210, 211), which were used in the DSSZ (115) global analysis of nuclear PDFs and resulted in reduced nuclear effects for the gluon PDF. The latest LHC measurements of Rp+Pb for high-pTneutral pion production come from the LHCb experiment (212), complementing the ALICE midrapidity data with forward/backward measurements. While the forward-rapidity data agree with the NLO predictions with nuclear PDFs,there appears tobea slight normalizationdifferencebetweenthe predictionsandtheLHCb data at negative rapidities. The preliminary LHCb data for ηmesons look consistent with nuclear PDFs (213). These data are not yet included in global fits. The LHCb forward pion data also agree well with the LHCb forward charged hadron (h±) data (214). The corresponding backward data are, however, in disagreement with the nuclear PDF predictions, which hints that the baryon production in the lead-going direction cannot be described solely within the factorization. The same issue is visible at midrapidity as well (215–218) and is more pronounced at RHIC (197, 198). Moreover, it has been noticed that even in simpler pp collisions at LHC energies, the collinear factorization around midrapidity appears to be applicable only at pT≳10 GeV for h±production (219). As a result, only the production of mesons is considered in global fits of nuclear PDFs. 5.4. Jets Jet measurements in p+Pb collisions probe the intermediate- to large-xregime of nuclear PDFs at large interaction scales (Q2≳103GeV2). A complication in p+Pb compared with pp collisions is the significantly larger background from the underlying event.Indeed, in Glauber-type models, an average p+Pb collision contains around 7 ±5pN interactions (220). To reduce the model dependence regarding how the multiparton interactions (MPIs) are dealt with, the jet pTmust therefore be large enough, or the jet cone must be small enough, to reduce the probability of particles from MPIs to occupy the same phase space. However, at cone sizes that are too small, the jet cross sections become unstable due to an incomplete cancellation of infrared divergences. Also, the hadronization corrections, which tend to widen the partonic jets, grow. While in pp collisions all nonperturbative corrections are applied to the theoretical predictions,the p+Pb data have already been subtracted for the backgrounds from MPIs.This works rather well; the obtained ratios Rp+Pb are broadly consistent with the expectations from nuclear PDFs. Currently, the most constraining data are the Run 1 CMS dijet data differential in the average pT(pave T) and rapidity (ηdijet) of the two jets (221). They supersede the earlier dijet data (222), which were included in the EPPS16 analysis. The cross sections are normalized to the rapidityintegrated cross section, so that most of the systematic uncertainties cancel. The resulting spectra in pp collisions are then so precise that they challenge the theoretical description, with the NLO 72 Klasen •Paukkunen Downloaded from www.annualreviews.org. Guest (guest) IP: 130.234.241.32 On: Mon, 21 Oct 2024 05:55:46 NS74_Art03_Klasen ARjats.cls August 26, 2024 9:52 be consistent with collinear factorization and process-independent nuclear PDFs. The relations between the observations discussed above remain open questions at this moment. In the case of heavy ion collisions, the formation of a QGP is now a generally accepted phenomenon. Nevertheless, even in heavy ion collisions, the LHC data for electroweak boson (346–349) and high-pTdirect photon production (350–352) are consistent with collinear factorization and process-independent nuclear PDFs (353). Thus, there is no reason to believe that the initial state of heavy ion collisions would not be dictated by nuclear PDFs. This idea has been pursued in the Eskola–Kolhinen–Ruuskanen–Tuominen model of heavy ion collisions, the most recent versions of which (354) apply NLO perturbative QCD calculations and impactparameter-dependent nuclear PDFs (355) to compute the initial conditions for the subsequent fluid-dynamical evolution of the system. Nuclear PDFs also find use in the field of neutrino astronomy and cosmic-ray physics (356– 358). Interactions of neutrinos coming from outer space can be measured using large neutrino telescopes such as IceCube, KM3NeT, and Baikal, where the neutrinos interact with water or ice. Precise theoretical calculations of the cross sections require nuclear PDFs as an input.In addition, protons from astrophysical sources can collide with the air molecules in the atmosphere, and such interactions can produce neutrinos. Precise calculations of the cross sections for these secondary neutrinos require nuclear PDFs as well. 7. CONCLUSION During the last 25 years of research in nuclear PDFs, the field has undergone enormous development.Methodologically,simple fits,performed byeye atLO,have maturedinto rigorousstatistical analyses, including machine learning techniques, at NLO and NNLO with full error estimates. Nonetheless, global nuclear PDF analyses are still driven by experimental measurements, and in this respect the p+Pb collisions carried out during the past decade at the LHC have opened up a wide, previously unexplored regime in terms of both kinematics and processes. SUMMARY POINTS 1. Despite the theoretical and experimental advances, there are still significant differences among the independent global analyses of nuclear parton distribution functions (PDFs) in terms of both the extracted nuclear modifications of PDFs and the absolute nuclear PDFs. In several places, the central values of a given analysis can be outside the error bands of the others, and in some cases even the error bands of two given analyses do not overlap. The widths of the error bands also can be very different. The most significant factors behind the observed differences can be attributed to (a) the assumed form of the nonperturbative parameterization of nuclear PDFs at low Q2, (b) the data selection, (c) the fitting of absolute cross sections versus ratios of cross sections, (d) the theoretical treatment of heavy flavors, and (e) the use of different baseline free-proton PDFs. These differences also lead to visible effects in observables. This underscores the need to carry out the global analysis independently in several groups.To faithfully chart the theoretical uncertainties in quantities that depend on nuclear PDFs, it is thus recommended to use more than one set of nuclear PDFs. 2. At the moment,next-to-leading-order (NLO) accuracy is the standard in the field of nuclear PDFs; the full next-to-next-to-leading-order (NNLO) accuracy is limited by the www.annualreviews.org •Nuclear PDFs After the First Decade of LHC Data 79 Downloaded from www.annualreviews.org. Guest (guest) IP: 130.234.241.32 On: Mon, 21 Oct 2024 05:55:46 NS74_Art03_Klasen ARjats.cls August 26, 2024 9:52 existence or public availability of cross section codes.Most pA data can be well described within the NLO calculations, but in the case of some observables there is a confirmed (Drell–Yan process below the Zpeak) or conjectured (prompt photons, jets) need for NNLO quantum chromodynamics. The NNLO accuracy also reduces the theoretical uncertainties in observables sensitive to the effects of partonic saturation, offers a standard candle to Glauber modeling of heavy ion collisions through precise predictions for electroweak observables, and should lead to more precise predictions for astrophysical applications. 3. In the long run, the global analysis of nuclear PDFs should be extended to include the proton and the deuteron. To date, most of the free-proton fits still use heavytarget—particularly neutrino deep inelastic scattering—data to constrain the full flavor decomposition. However, these same data can be (and are) also taken as constraints on the nuclear modifications of PDFs.Fully charting the interplay between the two requires simultaneous extraction of the free-proton and nuclear PDFs. DISCLOSURE STATEMENT The authors are not aware of any affiliations, memberships, funding, or financial holdings that might be perceived as affecting the objectivity of this review. ACKNOWLEDGMENTS The authors thank their nCTEQ and EPPS colleagues for their collaboration and useful discussions, E. Nocera for providing the nNNPDF3.0 values for Figure 7, and V. Guzey for providing the nCTEQ15HQ values for Figure 8b. M.K. thanks his ALICE colleagues for their collaboration and acknowledges funding from the German Federal Ministry of Education and Research (BMBF; project 05P21PMCAA) and the German Research Foundation (DFG; GRK 2149 and SFB 1225 “Isoquant,” project 273811115). H.P. acknowledges funding from the Academy of Finland through the Center of Excellence in Quark Matter (project 346326). The results shown in Figure 6 and in Figure 8ahave been computed for this review using computing resources of the Finnish IT Center for Science (CSC; project jyy2580). LITERATURE CITED 1. Frankfurt LL, Strikman MI. Phys. Rep. 160:235 (1988) 2. Arneodo M. Phys. Rep. 240:301 (1994) 3. Gelis F, Iancu E, Jalilian-Marian J, Venugopalan R. Annu. Rev. Nucl. Part. 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