Chapter 31641

The Nucleus: A Laboratory for1642

QCD1643

3.1 Introduction1644

QCD, the accepted theory of strong interactions, is in general very successful in describing1645

a broad range of hadronic and nuclear phenomena. One of the main achievements in our1646

understanding of QCD is the running of the strong coupling constant and related to it1647

asymptotic freedom, which is a name for the theoretically predicted and experimentally1648

established fact that quarks and gluons are almost free at very short (asymptotic) distances1649

inside the hadrons [116, 117]. QCD is often studied in the deep inelastic scattering (DIS)1650

experiments, in which one probes the inner structure of the proton or nucleus by scattering1651

a small probe (a lepton) on it. The lepton probes the quark distribution in the proton or1652

nucleus by exchanging a photon with it. The past DIS experiments were very successful in1653

determining the quark structure of the proton and of some light nuclei.1654

Despite the many successes in our understanding of QCD, some profound mysteries1655

remain. One of them is quark confinement: quarks can not be free (for a long time) in1656

nature and are always confined inside bound states – the hadrons. Another one is the mass1657

of the proton (and other hadrons), which, at 938 MeV is much larger than the sum of the1658

valence quark masses (about 10 MeV). Both of these problems at the moment can only1659

be tackled by numerical QCD simulations on the lattice. The current consensus is that1660

the gluons are responsible for both the quark confinement and much of the hadronic mass.1661

The gluons, which bind quarks together into mesons (bound states of a quark and an anti-1662

quark) and baryons (bound states of three quarks), significantly contribute to the masses1663

of hadrons. At the same time gluons are significantly less well-understood than quarks.1664

Unlike photons, the carriers of the electromagnetic force, gluons interact with each other.1665

The underlying non-linear dynamics of this self-interaction is hard to put under theoretical1666

control. Gluons are quite little-studied for particles providing over 98% of the proton and1667

neutron masses, generating much of the visible matter mass in the Universe.1 In addition,1668

it is known that gluons play a dominant role in high energy DIS, hadronic and nuclear1669

collisions, being responsible for much of the particle production and total cross sections1670

1One may compare the gluons to the Higgs boson, the search for which received a lot of attention inrecent decades: while the recently discovered Standard Model Higgs account for the masses of all the knownquarks along with the W± and Z bosons, this would still add up to only about 5% of the mass in the visibleUniverse.

58

in these processes. In high-energy heavy-ion collisions it is the gluons which are likely to1671

be responsible for production and thermalization of the medium made out of deconfined1672

quarks and gluons, known as the quark-gluon plasma (QGP). Clearly any progress in our1673

understanding of gluon dynamics would profoundly improve our knowledge of the strong1674

force, allowing us to better control and deeper understand this fundamental interaction.1675

In this chapter, we illustrate that DIS experiments on large nuclei (heavy ions) at high1676

energies are the best way to study gluon dynamics. We show that a large number of nu-1677

cleons in a heavy ion likely results in strong gluon fields in its wave function probed at1678

high energy, possibly leading to the phenomenon of parton (gluon) saturation, also known1679

as the Color Glass Condensate (CGC). The transition to this non-linear regime is charac-1680

terized by the saturation momentum Qs, which can be large for heavy ions. Our current1681

theoretical understanding suggests that this strong gluon field combines complex non-linear1682

QCD dynamics with a perturbatively large momentum scale Qs allowing one to perform1683

small-coupling theoretical calculations due to the asymptotic freedom property of QCD. An1684

electron-ion collider (EIC) would allow us to probe the wave functions of high energy nuclei1685

with an energetic electron: by studying these interactions one may probe the strong gluon1686

fields of the CGC. While experiments at HERA, RHIC, and LHC found evidence consistent1687

with saturation, an EIC would have the potential to seal the case, completing the discovery1688

process started at those accelerators.1689

Nuclei are made out of nucleons, which in turn, are bound states of the fundamental1690

constituents probed in high energy scattering or at short distance, namely quarks and1691

gluons. The binding of nucleons into a nucleus must be sensitive to how these quarks and1692

gluons are confined into nucleons, and must influence how they distribute inside the bound1693

nucleons. The European Muon Collaboration (EMC) discovery at CERN which revealed a1694

peculiar pattern of nuclear modification of the DIS cross section as a function of Bjorken1695

x, confirmed by measurements at several facilities in the following two decades, shows clear1696

evidence that the momentum distributions of quarks in a fast-moving nucleus are strongly1697

affected by the binding and the nuclear environment. With its much wider kinematic reach1698

in both x and Q, and the unprecedented high luminosity, the EIC not only can explore1699

the influence of the binding on the momentum distribution of sea quarks and gluons, but1700

also, for the first time, determine the spatial distribution of quarks and gluons in a nucleus1701

by diffractive or exclusive processes. In addition, the wealth of semi-inclusive probes at1702

the EIC provides direct and clean to the the fluctuations of color or density of quarks and1703

gluons in nuclei.1704

EIC would be the world’s first dedicated electron-nucleus (eA) collider. It would be1705

an excellent laboratory for exploring QCD dynamics. The experimental program of the1706

machine is targeted to answer the following fundamental questions concerning the dynamics1707

of quarks and gluons in nuclei:1708

• What is the role of strong gluon fields, parton saturation effects, and col-1709

lective gluon excitations in nuclei? Can we complete the discovery of the gluon1710

saturation/CGC regime, tantalizing hints of which may have been seen at HERA,1711

RHIC, and LHC? Accomplishing the discovery of a new regime of QCD would have1712

a profound impact on our understanding of strong interactions.1713

• Can we experimentally find the evidence of nonlinear QCD dynamics in1714

the high-energy scattering off nuclei? One of the main predictions of saturation1715

physics is that the x-dependence of DIS cross sections and structure functions, along1716

59

with other observables, is described by nonlinear evolution equations. Discovery of the1717

saturation regime would not be complete without unambiguous experimental evidence1718

in favor of these nonlinear equations.1719

• What is the momentum distribution of gluons and sea quarks in nuclei?1720

What is the spatial distribution of gluons and sea quarks in nuclei? The1721

physics of multiple rescatterings at larger-x, along with, if found, parton saturation,1722

would allow us to reconstruct the momentum and impact parameter distributions of1723

gluons and sea quarks in nuclei. At small-x the transverse momentum distribution1724

may allow us to identify the saturation scale Qs.1725

• Are there strong color (quark and gluon density) fluctuations inside a1726

large nucleus? How does the nucleus respond to the propagation of a1727

color charge through it? Our understanding of the spatial and momentum space1728

distributions of quarks and gluons inside the nuclei would not be complete without1729

studying their fluctuations. The typical size of color fluctuations can be measured1730

by sending a quark probe through the nucleus. Conversion of the quark probe into1731

a hadron (hadronization) may be affected by the nuclear environment, giving us a1732

chance at a better understanding of the process.1733

The big questions listed above can be answered by performing a set of measurements1734

in DIS on heavy ions at the EIC. The measurements relevant for the small-x eA physics1735

are described in Sec. 3.2, while those pertaining to the large-x eA physics are discussed in1736

Sec. 3.3. Some of these measurements have analogs in ep collisions but have never been1737

performed in nuclei; for these, eA collisions will allow us to understand universal features1738

of the physics of the nucleon and the physics of nuclei. Other measurements have no analog1739

in ep collisions and nuclei provide a completely unique environment to explore these.1740

The EIC would have a capability of colliding many ion species at a wide range of1741

collision energies. With its high luminosity and detector coverage, as well as its high collision1742

energies, the EIC could probe the confined motion as well as spatial distributions of quarks1743

and gluons inside a nucleus at a unprecedented resolution — one tenth of a femtometer1744

or better, and could detect soft gluons whose energy in the rest frame of the nucleus is1745

less than one tenth of the averaged binding energy needed to hold the nucleons together1746

to form the nucleus. With large nuclei, EIC could reach the saturation regime that may1747

only be reached by electron-proton collisions with a multi-TeV proton beam. The kinematic1748

acceptance of an EIC compared to all other data collected in DIS on nuclei and in Drell-Yan1749

(DY) experiments in the world is shown in Fig. 3.1. Clearly an EIC would greatly extend1750

our knowledge of strong interactions in a nuclear environment.1751

60

1

10

10-3

103

10-2

102

10-1 110-4

x

Q2 (

Ge

V2)

0.1

EIC √s =

90 G

eV, 0.0

1 ≤ y ≤ 0

.95

EIC √s =

45 G

eV, 0.0

1 ≤ y ≤ 0

.95

Measurements with A ≥ 56 (Fe):

eA/μA DIS (E-139, E-665, EMC, NMC)

νA DIS (CCFR, CDHSW, CHORUS, NuTeV)

DY (E772, E866)

perturbative

non-perturbative

Figure 3.1: Kinematic acceptance in x and Q2 of completed lepton-nucleus (DIS) and Drell-Yan(DY) experiments (all fixed target) compared to EIC energies. The acceptance bands for theEIC are defined by Q2 = x y s with 0.01 ≤ y ≤ 0.95 and values of s shown.

61

Diffractive Scattering1752

1753

Diffractive scattering has made a spectacular comeback with the observation of an un-1754

expectedly large cross-section for diffractive events at the HERA ep collider. At HERA,1755

hard diffractive events, e(k) +N(p)→ e′(k′) +N(p′) +X, were observed where the proton1756

remained intact and the highly virtual photon fragmented into a final state X that was sep-1757

arated from the scattered proton by a large rapidity gap without any particles. These events1758

are indicative of a color neutral exchange in the t-channel between the virtual photon and1759

the proton over several units in rapidity. This color singlet exchange has historically been1760

called the Pomeron, which had a specific interpretation in Regge theory. An illustration of1761

a hard diffractive event is shown in Fig. 3.2.1762

k

k'

p'p

q

gap

Mx

1763

Figure 3.2: Kinematic quantities for the de-scription of a diffractive event.

1764

The kinematic variables are similar to1765

those for DIS with the following additions:1766

t = (p− p′)2 is the square of the momentum1767

transfer at the hadronic vertex. The1768

variable t here is identical to the one1769

used in exclusive processes and gen-1770

eralised parton distributions (see the1771

Sidebar on page 43).1772

M2X = (p− p′ + k − k′)2 is the squared1773

mass of the diffractive final state.1774

η = − ln(tan(θ/2)) is the pseudorapidity of1775

a particle whose momentum has a rel-1776

ative angle θ to the proton beam axis.1777

For ultrarelativistic particles the pseu-1778

dorapidity is equal to the rapidity, η ∼1779

y = 1/2 ln((E + pL)/(E + pL)).1780

At HERA gaps of several units in rapidity have been observed. One finds that roughly1781

15% of the deep inelastic cross-section corresponds to hard diffractive events with invariant1782

masses MX > 3 GeV. The remarkable nature of this result is transparent in the proton1783

rest frame: a 25 TeV electron slams into the proton and ≈ 15% of the time, the proton is1784

unaffected, even though the virtual photon imparts a high momentum transfer on a quark1785

or antiquark in the target. A crucial question in diffraction is the nature of the color neutral1786

exchange between the proton and the virtual photon. This interaction probes, in a novel1787

fashion, the nature of confining interactions within hadrons.1788

The cross-section can be formulated analogously to inclusive DIS by defining the diffrac-1789

tive structure functions FD2 and FDL as1790

d4σ

dxB dQ2 dM2X dt

=4πα2

Q6

[(1− y +

y2

2

)FD,42 (x,Q2,M2

X , t)−y2

2FD,4L (x,Q2,M2

X , t)

].

In practice, detector specifics may limit the measurements of diffractive events to those1791

where the outgoing proton (nucleus) is not tagged, requiring instead a large rapidity gap1792

∆η in the detector. t can then only be measured for particular final states X, e.g. for J/Ψ1793

mesons, whose momentum can be reconstructed very precisely.1794

62

3.2 Physics of High Gluon Densities in Nuclei1795

Conveners: Yuri Kovchegov and Thomas Ullrich1796

In this Sec. we present a description of the physics one would like to access with the1797

small-x EIC program, along with the measurements needed to answer the related funda-1798

mental questions from the beginning of this Chap.: one needs to measure nuclear structure1799

functions F2 and FL (see the Sidebar on page 21) as functions of Bjorken-x variable and1800

photon virtuality Q2 (see the Sidebar on page 20), which allows us to extract quark and1801

gluon distribution functions of the nuclei, along with the experimental evidence for the non-1802

linear QCD effects; one needs to determine the saturation scale Qs characterizing the CGC1803

wave function by measuring two-particle correlations; the distribution of gluons, both in1804

position and momentum spaces, can be pinpointed by the measurement of the cross section1805

of elastic vector meson production; the cross sections for diffractive (quasi-elastic) events1806

are most sensitive to the onset of the non-linear QCD dynamics.1807

3.2.1 Gluon Saturation: a New Regime of QCD1808

Nonlinear Evolution1809

The proton is a bound state of three “valence” quarks: two up quarks and one down quark.1810

The simplest view of a proton reveals three quarks interacting via the exchanges of gluons,1811

which “glue” the quarks together. But experiments probing proton structure at the HERA1812

collider at Germany’s DESY laboratory, and the increasing body of evidence from RHIC1813

and LHC, suggest that this picture is far too simple. Countless other gluons and a “sea” of1814

quarks and anti-quarks pop in and out of existence within each hadron. These fluctuations1815

can be probed in high energy scattering experiments: due to Lorentz time dilation, the1816

more we accelerate a proton and the closer it gets to the speed of light, the longer are the1817

lifetimes of the gluons that arise from the quantum fluctuations. An outside “observer”1818

viewing a fast moving proton would see the cascading of gluons last longer and longer the1819

larger the velocity of the proton. So, in effect, by speeding the proton up, one can slow1820

down the gluon fluctuations enough to “take snapshots” of them with a probe particle sent1821

to interact with the high-energy proton.1822

In DIS experiments one probes the proton wave function with a lepton, which interacts1823

with the proton by exchanging a (virtual) photon with it (see the Sidebar on page 20).1824

The virtuality of the photon Q2 determines the size of the region in the plane transverse1825

to the beam axis probed by the photon: by uncertainty principle the region’s width is1826

∆rT ∼ 1/Q. Another relevant variable is Bjorken x, which is the fraction of the proton1827

momentum carried by the struck quark. At high energy x ≈ Q2/W 2 is small (W 2 is the1828

center-of-mass energy squared of the photon-proton system): therefore, small x corresponds1829

to high energy scattering.1830

The proton wave function depends on both x and Q2. An example of such dependence1831

is shown in Fig. 3.3, extracted from the data measured at HERA for DIS on a proton. Here1832

we plot the x-dependence of the parton (quark or gluon) distribution functions (PDFs). At1833

the leading order PDFs can be interpreted as providing the number of quarks and gluons1834

with a certain fraction x of the proton’s momentum. In Fig. 3.3 one can see the PDFs of1835

the valence quarks in the proton, xuv and xdv: they decrease with decreasing x. The PDFs1836

of the “sea” quarks and gluons, denoted by xG and xS in Fig. 3.3, appear to grow very1837

strongly towards the low x. In fact they had to be scaled down by a factor of 20 to even fit1838

63

0.2

0.4

0.6

0.8

1

10-4 10-3 10-2 10-1 1

experimental uncertainty

model uncertainty

parametrization uncertainty

HERAPDF1.7 (prel.)

HERAPDF1.6 (prel.)

x

xf(

x, Q

2)

Q2 = 10 GeV2

vxu

vxd

xS (× 0.05)

xG (× 0.05)

HERA

Figure 3.3: Proton parton distribution functions plotted a functions of Bjorken x. Note that thegluon and sea quark distributions are scaled down by a factor of 20. Clearly gluons dominate atsmall-x.

into the figure. One can also observe that the gluon distribution dominates over the valence1839

and “sea” quark ones already at a moderate x below x = 0.1. Remembering that low-x1840

means high energy, we conclude that the part of the proton wave function responsible for1841

the interactions in high energy scattering consists mainly of gluons.1842

The small-x proton wave function is dominated by gluons, which are likely to populate1843

the transverse area of the proton, creating a high density of gluons. This is shown in Fig. 3.4,1844

which illustrates how at the lower Bjorken-x (right panel) the partons (mainly gluons) are1845

much more numerous inside the proton then at larger-x (left panel), in agreement with1846

Fig. 3.3. This dense small-x wave function of an ultrarelativistic proton or nucleus is1847

referred to as the Color Glass Condensate (CGC) [118].1848

To understand the onset of the dense regime one usually employs QCD evolution equa-tions. The main principle is as follows: while the current state of the QCD theory does notallow for a first-principles calculation of the quark and gluon distributions, the evolutionequations, loosely-speaking, allow one to determine these distributions at some values of(x,Q2) if they are initially known at some other (x0, Q

20). The most widely used evolution

equation is the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equation [10, 11, 9]: ifthe PDFs are specified at some initial virtuality Q2

0, DGLAP equation allows one to findthe parton distributions at Q2 > Q2

0 at all x where DGLAP evolution is applicable. Theevolution equation that allows one to construct the parton distributions at low-x, giventhe value of it at some x0 > x and all Q2, is the Balitsky-Fadin-Kuraev-Lipatov (BFKL)evolution equation [119, 120]. This is a linear evolution equation, which is illustrated bythe first term on the right hand side of Fig. 3.5. The wave function of a high-energy protonor nucleus containing many small-x partons is shown on the left of Fig. 3.5. As we makeone step of evolution by boosting the nucleus/proton to higher energy in order to probe itssmaller-x wave function, either one of the partons can split into two partons, leading to an

64

many newsmaller partonsare produced

Proton

(x, Q2)

Proton

(x0, Q2)

x0 >> x

Low Energy High Energy

parton

“Color Glass Condensate”

Figure 3.4: Proton wave function at small-x (shown on the right) contains a large number ofgluons (and quarks) as compared to the same wave function at a larger x = x0 (shown on theleft). The figure is a projection on the plane transverse to the beam axis (the latter is shownby arrows coming “out of the page”, with the length of the arrows reflective of the momentumof the proton).

increase in the number of partons proportional to the number of partons N at the previousstep,

∂ N(x, rT )

∂ ln(1/x)= αsKBFKL ⊗ N(x, rT ), (3.1)

with KBFKL an integral kernel and αs the strong coupling constant. In DIS at high energy,1849

the virtual photon splits into a quark-antiquark dipole which interacts with the proton:1850

the dipole scattering amplitude N(x, rT ) probes the gluon distribution in the proton at the1851

transverse distance rT ∼ 1/Q.2 Note that a Fourier transform of N(x, rT ) is related to1852

the gluon transverse momentum distribution (TMD) f(x, kT ) from Chap. 2. The BFKL1853

evolution leads to the power-law growth of the parton distributions with decreasing x, such1854

that N ∼ (1/x)λ with λ a positive number [119]. This behavior may account for the increase1855

of gluon density at small-x in the HERA data of Fig. 3.3.1856

splitting recombination

Figure 3.5: Nonlinear small-x evolution of a hadronic or nuclear wave functions. All partons(quarks and gluons) are denoted by straight solid lines for simplicity.

The question arises whether the gluon and quark densities can grow without limit atsmall-x. While there is no strict bound on the number density of gluons in QCD, thereis a bound on the scattering cross sections stemming from unitarity. Indeed a proton (or

2In general, the dipole amplitude also depends on the impact parameter bT of the dipole (cf. Sec. 2.4.6):for simplicity we suppress this dependence in N(x, rT ).

65

nucleus) with a lot of “sea” gluons, is more likely to interact in high energy scattering,which leads to larger scattering cross sections: therefore the bound on cross sections shouldhave implications for the gluon density. The cross section bound arises due to the blackdisk limit known from quantum mechanics: high-energy total scattering cross section of aparticle on a sphere of radius R is bounded by

σtot ≤ 2π R2. (3.2)

In QCD the black disk limit translates into the Froissart–Martin unitarity bound, which1857

states that the total hadronic cross section can not grow faster than ln2 s at very high1858

energies with s the center-of-mass energy squared [121]. The cross section resulting from1859

the BFKL growth of the gluon density in the proton or nucleus wave function grows as a1860

power of energy, σtot ∼ sλ, and clearly violates both the black disk limit and the Froissart–1861

Martin bound at very high energy.1862

We see that something has to modify the BFKL evolution at high energy to prevent itfrom becoming unphysically large. The modification is illustrated on the far right of Fig. 3.5:at very high energies (leading to high gluon densities) partons may start to recombine witheach other on top of the splitting. The recombination of two partons into one is proportionalto the number of pairs of partons, which, in turn, scales as N2. We end up with the followingnon-linear evolution equation:

∂ N(x, rT )

∂ ln(1/x)= αsKBFKL ⊗ N(x, rT )− αs [N(x, rT )]2. (3.3)

This is the Balitsky-Kovchegov (BK) evolution equation [122, 123, 124, 125], which is valid1863

for QCD in the limit of the large number of colors Nc.3 Generalization of Eq. (3.3) be-1864

yond the large-Nc limit is accomplished by the Jalilian-Marian–Iancu–McLerran–Weigert–1865

Leonidov–Kovner (JIMWLK) [128, 129, 130, 131, 118] evolution equation, which is a func-1866

tional differential equation.1867

The physical impact of the quadratic term on the right of Eq. (3.3) is clear: it slows down1868

the small-x evolution, leading to parton saturation, when the number density of partons1869

stops growing with decreasing x. The corresponding total cross sections satisfy the black1870

disk limit of Eq. (3.2). The effect of gluon mergers becomes important when the quadratic1871

term in Eq. (3.3) becomes comparable to the linear term on the right-hand-side. This gives1872

rise to the saturation scale Qs, which grows as Q2s ∼ (1/x)λ with decreasing x [126, 132, 133].1873

Classical Gluon Fields and the Nuclear “Oomph” Factor1874

We have argued above that parton saturation is a universal phenomenon, valid both for1875

scattering on a proton or a nucleus. Here we demonstrate that nuclei provide an extra1876

enhancement of the saturation phenomenon, making it easier to observe and study experi-1877

mentally.1878

Imagine a large nucleus (a heavy ion), which was boosted to some ultrarelativistic ve-1879

locity, as shown in Fig. 3.6. We are interested in the dynamics of small-x gluons in the wave1880

function of this relativistic nucleus. One can show that due to the Heisenberg uncertainly1881

principle the small-x gluons interact with the whole nucleus coherently in the longitudinal1882

(beam) direction: therefore, only the transverse plane distribution of nucleons is important1883

3An equation of this type was originally suggested by Gribov, Levin and Ryskin in [126] and by Muellerand Qiu in [127], though at the time it was assumed that the quadratic term was only the first non-linearcorrection with higher order terms expected to be present as well: in [122, 124] the exact form of the equationwas found, and it was shown that in the large-Nc limit Eq. (3.3) does not have any higher-order terms in N .

66

Boost

Figure 3.6: Large nucleus before and after an ultrarelativistic boost.

for the small-x wave function. As one can see from Fig. 3.6, after the boost, the nucleons,1884

as “seen” by the small-x gluons with large longitudinal wavelength, appear to overlap with1885

each other in the transverse plane, leading to high parton density. Large occupation num-1886

ber of color charges (partons) leads to classical gluon field dominating the small-x wave1887

function of the nucleus. This is the essence of the McLerran-Venugopalan (MV) model1888

[134]. According to the MV model, the dominant gluon field is given by the solution of1889

the classical Yang-Mills equations, which are the QCD analogue of Maxwell equations of1890

electrodynamics.1891

The Yang-Mills equations were solved for a single nucleus exactly [135, 136]; their so-1892

lution was used to construct unintegrated gluon distribution (gluon TMD) φ(x, k2T ) shown1893

in Fig. 3.7 (multiplied by the phase space factor of the gluon’s transverse momentum kT )1894

as a function of kT .4 Fig. 3.7 demonstrates the emergence of the saturation scale Qs: the

αs << 1αs ∼ 1 ΛQCD

know how to

do physics here?

ma

x. d

en

sity

Qs

kT

~ 1/kT

kT φ

(x,

kT2)

Figure 3.7: Unintegrated gluon distribution (gluon TMD) φ(x, k2T ) of a large nucleus due to

classical gluon fields (solid line). The dashed curve denotes the lowest-order perturbative result.

1895

majority of gluons in this classical distribution have transverse momentum kT ≈ Qs. Note1896

that the gluon distribution slows down its growth with decreasing kT for kT < Qs (from a1897

power-law of kT to a logarithm, as can be shown by explicit calculations): the distribution1898

4Note that in the MV model φ(x, k2T ) is independent of Bjorken-x: its x-dependence comes in though the

BK/JIMWLK evolution equations described above.

67

saturates, justifying the name of the saturation scale.1899

The gluon field arises from all the nucleons in the nucleus at a given location in the1900

transverse plane (impact parameter). As the nucleon density in the nucleus is approximately1901

constant away from the edges, the number of nucleons at a fixed impact parameter is1902

simply proportional to the thickness of the nucleus in the longitudinal (beam) direction,1903

which, in turn, is proportional to the nuclear radius R, which, for a large nucleus scales1904

as R ∼ A1/3 with the nuclear mass number A. The transverse momentum kT of the1905

gluon can be thought of as arising from many transverse momentum “kicks” acquired from1906

interactions with the partons in all the nucleons at a given impact parameter. Neglecting1907

the correlations between nucleons, which is justified for a large nucleus in the leading power1908

of A approximation, once can think of the “kicks” as being random: just like in the random1909

walk problem, after A1/3 random kicks the typical transverse momentum, and hence the1910

saturation scale, becomes Qs ∼√A1/3, such that Q2

s ∼ A1/3. We see that the saturation1911

scale for heavy ions QAs is much larger than the saturation scale of the proton Qps (at the1912

same x), since (QAs )2 ≈ A1/3 (Qps)2 [126, 127, 137, 134]. This enhancement factor A1/3 of the1913

saturation scale squared is often referred to as the nuclear “oomph” factor, since it reflects1914

the enhancement of saturation effects in the nucleus as compared to the proton. For the1915

gold nucleus with A = 197 the nuclear “oomph” factor is A1/3 ≈ 6.1916

Map of High Energy QCD and the Saturation Scale1917

We summarize our theoretical knowledge of high energy QCD discussed above in Fig. 3.8,1918

in which different regimes are plotted in the (Q2, Y = ln 1/x) plane. On the left of Fig. 3.81919

we see the region with Q2 ≤ Λ2QCD in which the strong coupling is large, αs ∼ 1, and1920

small-coupling approaches do not work. (ΛQCD is the QCD confinement scale.) In the1921

perturbative region, Q2 � Λ2QCD, where the coupling is small, αs � 1, we see the standard1922

DGLAP evolution and the linear small-x BFKL evolution, denoted by the horizontal and1923

vertical arrows correspondingly. The BFKL equation evolves gluon distribution toward1924

small-x, where parton density becomes large and parton saturation sets in. Transition1925

to saturation is described by the non-linear BK and JIMWLK evolution equations. Most1926

importantly this transition happens at Q2s � Λ2

QCD where the small-coupling approach is1927

valid.1928

Saturation/CGC physics provides a new way of tackling the problem of calculatinghadronic and nuclear scattering cross sections. It is based on the theoretical observation thatsmall-x hadronic and nuclear wave functions, and, therefore, the scattering cross sectionsas well, are described by an internal momentum scale, the saturation scale Qs [126]. Aswe argued above, the saturation scale grows with decreasing x (and, conversely, with theincreasing center-of-mass energy

√s) and with the increasing mass number of a nucleus A

(in the case of a nuclear wave function) approximately as

Q2s(x) ∼ A1/3

(1

x

)λ(3.4)

where the best current theoretical estimates of λ give λ = 0.2 – 0.3 [138], in agreement withthe experimental data collected at HERA [139, 140, 141, 142] and at RHIC [138]. Therefore,for hadronic collisions at high energy and/or for collisions of large ultrarelativistic nuclei,the saturation scale becomes large, Q2

s � Λ2QCD. Since for the total and particle production

cross sections Qs is usually the largest momentum scale in the problem, we expect it to be

68

Geometri

c

Scalin

g

Y =

ln

1/x

no

n-p

ert

urb

ative

re

gio

n

ln Q2

Q2s(Y)

saturationregion

Λ2QCD

αs << 1αs ~ 1

BK/JIMWLK

DGLAP

BFKL

Figure 3.8: Map of high energy QCD in the (Q2, Y = ln 1/x) plane.

the scale determining the value of the running QCD coupling constant, making it small,

αs(Q2s) � 1, (3.5)

and allowing for first-principles calculations of total hadronic and nuclear cross sections,1929

along with extending our ability to calculate particle production and to describe diffraction1930

in a small-coupling framework. For detailed descriptions of the physics of parton saturation1931

and CGC we refer the readers to the review articles [143, 144, 145, 146] and to an upcoming1932

book [147].1933

Eq. (3.4) can be written in the following simple pocket formula if one puts λ = 1/3,which is close to the range of λ quoted above. One has

Q2s(x) ∼

(A

x

)1/3

. (3.6)

From the pocket formula (3.6) we see that the saturation scale of the gold nucleus (A = 197)1934

is as large as that for a proton at the 197-times smaller value of x! Since lower values of x1935

can only be achieved by increasing the center-of-mass energy, which could be prohibitively1936

expensive, we conclude that at the energies available at the modern-day colliders one is1937

more likely to complete the discovery of the saturation/CGC physics started at HERA,1938

RHIC, and LHC by performing DIS experiments on nuclei.1939

This point is further illustrated in Fig. 3.9, which shows our expectations for the satu-1940

ration scale as a function of x coming from the saturation-inspired Model-I [148] and from1941

the prediction of the BK evolution equation (with higher order perturbative corrections1942

included in its kernel) dubbed Model-II [139, 140]. One can clearly see from the left panel1943

that the saturation scale for Au is larger than the saturation scale for Ca, which, in turn,1944

is much larger than the saturation scale for the proton: the “oomph” factor of large nuclei1945

is seen to be quite significant.1946

As we argued above, the saturation scale squared is proportional to the thickness of1947

the nucleus at a given impact parameter b. Therefore, the saturation scale depends on the1948

69

-510

-410

-310

-210

-110

1

10

-510

-410

-310

-210

-110

1

10

x x

Q2 (

Ge

V2)

Q2 (

Ge

V2)

Q2s,quark Model-I

b=0Au, median b

Ca, median b

p, median b

Q2s,quark, all b=0

Au, Model-II

Ca, Model-IICa, Model-I

Au, Model-I

xBJ × 300

~ A1/3

Au

Au

p

Ca

Ca

Figure 3.9: Theoretical expectations for the saturation scale as a function of Bjorken x for theproton along with Ca and Au nuclei.

impact parameter, becoming larger for small b ≈ 0 (for scattering through the center of the1949

nucleus, see Fig. 3.6) and smaller for large b ≈ R (for scattering on the nuclear periphery,1950

see Fig. 3.6). This can be seen in the left panel of Fig. 3.9 where most values of Qs are1951

plotted for median b by solid lines, while, for comparison, the Qs of gold is also plotted1952

for b = 0 by the dashed line: one can see that the saturation scale at b = 0 is larger than1953

at median b. The curves in the right panel of Fig. 3.9 are plotted for b = 0: this is why1954

they give higher values of Qs than the median-b curves shown in the left panel for the same1955

nuclei.1956

This A-dependence of the saturation scale, including a realistic impact parameter de-1957

pendence, is the raison d’etre for an electron-ion collider. Collisions with nuclei probe the1958

same universal physics as seen with protons at values of x at least two orders of magnitude1959

lower (or equivalently an order of magnitude larger√s). Thus the nucleus is an efficient1960

amplifier of the universal physics of high gluon densities allowing us to study the saturation1961

regime in eA at significantly lower energy than would be possible in ep. For example, as1962

can be seen from Fig. 3.9, Q2s ≈ 7 GeV2 is reached at x = 10−5 in ep collisions requiring a1963

collider providing a center-of-mass energy of almost√s ≈

√Q2s/x ≈ 1 TeV, while in eAu1964

collisions only√s ≈ 60 GeV is required to achieve comparable gluon density and the same1965

saturation scale.1966

To illustrate the conclusion that Qs is an increasing function of both A and 1/x we1967

show a plot of its dependence on both variables in Fig. 3.10 using Model-I of Fig. 3.9. One1968

can see again from Fig. 3.10 that larger Qs can be obtained by increasing the energy or by1969

increasing mass number A.1970

Measurements extracting the x, b and A dependence of the saturation scale provide very1971

useful information on the momentum distribution and space-time structure of strong color1972

fields in QCD at high energies. The saturation scale defines the transverse momentum1973

of the majority of gluons in the small-x wave function, as shown in Fig. 3.7, thus being1974

instrumental to our understanding of the momentum distributions of gluons. The impact1975

parameter dependence of the saturation scale tells us how the gluons are distributed in the1976

70

10-510-4

10-310-2

1

10

0.1

Λ2QCD

Q2 (

Ge

V2)

200 120 40

A xP

roto

n

Calc

ium

Gold

Parton Gas

Color Glass Condensate

Confinement Regime

EIC

Coverage

Figure 3.10: The theoretical expectations for the saturation scale at medium impact parameterfrom Model-I as a function of Bjorken-x and the nuclear mass number A.

transverse coordinate plane, clarifying the spatial distribution of the small-x gluons in the1977

proton or nucleus.1978

Nuclear Structure Functions1979

The plots in Figs. 3.8, 3.9 and 3.10 suggest a straightforward way of finding saturation/CGC1980

physics: if we perform the DIS experiment on a proton, or, better yet, on a nucleus, and1981

measure the DIS scattering cross section as a function of x and Q2, then, at sufficiently1982

low x and Q2 one may be able to see the effects of saturation. As explained in the Sidebar1983

on page 21, the total DIS cross section is related to the structure functions F2(x,Q2) and1984

FL(x,Q2) by a linear relation. One finds that the structure function F2 is more sensitive1985

to the quark distribution xq(x,Q2) of the proton or nucleus, while the structure function1986

FL measures the gluon distribution xG(x,Q2) [9, 149]. Saturation effects can thus be seen1987

in both F2 and FL at low x and Q2, although, since saturation is gluon-driven, one would1988

expect FL to manifest them stronger.1989

The nuclear effects on the structure functions can be quantified by the ratios

R2(x,Q2) ≡ FA2 (x,Q2)

AF p2 (x,Q2), RL(x,Q2) ≡ FAL (x,Q2)

AF pL(x,Q2)(3.7)

for the two structure functions, where the superscripts p and A label the structure functionsfor the protons and nuclei correspondingly. Ratios like those in Eq. (3.7) can be constructedfor the quark and gluon nuclear PDFs too. The ratio for the gluon distribution comparesthe number of gluons per nucleon in the nucleus to the number of gluons in a single freeproton. Since the structure function FL measures the gluon distribution xG(x,Q2) [9, 149],the ratio RL(x,Q2) is close to the ratio RG(x,Q2) of the gluon PDFs in the nucleus and

71

the proton normalized the same way,

RG(x,Q2) ≡ xGA(x,Q2)

AxGp(x,Q2). (3.8)

10-4

10-3

10-2

10-1

1

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

RGP

b(x

, Q

2=

1.6

9 G

eV

2)

x

EPS09LO

EKS98

rcBK (Q2 = 1.85 GeV2)

HKN07 (LO)

nDS (LO)

Figure 3.11: Theoretical predictions for RG(x,Q2) plotted at Q2 = 1.69 GeV2 for a Pb nucleus:the models corresponding to different curves are explained in the plot legend. The models are:EPS09 [150], EKS 98 [151] (based on the leading-order (LO) global DGLAP analysis), HKN07 [152], nDS [153] (next-to-leading-order (NLO) DGLAP analysis), and rcBK [139], plottedfor Q2 = 1.85 GeV2 (based on BK nonlinear evolution with the running-coupling corrections(rcBK) [154, 155, 156, 157], referred to as Model-II in Sec. 3.2.1). The light-gray shaded areadepicts the uncertainty band of EPS09, while the blue shaded area indicates the uncertaintyband of the rcBK approach.

A sample of theoretical predictions for the ratio RG(x,Q2) for the gluon PDFs is plotted1990

in Fig. 3.11, comprising several DGLAP-based models along with the saturation-based1991

prediction. Note that DGLAP equation, describing evolution in Q2, can not predict the x1992

dependence of distribution functions at low-x without the data at comparable values of x1993

and at lower Q2: hence the DGLAP-based “predictions” in Fig. 3.11 strongly suffer from1994

the uncertainty in various ad hoc parameterizations of the initial conditions for DGLAP1995

evolution. Conversely, the saturation prediction is based on the BK equation (3.3), which1996

is an evolution equation in x, generating a very specific x-dependence of the distribution1997

functions that follows from QCD: this leads to a narrow error band for the saturation1998

prediction.1999

All existing approaches predict that the ratio RG would be below one at small-x: this is2000

the nuclear shadowing phenomenon [158], indicating that the number of small-x gluons per2001

nucleon in a nucleus is lower than that in a free proton. In the DGLAP-based description2002

of the nuclear PDFs shadowing is included in the parameterizations of the initial conditions2003

for DGLAP evolution. In the saturation/CGC approach gluon mergers and interactions2004

dynamically lead to the decrease in the number of gluons (and other partons) per nucleon2005

as compared to that in a single proton: this results in the shadowing of PDFs and reduction2006

of structure functions as well.2007

One can clearly see from Fig. 3.11 that new data is desperately needed to constrain the2008

DGLAP-based prediction and/or to test the prediction of the saturation physics. It is also2009

clear that such data would eliminate some of the predictions shown in Fig. 3.11, allowing us2010

to get closer to finding the model describing the correct physics. Still, as one can infer from2011

Fig. 3.11, due to the multitude of theoretical predictions, the RG (or RL) measurement2012

72

alone may only rule out some of them, leaving several predictions in agreement with the2013

data within the experimental error bars. As we detail further in Sec. 3.2.2 one would need2014

other measurements, like measurements of R2, FA2 , FAL , along with those described below2015

in Sec. 3.2.2, to uniquely determine the physics involved in high-energy DIS on the nucleus.2016

Nuclear effects in the structure functions can also be quantified using their expansionin powers of 1/Q2 [159]. The standard linear perturbative QCD approaches calculate theleading term in 1/Q2 expansion of structure functions, the order-1 contribution, referred toas the ’leading twist’ term. The multiple rescatterings of Sec. 3.2.1 along with the gluonmergers of Sec. 3.2.1 contribute to all orders in the 1/Q2 expansion. Of particular interestis their contribution to the non-leading powers of 1/Q2, known as ’higher twists’: the mainparts of those corrections are enhanced by the nuclear “oomph” factor A1/3 and by a powerof (1/x)λ, coming in as

∼Λ2QCD A

1/3

Q2

(1

x

)λ. (3.9)

We see that the telltale sign of saturation physics are the higher twist corrections, which2017

are enhanced in DIS on a nucleus, and at smaller-x.52018

(FL -

FLle

ad

ing

tw

ist )/

FL

(FL -

FLle

ad

ing

tw

ist )/

FL

Au (A=197)proton

-5-4

-3-2

1

2

3-10

-8

-6

-4

-2

0

log10(x) log 10

(Q2 )

log10(x) log 10

(Q2 )-5

-4-3

-2

1

2

3-10

-8

-6

-4

-2

0

Figure 3.12: Plots of the ratio from Eq. (3.10) for ep and eAu scattering from [160], demon-strating the sensitivity of nuclear structure function FL to the higher-twist effects. The plots godown to x = 10−5 as the smallest-x reachable at an EIC (see Fig. 3.1).

To illustrate the effect of higher twist corrections on the nuclear structure function weplot their relative contribution to FL defined by

FL − FL(leading twist)

FL(3.10)

in Fig. 3.12 as a function of x and Q2 as expected in the framework of the saturation-2019

inspired Golec-Biernat–Wusthoff (GBW) model [161, 142], which has been quite successful2020

in describing the HERA ep data. The left panel of Fig. 3.12 is for ep scattering, while the2021

right one is for eAu. Note that the ratio is negative in both plots, indicating that higher2022

5In fact, equating the correction in Eq. (3.9) to the leading-twist order-1 term gives the saturation scaleof Eq. (3.4) as the value of Q2 at which the higher-twist corrections become important.

73

twists tend to decrease the structure function. It is also clear from both plots that the effect2023

of higher twists becomes stronger at smaller-x, as expected from Eq. (3.9). Comparing2024

the two panels in Fig. 3.12 we see that the higher twist effects are also stronger in eAu2025

scattering due to nuclear enhancement. Fig. 3.12 demonstrates that the structure function2026

FL is rather sensitive to parton saturation. Experimentally it is impossible to single out2027

the higher-twist contribution if the Q2 of interest is too high, making it difficult to plot2028

the ratio from Eq. (3.10) to verify the prediction in Fig. 3.12. At lower Q2, experimental2029

separation of the leading twist contribution from the higher-twist terms may also become a2030

problem. Theoretical work is currently under way to enable the separation of higher twist2031

terms in FL (and F2), which is likely to make the ratio (3.10) an observable which could be2032

measured at an EIC.2033

Diffractive Physics2034

The phenomenon of diffraction is familiar to us from many areas of physics and is generally2035

understood to arise from the constructive or destructive interference of waves. Perhaps2036

the best analogy of diffraction in high-energy QCD comes from optics: imagine a standard2037

example of a plane monochromatic wave with the wave number k incident on a circular2038

screen of radius R (an obstacle). The diffractive pattern of the light intensity on a plane2039

screen behind the circular obstacle is shown in the left panel of Fig. 3.13 as a function of2040

the deflection angle θ, and features the well-known diffractive maxima and minima. The2041

positions of the diffractive minima are related to the size of the obstacle by θi ∼ 1/(k R)2042

for small-angle diffraction.2043

dσ

/dt

|t|

Coherent/Elastic

Incoherent/Breakup

t1 t2 t3 t4

Light

Intensity

θ2 θ3 θ4θ10 Angle

Figure 3.13: Left panel: diffractive pattern of light on a circular obstacle in wave optics. Rightpanel: diffractive cross section in high energy scattering. The elastic cross section in the rightpanel is analogous to the diffractive pattern in the left panel if we identify |t| ≈ k2 θ2.

Elastic scattering in QCD has a similar structure: imagine a hadron (a projectile)2044

scattering on a target nucleus. If the scattering is elastic both the hadron and the nucleus2045

will be intact after the collision. The elastic process is described by the differential scattering2046

cross section dσel/dt with the Mandelstam variable t describing the momentum transfer2047

between the target and the projectile. A typical dσel/dt is sketched by the solid line in2048

74

the right panel of Fig. 3.13 as a function of t. Identifying the projectile hadron with the2049

incident plane wave in the wave optics example, the target nucleus with the obstacle, and2050

writing |t| ≈ k2 θ2 valid for small angles, we can see that the two panels of Fig. 3.13 exhibit2051

analogous diffractive patterns and, therefore, describe very similar physics! The minima2052

(and maxima) of the cross section dσel/dt in the right panel of Fig. 3.13 are also related2053

to the inverse size of the target squared, |ti| ∼ 1/R2. This is exactly the same principle as2054

employed for spatial imaging of the nucleons as described in Sec. 2.3.2055

The essential difference between QCD and wave optics is summarized by two facts: (i)2056

The proton/nuclear target is not always an opaque “black disk” obstacle of geometric optics.2057

A smaller projectile, which interacts weaker due to color-screening and asymptotic freedom,2058

is likely to produce a different diffractive pattern from the larger stronger-interacting pro-2059

jectile. (ii) The scattering in QCD does not have to be completely elastic: the projectile or2060

target may break up. The event is still called diffractive if there is a rapidity gap, as de-2061

scribed in the Sidebar on 62. The cross section for the target breakup (leaving the projectile2062

intact) is plotted by the dotted line in the right panel of Fig. 3.13, and does not exhibit the2063

diffractive minima and maxima.2064

The property (i) is very important for diffraction in DIS in relation to the satura-2065

tion/CGC physics. As we have seen above, owing to the uncertainty principle, at higher Q22066

the virtual photon probes shorter transverse distances, and is less sensitive to saturation2067

effects. Conversely, the virtual photon in DIS with the lower Q2 is likely to be more sensitive2068

to saturation physics. Due to the presence of a rapidity gap, the diffractive cross section2069

can be thought of as arising from an exchange of several parton with zero net color between2070

the target and the projectile (see the Sidebar on page 62). In high-energy scattering, which2071

is dominated by gluons, this color neutral exchange (at the lowest order) consists of at least2072

two exchanged gluons. We see that compared to the total DIS cross section, which can be2073

mediated by a single gluon or quark exchange, the diffractive cross section includes more2074

interactions, and, therefore, is likely to be more sensitive to the saturation phenomena,2075

which, at least in the MV model, are dominated by multiple rescatterings. In fact, some2076

diffractive processes are related to the square of the gluon distribution xG. We conclude2077

that diffractive cross section is likely to be a very sensitive test of the saturation physics.2078

Of particular interest is the process of elastic vector meson (V ) production, e + A →2079

e+V +A. The cross-section dσ/dt for such process at lower Q2 is sensitive to the effects of2080

parton saturation [162], as we will explicitly demonstrate below. For a vector meson with2081

a sufficiently spread-out wave function (a large meson, like φ or ρ), varying Q2 would allow2082

one to detect the onset of the saturation phenomenon [162].2083

Diffraction can serve as a trigger of the onset of the black disk limit of Eq. (3.2). In thatregime, the total diffractive cross section σdiff (including all the events with rapidity gaps)would constitute 50% of the total cross section,

σdiff

σtot=

1

2. (3.11)

This may sound counter-intuitive: indeed, the naive expectation in QCD is that events with2084

gaps in rapidity are exponentially suppressed. It was therefore surprising to see that a large2085

fraction (approximately 15%) of all events reported by HERA experiments are rapidity gap2086

events [163]. This corresponds to a situation where the projectile electron slams into the2087

proton at rest with an energy 50,000 times the proton rest energy and in about 1 in 72088

such scatterings, nothing happens to the proton. In the black disk regime this ratio should2089

increase to 1 in 2 events.2090

75

3.2.2 Key Measurements2091

The main goal of the eA program at an EIC is to unveil the collective behavior of densely2092

packed gluons under conditions where their self-interactions dominate, a regime where non-2093

linear QCD supersedes “conventional” linear QCD. The plain fact that there is no data from2094

this realm of the nuclear wave function available is a already a compelling enough reason to2095

build an EIC. It is truly terra incognita. However, our goal is not only to observe the onset2096

of saturation but to explore its properties and reveal its dynamical behavior. As explained2097

above, the saturation scale squared for nuclei includes an “oomph” factor of A1/3 making2098

it larger than in the proton (cf. Eq. (3.6)); Fig. 3.14 demonstrates that. While at an EIC2099

a direct study of the saturation region in the proton is impossible (while remaining in the2100

perturbative QCD region where the coupling αs is small, i.e., above the horizontal dashed2101

line in the figure), this A1/3 enhancement may allow us to study the saturation region of2102

large nuclei, such as gold (Au). The borders of the kinematic reach of the EIC are shown2103

by the diagonal black lines in Fig. 3.14, labeled by the energies of the electron beam + per2104

nucleon energies of the nuclear beam, corresponding to different stages of the machine: the2105

actual kinematic reach regions are to the right of the border lines.2106

-510

-410

-310

-210

-110

1

10

Q 2s,quark Au, median b Ca, median b

p, median b

x

Q2 (

Ge

V2)

perturbative

non-perturbative

10 G

eV o

n 10

0 G

eV

20 G

eV o

n 10

0 G

eV

30 G

eV o

n 10

0 G

eV

5 G

eV o

n 10

0 G

eV

Figure 3.14: Kinematic reach in x and Q2 of the EIC for different electron beam energies,given by the regions to the right of the diagonal black lines, compared with predictions of thesaturation scale, Qs, in p, Ca, and Au from Model-I (see Sec. 3.2.1). Electron beam energiesup to 5 GeV can be achieved in the first EIC stage, stage-I, while higher energies (up to 30GeV) will only be achieved in stage-II. (Note that x < 0.01 in the figure.)

A wide range of measurements with EIC can distinguish between predictions in the2107

CGC, or other novel frameworks, and those following from the established DGLAP evolution2108

equations. However, these comparisons have to be made with care. Nonlinear models are2109

valid only at or below the saturation scale, Q2s, while perturbative QCD (pQCD) based on2110

the linear DGLAP evolution equation is strictly only applicable at large Q2. In the range2111

Q2 < Q2s, solely nonlinear theories such as the CGC can provide quantitative calculations. It2112

is only in a small window of approximately 1 . Q2 . 4 GeV2 where a comparison between2113

the two approaches can be made (see Fig. 3.14). Due to the complexity of high energy2114

76

nuclear physics, at the end the final insight will come from the thorough comparison of2115

models calculations with a multitude of measurements, each investigating different aspects2116

of the low-x regime. We will learn from varying the ion species, A, from light to heavy nuclei,2117

studying the Q2, x, and t dependence of the cross section in inclusive, semi-inclusive, and2118

exclusive measurements in DIS and diffractive events.2119

In what follows we discuss a small set of key measurements whose ability to extract2120

novel physics is beyond question. They serve primarily to exemplify the very rich physics2121

program available at an EIC. These “golden” measurements are summarized in Tab. 3.1,2122

where also their feasibility in stage-I (medium energy) and stage-II (full EIC energy) is2123

indicated. These measurements are discussed in further detail in the remainder of this Sec.2124

It should be stressed that the low-x physics program will only reach its full potential when2125

the beam energies are large enough to reach sufficiently deep into the saturation regime.2126

Ultimately this will only be possible in stage-II where x ∼ 10−4 can be reached at Q2 values2127

of 1–2 GeV2 as indicated in Fig. 3.14. Only the highest energies will give us enough lever2128

arm in Q2 to study the crossing into the saturation region allowing us at the same time the2129

comparison with DGLAP-based pQCD and CGC predictions.2130

Deliverables Observables What we learn Stage-I Stage-II

Integrated gluon F2,L Nuclear wave Gluons at Exploration

momentum function; 10−3 . x . 1 of the saturation

distributions GA(x,Q2) saturation regime

kT -dependent Di-hadron Non-linear QCD Onset of Nonlinear

gluons f(x, kT ); correlations evolution/universality; saturation; small-x

gluon correlations saturation scale Qs Qs measurement evolution

Spatial gluon Diffractive dissociation Nonlinear small-x saturation Spatial

distributions f(x, bT ); σdiff/σtot evolution; vs. non-saturation gluon

gluon correlations vector mesons & DVCS saturation dynamics; models distribution;

dσ/dt, dσ/dQ2 black disk limit Qs vs centrality

Table 3.1: Key measurements in eA collisions at an EIC addressing the physics of high gluon densities.

The statistical error bars depicted in the figures described in this section are derived2131

by assuming an integrated luminosity of∫Ldt = 10 fb−1/A for each species and include2132

experimental cuts (acceptance and momentum). Systematical uncertainties were estimated2133

in a few cases based on experience from HERA. Ultimately they will depend on the details2134

of detectors and machine and hence cannot be fully addressed at this time.2135

Structure Functions2136

As we mentioned above in Sec. 3.2.1, the differential unpolarized cross section for DIS is fully2137

described by a set of basic kinematic variables and two structure functions, F2(x,Q2) and2138

FL(x,Q2), that encapsulate the rich structure of quarks and anti-quarks (F2) and gluons2139

(FL). The structure function FL is directly proportional to the gluon distribution function,2140

FL(x,Q2) ∝ αs xG(x,Q2), at low x and not very small Q2 [9, 149]. The precise knowledge2141

of FL is mandatory for the study of gluons and their dynamics in nucleons and nuclei (see2142

the Sidebar on page 21).2143

As demonstrated in Sec. 3.2.1 and shown in Fig. 3.11, various models have different pre-2144

dictions for the gluon distribution ratio RG(x,Q2). The same is true for the ratios R2(x,Q2)2145

and RL(x,Q2), along with the nuclear structure functions FA2 (x,Q2) and FAL (x,Q2). These2146

77

observables can be measured in stage-I of the EIC as functions of x, Q2, and A. (For the2147

A-dependence one will need to perform machine runs with different types of nuclei, while2148

to extract FL one needs to vary the center-of-mass energy.) The multitude of theoretical2149

predictions should be counter-balanced by the multitude of possible data points for the four2150

observables in the 3-dimensional (x,Q2, A) parameter space. It is possible that the abun-2151

dance of data obtained with sufficient statistics would allow one to rule out many models,2152

hopefully pinpointing the one that best describes all the data to be obtained. Note however2153

that DGLAP-based models can not predict the A-dependence of PDFs and structure func-2154

tions without making additional data-driven assumptions: this is the origin of the broad2155

error bars of the EPS09 model in Fig. 3.11. However, this broad error band may also be2156

indicative of the ability of such models to indiscriminately describe a broad range of F22157

and FL data: in such cases, further experimental tests of DGLAP-based approaches can be2158

carried out using other observables described in the Secs. below.2159

A¹⁄³ A¹⁄³

rcBK

EPS09 (CTEQ)

Q2 = 2.7 GeV2, x = 10-3Q2 = 2.7 GeV2, x = 10-3

rcBK

EPS09 (CTEQ)

stat. errors enlarged (× 50)sys. uncertainty bar to scale

Cu AuSi

Beam Energies A ∫Ldt

5 on 50 GeV 2 fb-1 5 on 75 GeV 4 fb-1

5 on 100 GeV 4 fb-1

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

1.2

1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

1.2

R2 =

F2A/(A

F2p)

RL =

FLA/(A

FLp)

Beam Energies A ∫Ldt

5 on 50 GeV 2 fb-1 5 on 75 GeV 4 fb-1

5 on 100 GeV 4 fb-1

stat. errors enlarged (× 5)sys. uncertainty bar to scale

Figure 3.15: Left: Ratio R2 of the F2 structure function in a nucleus over that of the protonscaled by mass number A as a function of A1/3. The predictions from a CGC based calculation(rcBK) [139] and from a linear evolution using the latest nuclear PDFs (EPS09) and CTEQ6 forthe proton are shown [164, 150]. Right: Same for the longitudinal structure function FL (seetext for details).

To further illustrate this point, we show in Figure 3.15 two theoretical predictions for the2160

ratio R2 (RL), i.e., the ratio of the F2 (FL) structure function in a nucleus over that of the2161

proton scaled by mass number A. The calculations are shown as a function of A1/3 at Q2 =2162

2.7 GeV2 and x = 10−3. In the absence of any nuclear effects, both ratios R2 and RL should2163

be unity. Due to the lack of precise eA data, the models are not strongly constrained and we2164

use error bands to indicate the range of the referring predictions. In Fig. 3.15 we depict two2165

calculations for R2 (left) and RL (right): the calculation shown in blue is based on the CGC2166

framework (rcBK) [139] and therefore features an approximate A1/3 scaling of the saturation2167

scale squared (see Sec. 3.2.1), which allows us to make reasonably precise predictions for R22168

and RL; the second calculation (gray band) uses the linear NLO DGLAP evolution in pQCD2169

resulting in the nuclear parton distribution set EPS09 [164, 150]: it exhibits a broader error2170

band, similar to the case of RG in Fig. 3.11. Even in the linear DGLAP evolution, non-linear2171

effects may be absorbed into the non-perturbative initial conditions for the nuclear PDFs,2172

where the A-dependence is obtained through a fit to available data, resulting in the ability2173

of DGLAP-based approaches to indiscriminately describe a broad range of nuclear data.2174

This leads to the wide error bands of EPS09, especially for FL, clearly demonstrating the2175

78

lack of existing nuclear structure function data. Due to these large theoretical error bars,2176

the measurements of R2 and RL as functions of A1/3 may not allow to directly distinguish2177

between a non-linear (saturation) and linear (DGLAP) evolution approaches at a stage-I2178

EIC.2179

Shown along the line at unity by vertical notches in Fig. 3.15 are the statistical error2180

bars for an EIC stage-I measurement calculated using the Rosenbluth separation technique.2181

The statistical error bars were generated from a total of 10 fb−1/A of Monte Carlo data,2182

spread over three beam energies (see plot legend for details). The statistical error bars are2183

scaled up by a factor of 50 for R2 and a factor of 5 for RL; as the statistical errors are2184

clearly small, the experimental errors will be dominated by the systematic uncertainties2185

shown by the orange bars drawn to scale in the two panels of Fig. 3.15. Since the exact EIC2186

detector design has not been determined yet, the systematic errors were estimated based2187

on normalization uncertainties achieved by HERA. The measurements at an EIC (both2188

stage-I and stage-II) will have a profound impact on our knowledge of nuclear structure2189

functions (especially FL and the gluon distribution) as can be easily seen from the size of2190

the statistical and systematic errors bars for stage-I shown in Fig. 3.15. This measurement2191

together with the ones described below will constrain models to such an extent that the2192

“true” underlying evolution scheme can be clearly identified. It is also possible that stage-I2193

data would decrease the error band of DGLAP-based predictions, allowing for the R2 and2194

RL measurement in stage-II at a smaller x to discriminate between saturation and DGLAP2195

approaches. However it is also possible that, on its own, the R2 and RL measurements2196

may turn out to be insufficient to uniquely differentiate between DGLAP-based models2197

with nuclear “shadowing” in the initial conditions from the saturation/CGC effects; in such2198

a case the measurements presented below will be instrumental in making the distinction.2199

Note also that heavy quark flavors starting with the charm quarks, whose contribution can2200

be measured by the structure functions F charm2 and F charmL , may also be very helpful in2201

distinguishing between different approaches.2202

Di-Hadron Correlations2203

One of the experimentally easiest and compelling measurement in eA is that of di-hadron2204

azimuthal correlations in e A → e′ h1 h2 X processes. These correlations are not only2205

sensitive to the transverse momentum dependence of the gluon distribution but also to that2206

of gluon correlations for which first principles CGC computations are only now becoming2207

available. The precise measurements of these di-hadron correlations at an EIC would allow2208

one to extract the spatial multi-gluon correlations and study their non-linear evolution.2209

Saturation effects in this channel correspond to a progressive disappearance of the back-2210

to-back correlations of hadrons with increasing atomic number A. These correlations are2211

usually measured in the plane transverse to the beam axis (the ’transverse plane’), and are2212

plotted as a function of the azimuthal angle ∆ϕ between the momenta of the produced2213

hadrons in that plane. Back-to-back correlations are manifested by a peak at ∆ϕ = π (see2214

Fig. 3.16). In the conventional pQCD picture one expects, from momentum conservation,2215

that the back-to-back peak will persist as one goes from ep to eA. In the saturation2216

framework, due to multiple rescatterings and multiple gluon emissions, the large transverse2217

momentum of one hadron is balanced by the momenta of several other hadrons (instead of2218

just one back-to-back hadron), effectively washing out the correlation at ∆ϕ = π [165]. A2219

comparison of the heights and widths of the dihadron azimuthal distributions in eA and ep2220

collisions respectively would clearly mark out experimentally such an effect.2221

79

2 2.5 3 3.5 4 4.50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

eAueAu - sat

eAu - nosateCa

pTtrigger > 2 GeV/c

1 < pTassoc < p

Ttrigger

|η|<4

epQ2 = 1 GeV2

ΔϕΔϕ

C(Δ

ϕ)

Ce

Au(Δ

ϕ)

2 2.5 3 3.5 4 4.50

0.05

0.1

0.15

0.2EIC stage-II

∫ Ldt = 10 fb-1/A

Figure 3.16: Left: Saturation model prediction of the coincidence signal versus azimuthal angledifference ∆ϕ between two hadrons in ep, eCa, and eA collisions [166, 167]. Right: Comparisonof saturation model prediction for eA collisions with calculations from conventional non-saturatedmodel for EIC stage-II energies. Statistical error bars correspond to 10 fb−1/A integratedluminosity.

An analogous phenomenon has already been observed for di-hadrons produced at for-2222

ward rapidity in comparing dAu with pp collisions at RHIC (see Sec. 3.4.1). In that case,2223

di-hadron production is believed to proceed from valence quarks in the deuteron (proton)2224

scattering on small-x gluons in the target Au nucleons (proton). Lacking direct experimen-2225

tal control over x, the onset of the saturation regime is controlled by changing the centrality2226

of the collision, the di-hadron rapidity and the transverse momenta of the produced parti-2227

cles. (Note that the gluon density and, consequently, the saturation scale Qs depend on the2228

impact parameter and on rapidity/Bjorken-x.) Experimentally, a striking flattening of the2229

∆ϕ = π peak in dAu collisions as compared to pp collisions is observed in central collisions2230

[168, 169], but the peak reappears in peripheral collisions, in qualitative agreement with2231

the CGC predictions, since saturation effects are stronger in central collisions.2232

There are several advantages to studying di-hadron correlations in eA collisions ver-2233

sus dAu. Directly using a point-like electron probe, as opposed to a quark bound in a2234

proton or deuteron, is extremely beneficial. It is experimentally much cleaner as there is2235

no “spectator” background to subtract from the correlation function. The access to the2236

exact kinematics of the DIS process at EIC would allow for more accurate extraction of2237

the physics than possible at RHIC or LHC. Because there is such a clear correspondence2238

between the physics of this particular final state in eA collisions to the same in pA collisions,2239

this measurement is an excellent testing ground for universality of multi-gluon correlations.2240

The left plot in Fig. 3.16 shows prediction in the CGC framework for dihadron ∆ϕ2241

correlations in deep inelastic ep, eCa, and eAu collisions at stage-II energies [166, 167].2242

The calculations are made for Q2 = 1 GeV2. The highest transverse momentum hadron in2243

the di-hadron correlation function is called the “trigger” hadron, while the other hadron is2244

referred to as the “associate” hadron. The “trigger” hadrons have transverse momenta of2245

ptrigT > 2 GeV/c and the “associate” hadrons were selected with 1 GeV/c < passocT < ptrigT .2246

The CGC based calculations show a dramatic “melting” of the back-to-back correlation peak2247

with increasing ion mass. The right plot in Fig. 3.16 compares the prediction for eA with2248

a conventional non-saturated correlation function. The latter was generated by a hybrid2249

Monte Carlo generator, consisting of PYTHIA-6 [170] for parton generation, showering and2250

80

fragmentation and DPMJetIII [171] for the nuclear geometry. The EPS09 [150] nuclear2251

parton distributions were used to include leading twist shadowing. The shaded region2252

reflects uncertainties in the CGC predictions due to uncertainties in the knowledge of the2253

saturation scale, Qs. This comparison nicely demonstrates the discrimination power of these2254

measurement. In fact, already with a fraction of the statistics used here one will be able to2255

exclude one of the scenarios conclusively.2256

log10(xg)

Je

Au

-3 -2.5 -2 -1.5 -1

1

eAu - sat

eAu - nosat

Q2 = 1 GeV2

pTtrigger

> 2 GeV/c

1 < pTassoc

< pTtrigger

|η|<4

EIC stage-II

∫ Ldt = 10 fb-1/A

10-3 10-2

1

10-1 peripheral

central

xAfrag

Jd

Au

RHIC dAu, √s = 200 GeV

Figure 3.17: Left: Relative yield of di-hadrons in eAu compared to ep collisions, JeAu, plottedversus xg, the relative momentum fraction of the probed gluon. Shown are predictions forlinear (nosat) and non-linear (sat) QCD models for EIC stage-II energies. Statistical errorbars correspond to 10 fb−1/A integrated luminosity. Right: The corresponding measurement

in√s = 200 GeV per nucleon dAu collisions at RHIC [168]. Here xfracA is a only a rough

approximation of the average momentum fraction of the struck parton in the Au nucleus. Curvesdepict calculations in the CGC framework.

The left panel of Fig. 3.17 depicts the predicted suppression through JeAu, the relativeyield of correlated back-to-back hadron pairs in eAu collisions compared to ep collisionsscaled down by A1/3 (the number of nucleons at a fixed impact parameter)

JeA =1

A1/3

σpaireA /σeA

σpairep /σep. (3.12)

Here σ and σpair are the total inelastic and the dihadron pair production cross sections (or2257

normalized yields). The absence of collective nuclear effects in the pair production cross2258

section σpaireA would correspond to JeA = 1,6 while JeA < 1 would signify suppression of2259

di-hadron correlations. In the left panel of Fig. 3.17, JeAu is plotted as a function of the2260

longitudinal momentum fraction of the probed gluon xg. Compared to the measurement2261

shown in the right panel of Fig. 3.16 this study requires the additional ep baseline mea-2262

surement but has the advantage of several experimental uncertainties canceling out. It is2263

instructive to compare this plot with the equivalent measurement in dAu collisions at RHIC2264

[168] shown in the right panel of Fig. 3.17. In dAu collisions JdAu is defined by analogy2265

to Eq. (3.12) with A1/3 in the denominator replaced by the number of the binary nucleon–2266

nucleon collisions Ncoll at a fixed impact parameter [168]. Unlike DIS, where Bjorken x is2267

6Without collective nuclear effects the hadron pairs are produced in independent electron–nucleon scat-terings, such that σpair

eA = Aσpairep . The cross section for inelastic eA collisions, σeA, is related to the

probability for the incoming electron (or, more precisely, γ∗ → qq) to get the first inelastic collision, whichusually takes place on the nuclear surface: hence σeA = A2/3 σep. Combining these results we get JeA = 1[172].

81

known precisely, the exact momentum fraction of the struck parton cannot be measured in2268

hadron-hadron collisions. Instead JdAu in Fig. 3.17 is plotted against xfracA , which is only2269

a rough approximation of the average momentum fraction of the struck parton in the Au2270

nucleus, derived from the kinematics of the measured hadrons assuming they carry the full2271

parton energy. The two curves in the right panel of Fig. 3.17 represent the CGC calculations2272

from [166, 167] and appear to describe the data rather well. This example nicely demon-2273

strates on the one hand the correspondence between the physics in pA(dA) and eA collisions2274

but on the other hand the lack of precise control in pA that is essential for precision studies2275

of saturation phenomena.2276

Measurements of Diffractive Events2277

Diffractive interactions result when the electron probe in DIS interacts with a proton or2278

nucleus by exchanging several partons with zero net color. This exchange, which in QCD2279

may be visualized as colorless combination of two or more gluons, is commonly referred to2280

as the “Pomeron” (see the Sidebar on page 62).2281

The HERA physics program of ep collisions surprisingly showed a large fraction of2282

diffractive events contributing about 15% to the total DIS cross section [163]. One of2283

the key signatures of these events is an intact proton traveling at near-to beam energies,2284

together with a gap in rapidity before some final-state particles are produced at mid-rapidity2285

(i.e., at 90◦ angle to the beam axis). While linear pQCD is able to describe some aspects2286

of diffraction it fails to describe other major features without introducing new types of2287

structure functions, the diffractive structure functions (see the Sidebar on page 62), which2288

describe the rapidity gap. A striking example is the fact that the ratio of the diffractive2289

to the total cross section is constant with energy, an observation not easily reconciled in a2290

conventional pQCD scenario without introducing the diffractive structure functions [163].2291

As may therefore be anticipated and as we have argued above, the strongest hints for a2292

manifestations of new, non-linear effects in eA collisions are likely to come from diffractive2293

measurements.2294

What makes the diffractive processes so interesting is that they are most sensitive to2295

the underlying gluon distribution, and that they are the only known class of events that2296

allows us to gain insight into the spatial distribution of gluons in nuclei. The reason for this2297

sensitivity is that the diffractive structure functions (see the Sidebar on page 62) depend,2298

in a wide kinematic range, quadratically on the gluon momentum distribution and not2299

linearly as in DIS. However, while the physics goals are golden, the technical challenges are2300

formidable but not insurmountable and require careful planing of detector and interaction2301

region. Diffractive events are characterized by a rapidity gap, i.e. an angular region in the2302

direction of the scattered proton or nucleus without particle flow. Detecting events with2303

rapidity gaps requires a largely hermetic detector.2304

As discussed earlier (see Sec. 3.2.1) we distinguish two kinds of diffractive events: co-2305

herent (nucleus stays intact) and incoherent (nucleus excites and breaks up). Both contain2306

a rich set of information. Coherent diffraction is sensitive to the space-time distribution of2307

the partons in the nucleus, while incoherent diffraction (dominating at larger t and thus2308

small impact parameter bT ) is most sensitive to high parton densities where saturation ef-2309

fects are stronger. In ep collisions the scattered intact protons can be detected in a forward2310

spectrometer placed many meters down the beam line. This is not possible for nuclei that,2311

due to their large mass, stay too close to the ion beam. However, studies showed that the2312

nuclear breakup in incoherent diffraction can be detected with close to 100% efficiency by2313

82

measuring the emitted neutrons in a zero degree calorimeter placed after the first dipole2314

magnet that bends the hadron beam. This tagging scheme could be further improved by2315

using a forward spectrometer to detect charged nuclear fragments. A rapidity gap and the2316

absence of any break-up fragments was found sufficient to identify coherent events with very2317

high efficiency.2318

In the following we present several measurement focusing on the discrimination power2319

between non-linear saturation models and prediction from conventional linear QCD DGLAP2320

evolution. Saturation models incorporate the effects of linear small-x evolution for Q > Qs2321

and saturation nonlinear evolution effects for Q < Qs.2322

0

Q2 = 10 GeV2

x = 6.6×10-3 eAu stage-I∫Ldt = 10 fb-1/A

eAu - Saturation Modelep - Saturation ModeleAu - Non-Saturation Model (LTS)ep - Non-Saturation Model (LTS)

Q2 = 5 GeV2

x = 3.3×10-3

eAu stage-I∫Ldt = 10 fb-1/A

10-2

10-3

0.5

1

1.5

2

2.5ra

tio

(e

Au

/ep

)(1

/σto

t) d

σd

iff/

dM

x2 (G

eV

-2)

101

Mx2

(GeV2)

10-2

10-3

0.5

1

1.5

2

2.5

ratio

(e

Au

/ep

)(1

/σto

t) d

σd

iff/

dM

x2 (G

eV

-2)

101

Mx2

(GeV2)

stat. errors & syst. uncertainties enlarged (× 10)stat. errors & syst. uncertainties enlarged (× 10)

no-sat (LTS) no-sat (LTS)

sat sat

Figure 3.18: Top of each panel: the ratio of diffractive over total cross sections, plotted as afunction of the invariant mass of the produced particles M2

X for stage-I EIC kinematics. Thebottom of each panel contains the double ratio [(dσdiff/dM

2X)/σtot]eA/[(dσdiff/dM

2X)/σtot]ep

plotted as a function of M2X for the same kinematics as used at the top of each panel. The

statistical error bars for the integrated luminosity of 10 fb−1/A are shown as well, multiplied bya factor of 10 for increased visibility. The non-monotonicity of the saturation curve in the lowerpanels is due to crossing the cc threshold; this threshold is not included in the LTS prediction.

Ratio of diffractive and total cross sections. Figure 3.18 depicts predictions for one2323

of the simplest inclusive measurements that can be performed with diffractive events: the2324

measurement of the ratio of the coherent diffractive cross section over the total cross sec-2325

tion in ep and eA collisions is shown at the top of each panel. It is plotted here as a2326

function of the diffractive mass of the produced final state particles, M2X (see the Sidebar2327

on page 62), for various fixed x and Q2 values. The red curves represent the predictions2328

of the saturation model [173, 174] based on Model-I of Sec. 3.2.1 combined with the the-2329

oretical developments of [175, 142, 176], while the blue curves represent the leading-twist2330

shadowing (LTS) model [177, 178]. The bottom part of each panel depicts the double ratio2331

[(dσdiff/dM2X)/σtot]eA/[(dσdiff/dM

2X)/σtot]ep. The curves in Fig. 3.18 are plotted for the2332

range of x and Q2 values which will be accessible in stage-I at an EIC. The ep curves in2333

both approaches are in a reasonable agreement with the available HERA [173, 2]. The2334

statistical error bars, shown in the bottom parts of the panels in Fig. 3.18 are very small,2335

and had to be scaled up by a factor of 10 to become visible: we conclude that the errors2336

of the actual measurement would be dominated by the systematic uncertainties dependent2337

on the quality of the detector and of the luminosity measurements. The size of the error2338

83

bars shows that the two scenarios can be clearly distinguished over a wide x and Q2 range,2339

allowing for a clear stage-I measurement aimed at finding evidence of parton saturation.2340

Note that in the saturation predictions plotted in Fig. 3.18, the nuclear effects, respon-2341

sible for the difference between the eAu and ep curves, are stronger at large Q2: the effect2342

of saturation is to weaken the A-dependence in the σdiff/σtot ratio at low Q2. Also, in2343

agreement with the expectation that diffraction would be a large fraction of the total cross2344

section with the onset of the black disk limit (see Eq. (3.11)), the ratio (dσdiff/dM2X)/σtot2345

plotted in Fig. 3.18 both for ep and eAu grows with decreasing Q2, getting larger as one2346

enters the saturation region. (This last property is true for the non-saturation LTS scenario2347

plotted in Fig. 3.18 as well, since the black disk limit physics is incorporated in that model2348

too.)2349

0.2

0.4

0.6

0.8

1

1.2

Coherent events only∫Ldt = 10 fb-1/Ax < 0.01

Experimental Cuts:|η(edecay)| < 4p(edecay) > 1 GeV/c

Coherent events only∫Ldt = 10 fb-1/Ax < 0.01

1 2 3 4 5 6 7 8 9 10

(1/A

4/3

) σ

(eA

u)/

σ(e

p)

Q2 (GeV2)

no saturation

saturation (bSat)

no saturation

saturation (bSat)

e e

e + p(Au) → e’ + p’(Au’) + J/ψ

1 2 3 4 5 6 7 8 9 10

(1/A

4/3

) σ

(eA

u)/

σ(e

p)

Q2 (GeV2)

2.2

2.0

1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

0

e + p(Au) → e’ + p’(Au’) + φ

K K

Experimental Cuts:|η(Kdecay)| < 4p(Kdecay) > 1 GeV/c

Figure 3.19: Ratios of the cross sections for exclusive J/ψ (left panel) and φ (right panel)meson production in coherent diffractive eA and ep collisions as a function of Q2. Shown areprediction for saturation and non-saturation models. The ratios are scaled by 1/A4/3.

Diffractive vector meson production. Diffractive vector meson production, e + A →2350

e′+A′+V where V = J/ψ, φ, ρ, or γ, is a unique process, for it allows the measurement of the2351

momentum transfer, t, at the hadronic vertex even in eA collisions where the 4-momentum2352

of the outgoing nuclei cannot be measured. Since only one new final state particle is2353

generated, the process is experimentally clean and can be unambiguously identified by the2354

presence of a rapidity gap. The study of various vector mesons in the final state allows to2355

systematically test the saturation regime [162]. The J/ψ is the vector meson least sensitive2356

to the saturation effects due to the small size of its wave function. Larger mesons such as2357

φ or ρ are considerably more sensitive to saturation effects [179].2358

The two panels in Fig. 3.19 show the ratios [dσ(eAu)/dQ2]/[dσ(ep)/dQ2] (scaled down2359

by A4/3) of the cross sections σ(eAu) and σ(ep) for exclusive J/ψ (left panel) and φ (right2360

panel) production in coherent diffractive events for eAu and ep collisions correspondingly.2361

The ratios are plotted as functions of Q2 for saturation and non-saturation models. The2362

parameters of both models were tuned to describe the ep HERA data [179, 148]. Beam2363

energies correspond to stage-II of an EIC. All curves were generated with the Sartre event2364

generator [180], an eA event generator specialized for diffractive exclusive vector meson2365

production based on the bSat [179] dipole model. We limit the calculation to 1 < Q2 < 102366

84

GeV2 and x < 0.01 to stay within the validity range of saturation and non-saturation2367

models. The produced events were passed through an experimental filter and scaled to2368

reflect an integrated luminosity of 10 fb−1/A. The basic experimental cuts are listed in the2369

legends of the panels in Fig. 3.19. As expected the difference between the saturation and2370

non-saturation curves is small for the smaller-sized J/ψ (< 20%), which is less sensitive2371

to saturation effects, but is substantial for the larger φ, which is more sensitive to the2372

saturation region. In both cases the difference is larger than the statistical errors. In fact,2373

the small errors for diffractive φ production indicate that this measurement can already2374

provide substantial insight into the saturation mechanism after a few weeks of EIC running.2375

(This measurement could already be feasible at a stage-I EIC: however, since in stage-I the2376

typical values of x would be larger, the saturation effects would be less pronounced.) For2377

large Q2 the two ratios asymptotically approach unity.2378

|t | (GeV2) |t | (GeV2)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

)2

/d

t (n

b/G

eV

e’ +

Au

’ +

J/ψ

)→

(e +

Au

σd

)2

/d

t (n

b/G

eV

e’ +

Au

’ +

φ)

→(e

+ A

u

σd

J/ψ φ

∫Ldt = 10 fb-1/A1 < Q2 < 10 GeV2

x < 0.01|η(edecay)| < 4p(edecay) > 1 GeV/cδt/t = 5%

∫Ldt = 10 fb-1/A1 < Q2 < 10 GeV2

x < 0.01|η(Kdecay)| < 4p(Kdecay) > 1 GeV/cδt/t = 5%

104

103

102

10

1

10-1

10-2

105

104

103

102

10

1

10-1

10-2

coherent - no saturationincoherent - no saturationcoherent - saturation (bSat)incoherent - saturation (bSat)

coherent - no saturationincoherent - no saturationcoherent - saturation (bSat)incoherent - saturation (bSat)

Figure 3.20: dσ/dt distributions for exclusive J/ψ (left) and φ (right) production in coherent andincoherent events in diffractive eAu collisions. Predictions from saturation and non-saturationmodels are shown.

As explained earlier in Sec. 3.2.1 coherent diffractive events allow one to learn about the2379

shape and the degree of “blackness” of the black disk: this enables one to study the spatial2380

distribution of gluons in the nucleus. Exclusive vector meson production in diffractive2381

eA collisions is the cleanest such process, due to the low number of particles in the final2382

state. This would not only provide us with further insight into saturation physics but also2383

constitute a highly important contribution to heavy-ion physics by providing a quantitative2384

understanding of the initial conditions of a heavy ion collision as described in Sec. 3.4.2. It2385

might even shed some light on the role of glue and thus QCD in the nuclear structure of light2386

nuclei (see Sec. 3.3). As described above, in diffractive DIS the virtual photon interacts2387

with the nucleus via a color-neutral exchange, which is dominated by two gluons at the2388

lowest order. It is precisely this two gluon exchange which yields a diffractive measurement2389

of the gluon density in a nucleus.2390

Experimentally the key to the spatial gluon distribution is the measurement of the2391

dσ/dt distribution. As follows from the optical analogy presented in Sec. 3.2.1, the Fourier-2392

transform of (the square root of) this distribution is the source distribution of the ob-2393

ject probed, i.e., the dipole scattering amplitude N(x, rT , bT ) on the nucleus with r2T ∼2394

85

1/(Q2 + M2V ), where MV is the mass of the vector meson [162] (see also the Sidebar on2395

page 43).2396

Figure 3.20 shows the dσ/dt distribution for J/ψ on the left and φ mesons on the2397

right. The coherent distribution depends on the shape of the source while the incoherent2398

distribution provides valuable information on the fluctuations or “lumpiness” of the source2399

[173]. As discussed above we are able to distinguish both by detecting the neutrons emitted2400

by the nuclear breakup in the incoherent case. Again we compare prediction of saturation2401

and non-saturation models. As for the previous figures the curves were generated with the2402

Sartre event generator and had to pass through an experimental filter. The experimental2403

cuts are listed in the figures.2404

Since the J/ψ is smaller than the φ, as expected one sees little difference between the2405

saturation and no saturation scenarios for exclusive J/ψ production but a pronounced effect2406

for the φ. For the former the statistical errors after the 3rd minimum become excessively2407

large requiring substantial more than the used integrated luminosity of 10 fb−1/A. The2408

situation is more favorable for the φ where enough statistics up to the 4th minimum is2409

available. The ρ meson has even higher rates and is also quite sensitive to saturation2410

effects: however, it suffers currently from large theoretical uncertainties in the knowledge2411

of its wave function making calculations less reliable.2412

86

3.3 Quarks and gluons in the nucleus2413

Conveners: William Brooks and Jianwei Qiu2414

In this section we present a few key measurements that will allow us to answer the2415

fundamental questions from the beginning of this chapter and to explore the properties of2416

quarks and gluons and their interactions in a nuclear environment. In Table 3.2, we list2417

the key measurements to be carried out at an EIC. The measurement of nuclear structure2418

functions with various ion beams at intermediate-x will allow us to see the first glimpses of2419

collective nuclear effects at the partonic level and the onset of the breakdown of DGLAP2420

evolution. The semi-inclusive production of energetic hadrons will allow us to study nuclear2421

matter’s response to a fast moving color charge as well as the mass of the particle carrying2422

the charge. With the unique and well-determined leptonic and hadronic planes of semi-2423

inclusive DIS scattering, as shown in the Sidebar on page 34, the azimuthal angle φ between2424

these planes determines the path of the produced hadron. The nuclear modification to2425

the angular φ modulation of the production rate as well as the transverse momentum2426

broadening of the produced hadrons provides a sensitive probe to the color and parton2427

density fluctuations inside the nucleus [181], which is very important for understanding the2428

initial condition of relativistic heavy ion collisions.2429

Deliverables Observables What we learn Stage-I Stage-II

Collective Ratios R2 Q2 evolution, Shadowing, Q2 evolution

nuclear effects from inclusive DIS Initial conditions Onset of beyond

at intermediate-x for small-x evolution DGLAP violation DGLAP

Transport Production of light Hadronization, Mass dependence Medium effect of

coefficients in and heavy hadrons, multiple scattering, of hadronization heavy quarkonium

nuclear matter and jets in SIDIS energy loss mechanism and energy loss production

Fluctuations φ-modulation of Color fluctuations; φ-modulation φ-modulation

of the nuclear hadron production connection to of light hadron of heavy

density in SIDIS heavy ion physics production mesons

Table 3.2: Key measurements in eA collisions at an EIC to explore the dynamics of quarks and gluons

in a nucleus in the non-saturation regime.

3.3.1 The distributions of quarks and gluons in a nucleus2430

The momentum distribution of quarks and gluons inside a fast moving proton was best2431

measured by lepton DIS on a proton beam at HERA. Although the scattering could take2432

place between the lepton and a single quark (or gluon) state as well as a multiple quark-2433

gluon state of the proton, the large momentum transfer of the scattering, Q, localizes the2434

scattering, suppresses the contribution from multiple scattering, and allows to express the2435

complex DIS cross sections in terms of a set of momentum distributions of quarks and2436

gluons - probability density distributions to find a parton (quark, antiquark or gluon) to2437

carry the momentum fraction x of a fast moving hadron. Actually, it is a triumph of QCD2438

that one set of universal parton distributions, extracted from HERA data, plus calculable2439

scatterings between quarks and gluons, can successfully interpret all existing data of high2440

energy proton collisions with a momentum transfer larger than 2 GeV (corresponding to2441

hard scatterings taking place at a distance less than one tenth of a femtometer).2442

87

Are the quarks and gluons in a nucleus confined within the individual nucleons? or does2443

the nuclear environment significantly affect their distributions? The EMC experiment at2444

CERN [182] and experiments in the following two decades clearly revealed that the momen-2445

tum distribution of quarks in a fast moving nucleus is not a simple superposition of their2446

distributions within nucleons. Instead, the measured ratio of nuclear over nucleon structure2447

functions, as defined in Eq. (3.7), follows a non-trivial function of Bjorken x, significantly2448

different from unity, and shows the suppression as x decreases, as shown in Fig. 3.21. The2449

observed suppression at x ∼ 0.01, which is often referred to as the phenomenon of nu-2450

clear shadowing, is much stronger than what the Fermi motion of nucleons inside a nucleus2451

could account for. This discovery sparked a world-wide effort to study the properties of2452

quarks and gluons, and their dynamics in the nuclear environment both experimentally and2453

theoretically.2454

0.6

0.7

0.8

0.9

1

1.1

1.2

0.0001 0.001 0.01 0.1 1

F2C

a / F

2D

x

EIC

EMC E136

NMC E665

Stage I

EIC Stage II

0.5

Figure 3.21: The ratio of nuclear over nucleon F2 structure function, R2, as a function ofBjorken x, with data from existing fixed target DIS experiments at Q2 > 1 GeV2, along withthe QCD global fit from EPS09 [150]. Also shown are the respective coverage and resolutionof the same measurements at the EIC at Stage-I and Stage-II. The purple error band is theexpected systematic uncertainty at the EIC assuming a ±2% (a total of 4%) systematic error,while the statistical uncertainty is expected to be much smaller.

Using the same very successful QCD formulation at the leading power in Q for proton2455

scattering, and using the DGLAP evolution for the scale dependence of parton momentum2456

distributions, several QCD global analyses have been able to fit the observed non-trivial2457

nuclear dependence of existing data by attributing all observed nuclear dependence including2458

its x-dependence and nuclear atomic weight A-dependence to a set of nucleus-dependent2459

quark and gluon distributions at an input scale Q0 & 1 GeV [153, 152, 150]. As an example,2460

the fitting result of Eskola et al is plotted along with the data on the ratio of the F22461

structure function of calcium divided by that of deuterium in Fig. 3.21, where the dark2462

blue band indicates the uncertainty of the EPS09 fit [150]. The success of the QCD global2463

analyses clearly indicates that the response of the nuclear cross section to the variation2464

88

of the probing momentum scale Q & Q0 is insensitive to the nuclear structure, since the2465

DGLAP evolution itself does not introduce any nuclear dependence. However, it does not2466

answer the fundamental questions: why are the parton distributions in a nucleus so different2467

from those in a free nucleon at the probing scale Q0? how do the nuclear structure and2468

QCD dynamics determine the distributions of quarks and gluons in a nucleus?2469

The nucleus is a “molecule” in QCD, made of nucleons, which, in turn, are bound states2470

of quarks and gluons. Unlike the molecule in QED, nucleons in the nucleus are packed2471

next to each other, and there are many soft gluons inside nucleons when probed at small2472

x. The DIS probe has a high resolution in transverse size ∼ 1/Q. But, its resolution in2473

the longitudinal direction, which is proportional to 1/xp ∼ 1/Q, is not necessary sharp in2474

comparison with the Lorentz contracted size of a light-speed nucleus, ∼ 2RA(m/p), with2475

nuclear radius RA ∝ A1/3 and the Lorentz contraction factor m/p and nucleon mass m.2476

That is, when 1/xp > 2RA(m/p), or at a small x ∼ 1/2mRA ∼ 0.01, the DIS probe could2477

interact coherently with quarks and gluons of all nucleons at the same impact parameter of2478

the largest nucleus moving nearly at the speed of light, p� m. The destructive interference2479

of the coherent multiple scattering could lead to a reduction of the DIS cross section [178].2480

Although such coherent multiple scattering is expected to be suppressed by the powers of2481

1/Q, the resolution of the probe size, it can be enhanced by the number of nucleons at the2482

same impact parameter in a nucleus and the large number of soft gluons in nucleons, which2483

naturally leads to the observed phenomena of nuclear shadowing: more suppression when2484

x decreases, Q decreases, and A increases. But, none of these dependences could have been2485

predicted by the very successful leading power DGALP based QCD formulation.2486

The contribution of coherent multiple scattering to inclusive DIS cross section could2487

be systematically investigated in QCD in terms of corrections to the DGLAP based QCD2488

formulation in powers of 1/Q2 [183]. However, when the contributions from higher powers2489

are equally important, which corresponds to the new QCD phenomenon - the saturation at2490

an extremely high gluon density that was discussed in the last section, the power correction2491

approach to the coherent multiple scattering would be completely ineffective. The RHIC2492

data [168, 169] on the correlation in deuteron-gold collisions indicate that the saturation2493

phenomena might take place at x . 0.001 [168, 169]. Therefore, the region of 0.001 <2494

x < 0.1, at a sufficiently large probing scale Q, could be the most interesting place to see2495

the transition of a large nucleus from a diluted partonic system, whose response to the2496

resolution of the hard probe (the Q2-dependence) follows the linear DGLAP evolution, to2497

matter composed of condensed and saturated gluons, which is referred to as a Color Glass2498

Condensate (CGC) [134, 131].2499

This very important transition region with Bjorken x ∈ (0.001, 0.1) could be best ex-2500

plored by the EIC, as shown in Fig. 3.21. At the stage-I, the EIC will not only explore this2501

transition region, but also have a wide overlap with what have been and will be measured2502

by the fixed target experiments, as indicated by the yellow box in Fig. 3.21. At later stages2503

of the EIC operation, the coverage of the Bjorken x could be extended down to 10−4, as2504

shown by the blue box in Fig. 3.21 while maintaining a sufficiently large Q. The EIC will2505

have an ideal kinematic coverage for the systematically study of QCD dynamics in this very2506

rich transition region, as well as the new regime of saturated gluons.2507

If the nuclear effect of the DIS cross section, as shown in Fig. 3.21, is mainly due to the2508

abundance of nucleons at the same impact parameter of the nucleus (proportional to A1/3),2509

while the elementary scattering is still relatively weak, one would expect the ratio of nuclear2510

over nucleon structure functions to saturate when x goes below 0.01, or equivalently, the2511

nuclear structure function to be proportional to the nucleon structure functions, as shown,2512

89

for example, by the upper line of the grey area extrapolated from the current data in2513

Fig. 3.21. In this case, there is no saturation in nuclear structure functions since the proton2514

structure function is not saturated at this intermediate-x region, and the ratio could have2515

a second drop at a smaller x when nuclear structure functions enter the saturation region.2516

On the other hand, if the soft gluons are a property of the whole nucleus and the coherence2517

is strong, one would expect the ratio of the nuclear to nucleon structure function to fall2518

continuously as x decreases, as sketched by the lower line of the grey band, and eventually,2519

the ratio reaches a constant when both nuclear and nucleon structure functions are in the2520

saturation region. From the size of the purple error band in Fig. 3.21, which is the expected2521

systematic uncertainty at the EIC (while the statistical uncertainty is expected to be much2522

smaller), the EIC could easily distinguish these two extreme possibilities to explore the2523

nature of sea quarks and soft gluons in a nuclear environment.2524

With the unprecedented energy and luminosity of the lepton-nucleus collisions at the2525

EIC, the precision measurements of the Q dependence of the nuclear structure functions2526

could extract nuclear gluon distributions at small x which are currently effectively unknown,2527

and identify the momentum scaleQ0, below which the DGLAP based QCD formulation fails,2528

to discover the onset of the new regime of non-linear QCD dynamics. With its variety of2529

nuclear species, and the precise measurements of the x and Q dependence in this transition2530

region, the EIC is an ideal machine to explore the transition region and to provide immediate2531

access to the first glimpses of collective nuclear effects caused by coherent multi-parton2532

dynamics in QCD. Inclusive DIS measurements at the EIC provide an excellent and unique2533

test ground to study the transition to new and novel saturation physics.2534

3.3.2 Propagation of a fast moving color charge in QCD matter2535

The discovery of the quark-gluon plasma (QGP) in the collision of two heavy ions at the2536

relativistic heavy ion collider (RHIC) at Brookhaven National Laboratory made it possible2537

to study in a laboratory the properties of the quark-gluon matter at extremely high tem-2538

peratures and densities, which were believed to exist only a few microseconds after the Big2539

Bang. One of the key evidences of the discovery was the strong suppression of fast moving2540

hadrons produced in the relativistic heavy ion collisions [184], which is often referred as the2541

jet quenching [185]. It was found that the production rate of the fast moving hadrons in2542

a central gold-gold collision could be suppressed by as much as a factor of five of that in2543

a proton-proton collision at the same energy, and the same phenomenon was confirmed by2544

the heavy ion program at the LHC.2545

The fast moving hadrons at RHIC are dominantly produced by the fragmentation of2546

colored fast moving quarks or gluons that are produced during hard collisions at short2547

distances. The fragmentation (or in general, the hadronization) – the transition of a colored2548

and energetic parton to a colorless hadron – is a rich and dynamical process in QCD, and is2549

quantified by the fragmentation function Dparton→hadron(z) with z the momentum fraction2550

of the fast moving parton to be carried by the produced hadron in the DGLAP based2551

QCD formulation. Although QCD calculations are consistent with hadron production in2552

high energy collisions, knowledge about the dynamics of the hadronization process remains2553

limited and model dependent. It is clear that color is ultimately confined in these dynamical2554

processes. The color of an energetic quark or a gluon produced in high energy collisions has2555

to be neutralized so that it can transmute itself into hadrons. Even the determination of a2556

characteristic time scale for the color neutralization would shed some light on the properties2557

of color confinement and the answer to the question: what governs the transitions of quarks2558

90

0

0.5

1

1.5

2

2.5

3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

z

D(z

)

Figure 3.22: Left: A cartoon for the interactions of the parton moving through cold nuclearmatter when the produced hadron is formed outside (upper) and inside (lower) the nucleus.Right: Fragmentation function as function of z: from the charm quark to the D0 meson(solid) [188] and from up quark to π0 meson (dashed) [39].

and gluons to hadrons?2559

The collision of a fast moving parton with the QGP could induce gluon radiation to2560

reduce the parton’s forward momentum and energy, while the parton-to-hadron fragmen-2561

tation functions might not be affected since the energetic hadrons are likely to be formed2562

outside the QGP due to time dilation, as indicated by the cartoon in Fig. 3.22 (Left - upper2563

plot). The energy loss of the active parton would require a fragmentation function of a2564

larger z in order to produce a hadron with the same observed momentum as that produced2565

in proton-proton collisions without energy loss [186]. However, it has been puzzling [187]2566

that heavy meson production in the same experiments at RHIC seems to be suppressed as2567

much as the production of light mesons, although a heavy quark is much less likely to lose2568

its energy via medium induced radiation. It is critically important to have new and clean2569

measurements as well as the independent test of the energy loss mechanisms, in order to2570

have the full confidence on jet quenching as a hard probe of QGP properties.2571

Semi-inclusive DIS in eA collision provides a known and stable nuclear medium (“cold2572

QCD matter”), well-controlled kinematics of hard scattering, and a final state particle2573

with well-known properties. The time for the produced quark (or gluon) to neutralize its2574

color depends on its momentum and virtuality when it was produced. The process could2575

take place entirely inside the nuclear medium, or outside the medium, or somewhere in-2576

between, as indicated by the cartoon in Fig. 3.22 (Left) [189, 190]. Cold QCD matter could2577

be an excellent femtometer-scale detector of the hadronization process from its controllable2578

interaction with the produced quark (or gluon).By facilitating studies on how struck partons2579

propagate through cold nuclear matter and evolve into hadrons, as sketched in Fig. 3.222580

(Left), the EIC would provide independent and complementary information essential for2581

understanding the response of the nuclear medium to a colored fast moving (heavy or light)2582

quark. With its collider energies and thus the much larger range of ν, the energy of the2583

91

exchanged virtual photon, the EIC is unique for providing clean measurements of medium2584

induced energy loss when the hadrons are formed outside the nuclear medium, while it is2585

also capable of exploring the interplay between hadronization and medium induced energy2586

loss when the hadronization takes place inside the medium.2587

The amount of the medium induced energy loss and the functional form of the fragmen-2588

tation functions should be the most important cause for the difference of multiplicity ratio2589

of hadrons produced in a large nucleus and that produced in a proton, if the hadrons are2590

formed outside the nuclear medium. It was evident from hadron production in electron-2591

positron collisions that the fragmentation functions for light mesons, such as pions, have2592

a very different functional form with z from that of heavy mesons, such as D-mesons. As2593

shown in Fig. 3.22 (Right), the heavy D0-meson fragmentation function has a peak while2594

the pion fragmentation function is a monotonically decreasing function of z. The fact that2595

the energy loss matches the active parton to the fragmentation function at a larger value2596

of z leads to two dramatically different phenomena in the semi-inclusive production of light2597

and heavy mesons at the EIC, as shown in Fig. 3.23 [191]. The ratio of light meson (π)2598

production in electron-lead collisions over that in electron-deuteron collisions (red square2599

symbols) is always below unity, while the ratio of heavy meson (D0) production can be less2600

than as well as larger than unity due to the difference in hadronization.2601

0.30

0.50

0.70

0.90

1.10

1.30

1.50

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tio

of p

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icle

s p

rod

uce

d in

le

ad

ov

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pro

ton

Pions (lower energy)Pions (higher energy)Wang, pions (lower energy)Wang, pions (higher energy)

0.01 < y < 0.85, x > 0.1, 10 fb-1

Higher energy : 25 GeV2< Q

2< 45 GeV

2, 140 GeV < < 150 GeV

Lower energy : 8 GeV2< Q2

<12 GeV2, 32.5 GeV< < 37.5 GeV

1-systematicuncertainty

v

v

0.0 0.2 0.4 0.6 0.8 1.0Z

σ

0.30

0.50

0.70

0.90

1.10

1.30

1.50

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of p

art

icle

s p

rod

uce

d in

le

ad

ov

er

pro

ton

D0 mesons (lower energy)D0 mesons (higher energy)

0.01 < y < 0.85, x > 0.1, 10 fb-1

Higher energy : 25 GeV2< Q

2< 45 GeV

2, 140 GeV < < 150 GeV

Lower energy : 8 GeV2< Q2

<12 GeV2, 32.5 GeV < < 37.5 GeV

1-σsystematicuncertainty

v

v

0.0 0.2 0.4 0.6 0.8 1.0Z

Figure 3.23: Ratio of semi-inclusive cross sections for producing a single pion (Left - placeholder) and a single D0 (Right) in electron-lead collisions to the same produced in electron-deuteron collisions as a function of z at the EIC with two different photon energies ν = 35 GeVat Q2 = 10 GeV2 (solid symbols) and ν = 145 GeV at Q2 = 35 GeV2 (open symbols) (pTof the hadron is integrated). The solid lines are predictions of pure energy loss calculations forpion production (see the text).

In Fig. 3.23 the simulation results were plotted for the multiplicity ratio of semi-inclusive2602

DIS cross sections for producing a single pion (Left) and a single D0 (Right) in electron-2603

Lead collisions to the same produced in the electron-deuteron as function of z at the EIC2604

with two different photon energy ν = 35 GeV at Q2 = 10 GeV2 (solid line and square2605

symbols) and ν = 145 GeV at Q2 = 35 GeV2 (dashed line and open symbols). The2606

92

pT of the observed hadrons is integrated. The ratio for pions (red square symbols) was2607

taken from the calculation of [189], extended to lower z, and extrapolated from a Copper2608

nucleus to a Lead nucleus using the prescription of [190]. In this model approach, pions2609

are suppressed in electron-nuclei collisions due to a combination of the attenuation of pre-2610

hadrons as well as medium-induced energy loss. In this figure, the solid lines (red - ν =2611

145 GeV, and blue - ν = 35 GeV) are predictions of pure energy loss calculations using2612

the energy loss parameters of Ref. [192]. The large differences in the suppression between2613

the square symbols and solid lines are immediately consequences of the characteristic time2614

scale for the color neutralization and the details of the attenuation of pre-hadrons, as well2615

as the model for energy loss. With the size of the systematic errors shown by the yellow2616

bar on the left of the unity ratio, the multiplicity ratio of pion productions at the EIC will2617

provide an excellent and unique opportunity to study hadronization by using the nucleus2618

as a femtometer detector.2619

The dramatic difference between the multiplicity ratios of D0 meson production and2620

that of pions, as shown in Fig. 3.23 is an immediate consequence of the difference in the2621

fragmentation functions shown in Fig. 3.22 (Right). The enhancement of the ratio is caused2622

by the peak in the D0’s fragmentation function. The slope of the enhancement is sensitive2623

to the amount of energy loss, or equivalently, the transport coefficient, q of cold nuclear2624

matter, and the shape of the fragmentation function [191]. The energy loss used in the2625

simulation is a factor of 0.35 less than that of light quarks as derived in Ref. [193] by2626

taking into account the limited cone for gluon radiation caused by the larger charm quark2627

mass. The solid symbols are for x = 0.1 and Q2 = 10 GeV2. The dashed line shows the2628

effect of reducing the charm quark energy loss by 10% to show the sensitivity of the energy2629

loss at fixed ν and Q2. In the same figure we also show the same type of plot but for2630

ν = 145 GeV and Q2 = 35 GeV2. The expected reduction in the level of pion suppression2631

relative to ν = 35 GeV is visible and the shape of the D0 data is quite different from that2632

for ν = 35 GeV. This strong sensitivity of the shape to the value of ν will be a unique2633

and powerful tool to understand energy loss of heavy quarks in cold nuclear systems. The2634

discovery of such a dramatic difference in multiplicity ratios between light and heavy meson2635

production in Fig. 3.23 at the EIC would shed light on the hadronization process and on2636

what governs the transition from quarks and gluons to hadrons.2637

3.3.3 Strong color fluctuations inside a large nucleus2638

The transverse flow of particles is one of the key evidences for the formation of strongly2639

interacting QGP in relativistic heavy ion collision. It was recognized that fluctuations in2640

the geometry of the initial overlap zone of heavy ion collisions lead to the fluctuations of2641

vn, the coefficients of the Fourier expansion of the particle multiplicity with respect to the2642

reaction plane, and then the vn fluctuation, in particular, the v3 fluctuation leads to very2643

interesting features of two particle correlations observed in relativistic heavy ion collisions.2644

The initial-state density fluctuations seem to influence the formation and expansion of QGP.2645

An independent measurement of the fluctuations of quarks and gluons, or in general, the2646

color inside a large nucleus, is very interesting and important for understanding the nuclear2647

structure and QCD dynamics.2648

Multiple scattering between the produced parton and the nuclear medium can change2649

the momentum spectrum of the produced hadron. It can lead to a shift in the longitudinal2650

momentum (“energy loss”) as well as a broadening in averaged transverse momentum of2651

the hadron. The nuclear modification of the spectrum will be sensitive to the density of2652

93

partonic scattering centers along the path where the produced parton was passing through.2653

In semi-inclusive DIS, one can uniquely define the angle between the leptonic andhadronic scattering planes, as shown in the Sidebar on page 34. If the nuclear densityor the density of partonic scattering centers is not spherically symmetric in the transverseplane, the strength of the multiple scattering and the broadening will be sensitive to thepath or the angle between the two planes. Defining the azimuthal angle dependence of thebroadening as

〈∆p2T (φ)〉AN ≡ 〈p2

T (φ)〉A − 〈p2T (φ)〉N (3.13)

with the averaged transverse momentum squared,

〈p2T (φpq)〉A =

∫dp2

T p2T

dσeAdxBdQ2dp2

Tdφ

/dσeA

dxBdQ2(3.14)

The broadening could have a significant angular dependence in φpq - the angle between2654

the two planes - if there is strong enough fluctuation in the density of partonic scattering2655

centers, as observed by CLAS collaboration at Jefferson Lab [181].2656

With the higher energy and collider configuration, the EIC can measure the semi-2657

inclusive production of various hadrons with a much large range of momenta. It would2658

be of great interest to evaluate the nuclear dependence of the φ modulation of transverse2659

momentum distributions as well as the broadening, and, further, to see the connection to2660

the fluctuations believed to be responsible for the observed higher-order flow terms in heavy2661

ion collisions.2662

94

3.4 Connections to pA, AA and Cosmic Ray Physics2663

Conveners: Yuri Kovchegov and Thomas Ullrich2664

0.8

0.6

0.4

0.2

5 10 50 100 50010

Fra

ctio

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l co

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ibu

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n to

cro

ss-s

ectio

n

pT (GeV/c)

NLO, π0, y=0

calc. by W. Vogelsang ----- RHIC √s = 200 GeV

----- LHC √s = 5.5 TeV

gg qg

qq

ggqg

qq

Figure 3.24: Fractional contributions from gg, qg, and qq scattering processes to pion productionat mid-rapidity pp collisions at RHIC (black) and LHC (blue).

3.4.1 Connections to pA Physics2665

Both pA and eA collisions can provide excellent information on the properties of gluons2666

in the nuclear wave functions. It is therefore only logical to compare the strengths and2667

weaknesses of the two different programs in exploring the saturation regime.2668

In the beginning of the RHIC era, the dAu program was perceived as merely a useful2669

baseline reference for the heavy-ion program. It very soon turned out that due to a wise2670

choice of colliding energy, RHIC probes the transition region to a new QCD regime of2671

gluon saturation. While only marginal hints of non-linear effects were observed in DIS2672

experiments at HERA [194], it is fair to say that very tantalizing hints for gluon saturation2673

were observed in dA collisions at RHIC [195, 168, 169, 196, 197]. In the upcoming pA2674

program at the LHC, these effects will be even more pronounced. While pA and pp colliders2675

provide superior access to the low-x region, they have also some severe disadvantages which2676

impede systematic studies of saturation phenomena that we will describe below.2677

As shown in Fig. 3.24 already in pp collisions at mid-rapidity at RHIC and LHC the2678

bulk of particles produced originates from processes involving gluons; a simple manifesta-2679

tion of the dominance of gluons at low-x in hadrons (see Fig. 3.3). While it is unlikely2680

that saturation phenomena are observed at RHIC energies in pp collisions due the small2681

values of Qs even at the lowest accessible x, the amplified Qs scale in pA collision opens2682

the experimental accessible range where saturation effects become detectable. The relation2683

between rapidity y and transverse momentum pT of the final state partons/particles with2684

mass m and their fractional longitudinal momenta x1,2 is x1,2 = e±y√

(p2T +m2)/s. Hence,2685

at mid-rapidity (y = 0) at RHIC, only particle production with very small pT will be sensi-2686

95

tive to the saturation region in parton densities while at the LHC the region of transverse2687

momenta will be much larger. At RHIC saturation effects are largely absent at central2688

rapidities but become measurable at large forward rapidities (that is, for particles coming2689

out close to the incoming proton or deuteron direction with y = 2 − 4 corresponding to2690

small x2).2691

First hints for the onset of saturation dAu collisions at RHIC have been initially observed2692

in studying the rapidity dependence of the nuclear modification factor, RdAu, as a function2693

of pT for charged hadrons [195] and π0 mesons [197] and recently through forward-forward2694

hadron-hadron correlations [169, 168].2695

The nuclear modification factor for a pAu collision is defined by

RpA =1

Ncoll

dNpA/d2pT dy

dNpp/d2pT dy, (3.15)

where dN/d2pT dy is the produced hadron multiplicity in a given region of phase space2696

while Ncoll is the number of the binary nucleon–nucleon collisions. The nuclear modification2697

factor RpA is equal to 1 in absence of collective nuclear effects. Figure 3.25 shows RdAu

0

0

0.5

1

1

1

π0 〈η〉=4.0

h- 〈η〉=3.2

h- 〈η〉=2.2

1

0.8

0.6

0.4

0.2

02

2

pT (GeV/c)

RHIC √s = 200 GeV

pT (GeV/c)

Rd

Au

Rd

Au

3 4

π0 mesons 〈η〉=4.0

shadowing (KKP)

shadowing (Kretzer)

multiple scattering

Figure 3.25: Nuclear modification factor (RdAu) versus pT for minimum bias dAu collisionsmeasured at RHIC. The solid circles are for π0 mesons [197], the open circles and boxes arefor negative hadrons [195]. The error bars are statistical, the shaded boxes are point-to-pointsystematic errors. (Inset) RdAu for π0 mesons compared with pQCD calculations based oncollinear factorization. Note that none describes the magnitude of the effect.

2698

versus pT for minimum bias dAu collisions for charged hadrons measured by the BRAHMS2699

experiment [195] and π0 mesons by STAR [197]. While the inclusive yields of hadrons (π02700

mesons) at√s=200 GeV in pp collisions generally agree with pQCD calculations based2701

on DGLAP evolution and collinear factorization, in dAu collisions, the yield per binary2702

collision is suppressed with increasing η, decreasing to ∼30% of the pp yield at 〈η〉 = 4,2703

well below shadowing and multiple scattering expectations. The pT dependence of the dAu2704

yield is found to be consistent with gluon saturation picture of the Au nucleus (e.g., CGC2705

model calculations [198, 199, 200, 201, 202, 203]) although other interpretations cannot be2706

ruled out based on this observable alone [183, 204, 205].2707

96

STAR Preliminary

Unc

orre

cted

Coi

ncid

ence

Pro

babi

lity

(rad

-1) p+p → π0 π0 + X, √s = 200 GeV d+Au → π0 π0 + X, √s = 200 GeV d+Au → π0 π0 + X, √s = 200 GeV

STAR Preliminary STAR Preliminary

d+Au peripheral d+Au centralp+p

pT,L > 2 GeV/c, 1 GeV/c < pT,S < pT,L

〈ηL〉=3.2, 〈ηS〉=3.2

pT,L > 2 GeV/c, 1 GeV/c < pT,S < pT,L

〈ηL〉=3.2, 〈ηS〉=3.2pT,L > 2 GeV/c, 1 GeV/c < pT,S < pT,L

〈ηL〉=3.1, 〈ηS〉=3.2

Δφ Δφ Δφ

σ0.41 ± 0.010.68 ± 0.01

PeaksΔφ0π

σ0.46 ± 0.020.99 ± 0.06

PeaksΔφ0π

σ0.44 ± 0.021.63 ± 0.29

PeaksΔφ0π

-1 0 1 2 3 4 5 -1 0 1 2 3 4 5-1 0 1 2 3 4

0.02

0.15

0.01

0.05

0

0.03

0.025

0.02

0.015

0.01

0.005

0

0.016

0.014

0.012

0.01

0.008

0.006

0.004

0.002

0

Figure 3.26: Di-hadron correlation measured at forward rapidities at RHIC. Uncorrected coinci-dence signal versus azimuthal angle difference between two forward neutral pions in pp collisions(left) compared to peripheral (center) and central dAu collisions (right) [169]. Data are shownwith statistical errors and fit with a constant plus two Gaussian functions (in red).

More powerful technique than single inclusive measurements are two particle azimuthal2708

correlations as discussed in Section 3.2.2. In collinear factorization-based pQCD at leading2709

order, particle production in high energy hadronic interactions results from the elastic scat-2710

tering of two partons (2→ 2 scattering) leading to back-to-back jets. When high-pT hadrons2711

are used as jet surrogates, we expect the azimuthal correlations of hadron pairs to show a2712

peak at ∆φ = 0, and a ’back-to-back’ peak at π. When the gluon density increases, the2713

basic dynamics for the particle production is expected to change. Instead of elastic 2 → 22714

scattering, the particle production can proceed by the interaction of a probe parton from2715

the proton (deuteron) beam with multiple gluons from the heavy-ion beam. At sufficiently2716

high gluon densities, the transverse momentum from the fragments of the probing parton2717

may be compensated by several gluons with lower pT . Two particle azimuthal correlations2718

are expected to show a broadening of the back-to-back peak (loss of correlation: 2→ many2719

processes) and eventually to disappear. In the CGC framework, the hadronic wave-function2720

is saturated as a consequence of gluon recombination. At very low values of the longitudinal2721

momentum fraction x of the probed gluons the occupation numbers become large and the2722

probe scatters coherently off the dense gluon field of the target, which recoils collectively,2723

leading to a modification in ∆φ [165].2724

Figure 3.26 shows the (efficiency uncorrected) probability to find an associated π0 given2725

a trigger π0, both in the forward region measured by the STAR detector. Shown is the coin-2726

cidence signal versus azimuthal angle difference between the two pions in pp collisions (left)2727

compared to peripheral (center) and central dAu collisions (right) [169] (see also Fig. 3.172728

and the related discussion, along with [168]). All the distributions present two signal compo-2729

nents, surmounting a constant background representing the underlying event contribution2730

(larger in dAu). The near-side peak represents the contribution from pairs of pions belong-2731

ing to the same jet. It is not expected to be affected by saturation effects. The away-side2732

peak, represents the back-to-back contribution to the coincidence probability which should2733

disappear in going from pp to dAu if saturation sets in [165]. The data show that the width2734

of the near-side peak remains nearly unchanged from pp to dAu, and particularly from2735

peripheral to central dAu collisions. Central dAu collisions show a substantially reduced2736

97

away side peak that is significantly broadened. Again, pQCD calculations based on lin-2737

ear DGLAP evolution without coherent multiple scattering fail to describe this observation2738

while those including non-linear effects describe the data considerably well [206, 207, 199].2739

This measurement represent the strongest to-date hint for saturation phenomena and also2740

indicates that the kinematic range of the EIC is well suited to explore saturation physics2741

with great precision.2742

Although the results of the dAu program at RHIC show tantalizing evidence of satura-2743

tion phenomena, alternative explanation for each of the individual observation exist. The2744

unambiguous ultimate proof of existence of saturation can only come from an eAcollider.2745

While in eA collision the probe, the electron, is point-like and structure-less, in pA collisions2746

one has to deal with a probe whose structure is almost as complex as that of the target2747

nucleus to be studied. The Electron-Ion Colliders usefulness as a gluon “microscope” is2748

somewhat counter-intuitive since electrons do not directly interact with gluons. However,2749

the presence and dynamic of the gluons in the ion will modify the precisely understood2750

electromagnetic interaction of the electron with quarks in ways that allow us to infer the2751

gluon properties. Deeply inelastic eA collisions are dominated by one photon exchange (see2752

the Sidebar on page 20). The photon could interact with one parton to probe parton distri-2753

butions, as well as multiple partons coherently to probe multi-parton quantum correlations.2754

One of the major advantages of DIS is that it allows for the direct, model-independent, de-2755

termination of the momentum fraction x carried by the struck parton before the scattering2756

and Q2, the momentum transferred to the parton in the scattering process; only the control2757

of these variables ultimately will allow us a precise mapping of the gluon distributions and2758

their dynamics.2759

One may wonder whether the physics similar to what one can probe at an EIC could be2760

studied in the Drell-Yan process in a pA collider. Due to crossing symmetry the Drell-Yan2761

process can be related to DIS [208] with the invariant mass of the di-lepton pair M2 playing2762

the role of Q2. Owing to the very broad reach in x and M2, pA collisions at RHIC and even2763

more so at the LHC have clearly a significant discovery potential for the physics of strong2764

color fields in QCD. However, the di-lepton signal in pA is contaminated by the leptons2765

resulting from decays of heavy-flavor hadrons, such as J/ψ, up to a rather large invariant2766

masses of M2 = 16 GeV2 and even beyond [209]. This contamination does not allow one to2767

cleanly probe the saturation region of M2 < 16 GeV2. To avoid hadronic decay background2768

one may study large values of the net transverse momentum pT of the pair: however, this2769

would also push one away from the lower-pT saturation region.2770

Ultimately it will be the combination of strong pA and eA programs, each providing2771

complementary measurements, that will answer the questions raised above in full.2772

3.4.2 Connections to Ultrarelativistic Heavy-Ion Physics2773

Measurements over the last decade in heavy ion collision experiments at RHIC indicate2774

the formation of a strongly coupled plasma of quarks and gluons (sQGP). Striking results2775

include: (i) the strong collective flow of all mesons and baryons, and especially that of heavy2776

charm quarks, and (ii) the opaqueness of the hot and dense medium to hadron jets up to2777

p⊥ ∼ 20 GeV.2778

This sQGP appears to behave like a “near-perfect fluid” with a ratio of the shear viscosity2779

to entropy density, η/s, approaching zero [210, 211, 212, 213, 214]. Recent experiments at2780

the LHC with substantially higher energies and thus a hotter and longer lived plasma phase2781

confirm this picture [215].2782

98

Despite the significant insight that the QGP is a strongly correlated nearly perfect fluid,2783

little is understood about how the QGP is created and what its properties are. Qualitative2784

questions that the heavy ion community would like to answer include how the dynamics2785

of gluons in the nuclear wave functions generates entropy after the collision, what the2786

properties and dynamics of the pre-equilibrium state are, why the thermalization of the2787

system occurs rapidly, and whether the system is fully or only partially thermalized during2788

the evolution. Further, though it is widely accepted that the QGP medium is a strongly2789

correlated one, it is less clear whether the coupling is weak or strong. In the weak-coupling2790

scenario, the strongly correlated dynamics is generated by the scales that characterize the2791

electric and magnetic sectors of the hot fluid. In the strongly-coupled scenario progress2792

has been made by exploiting the Anti-de Sitter space/Conformal Field Theory (AdS/CFT)2793

correspondence [216, 217] of weakly coupled gravity (which is calculable) to strongly coupled2794

supersymmetric Yang-Mills theory with many features in common with QCD.2795

Quantitative questions the heavy ion community would like to answer include determin-2796

ing the shear viscosity of the medium averaged over its evolution, measuring the values of2797

other transport coefficients such as the bulk viscosity and the heavy quark diffusion coeffi-2798

cient, and, perhaps most importantly, identifying the equation of state of finite-temperature2799

QCD medium. Some of these questions can be addressed in numerical lattice QCD com-2800

putations. It is still not entirely clear how these results can be cross-checked and improved2801

upon in the environment of a rapidly evolving and incredibly complex heavy ion event.2802

Despite the significant progress in qualitative understanding of several aspects of this2803

matter there is still no comprehensive quantitative framework to understand all the stages2804

in the creation and expansion of the hot and dense QGP medium. In the following we2805

outline how an EIC can contribute to a better understanding of the dynamics of heavy2806

ion collisions, from the initial formation of partonic matter in bulk to jet quenching and2807

hadronization that probe the properties of the sQGP.2808

Initial Conditions in AA collisions2809

Understanding the dynamical mechanisms that generate the large flow in heavy ion collisions2810

is one of the outstanding issues in the RHIC program. Hydrodynamic modeling of RHIC2811

data is consistent with the system rapidly thermalizing at times of around 1 – 2 fm/c after2812

the initial impact of the two nuclei [218, 219, 220]. These hydrodynamic models are very2813

sensitive to the initial pre-equilibrium properties of the matter formed immediately after2814

the collision of the two nuclei.2815

Our current understanding based on the CGC framework suggests that the wave func-2816

tions of the nuclei, due to their large occupancy, can be described as classical fields, as2817

was explained above; the collision can therefore in a leading order be approximated by2818

the collision of “shock waves” of classical gluon fields [221, 222] resulting in produc-2819

tion of non-equilibrium gluonic matter. It is generally believed that the instability and2820

consequent exponential growth of these intense gluon fields would be the origin of early2821

thermalization [223, 224], though the exact mechanism for the process is not completely2822

understood. Alternatively, within the strong coupling paradigm, thermalization in heavy2823

ion collisions is achieved very rapidly, and there has been considerable recent work in this2824

direction [225, 226].2825

The properties of the nuclear wave functions will be studied in great detail in eA col-2826

lisions. They promise a better understanding of the initial state and its evolution into the2827

sQGP. Specifically, the saturation scale Qs, which can be independently extracted in eA2828

99

collisions, sets the scale for the formation and thermalization of strong gluon fields. Satu-2829

ration effects of these low-x gluon fields affect the early evolution of the pre-QGP system in2830

heavy ion collisions. Their spatial distribution governs the eccentricity of the collision vol-2831

ume and this affects our understanding of collective flow and its interpretation profoundly.2832

However, the features of these gluon fields, their momentum and spatial distributions at2833

energies relevant for RHIC are only vaguely known. More detailed information of relevance2834

to the properties of the initial state (such as the spatial distributions of gluons and sea2835

quarks) and thereby improved quantitative comparisons to heavy ion data, can be attained2836

with an EIC.2837

Figure 3.27: Spatial distribution of gluon fields of the incoming nuclei for a collision of leadions at

√s = 2760 GeV. The colors – from blue to red – denote increasing strength of gluon

correlations.

The high energy wave functions of nuclei can be viewed as coherent superpositions of2838

quantum states that are “frozen” configurations of large numbers of primarily gluons. How2839

these states decohere, produce entropy and subsequently interact is clearly essential to a2840

deep understanding of high energy heavy ion collisions. Remarkably, models based on the2841

CGC framework manage to describe particle production in AA collisions over a broad range2842

of energies and centralities extraordinary well. These models are constrained by HERA2843

inclusive and diffractive DIS data on ep collisions, and the limited fixed target eA DIS data2844

available. One such model is the IP-Sat model [227, 148] (Model-I from Sec. 3.2.1) Another2845

saturation model (Model-II from Sec. 3.2.1 ) is based on BK nonlinear evolution including2846

the running-coupling corrections (rcBK) [154, 155, 156, 157] and the impact parameter2847

independence [228].2848

The IP-Sat model can be used to construct nucleon color charge distributions event-by-2849

event; convoluting this with Woods-Saxon distributions of nucleons enables one to construct2850

Lorentz contracted two dimensional nuclear color charge distributions of the incoming nuclei2851

event-by-event. Such a nuclear color charge density profile is shown in Fig. 3.27 for a2852

heavy ion collision. The scale of transverse event-by-event fluctuations in Fig. 3.27 is 1/Qs,2853

not the nucleon size. The resulting model [232, 231] employs the fluctuating gluon fields2854

generated by the IP-Sat model to study the event-by-event evolution of gluon fields. Here,2855

the corresponding energy density distributions vary on the scale 1/Qs and are therefore2856

highly localized (as shown in the left panel of Fig. 3.28).2857

The right panel in Fig. 3.28 shows data for the centrality dependence of charged particle2858

production for heavy-ion collisions at√sNN = 200 GeV and 2760 GeV compared to both2859

Model-I (IP-Sat with fluctuations) and Model-II (based on rcBK evolution). Both models2860

do an excellent job of describing the data. (Note that Model-II is a prediction.) The pale2861

bands shown in this figure are the event-by-event fluctuations of the multiplicity in the2862

100

8

8

6

6

4

4

22

0

0

-8

-8

-6

-6

-4

-4

-2

-2 y (fm

)

x (fm) 0

2

4

6

8

10

12

14

0 50 100 150 200 250 300 350 400

Centrality (Npart)

2d

Nch/d

η/N

pa

rt

Au+Au, 200 GeV

Pb+Pb, 2.76 TeV

CGC framework:

rcBK 200 GeV

rcBK 2.76 TeV

IP-Sat + fluct. 200 GeV

IP-Sat + fluct. 2.76 TeV

Figure 3.28: Left panel: Spatial variation of the energy density in a single heavy ion event(based on IP-Sat model with fluctuations). The variations occur on distance scales 1/Qs,much smaller than the nucleon size. Right panel: Centrality dependence of the multiplicityat√sNN = 200 GeV and 2760 GeV. (AuAu [229] data from RHIC, PbPb data [230] from

LHC.). The experimental data are compared to results from two model realizations in the CGCframework. Solid curves represents the results from kT -factorization with running-coupling BKunintegrated gluon distributions [228] (Model-II from before) while dashed curves represents theresult in the IP-Sat model with fluctuations [231] (Model-I). The pale blue (LHC) and pink bands(RHIC) denote the referring range of event-by-event values of the single inclusive multiplicity.

Model-I. The successful descriptions of the energy and centrality dependence of multiplicity2863

distributions at RHIC and LHC are strong indications that the CGC provides the right2864

framework for entropy production. Therefore a fuller understanding of the small x formal-2865

ism promises to enable us to separate these initial state effects from final state entropy2866

production during the thermalization process and thereby constrain mechanisms (by their2867

centrality and energy dependence) that accomplish this.2868

The other bulk quantity very sensitive to the properties of the initial state is the collectiveflow generated in heavy ion collisions. A useful way to characterize flow [233] is throughmeasured harmonic flow coefficients vn defined through the expansion of the azimuthalparticle distribution as

dN

dφ=N

2π

(1 +

∑n

2 vn cos(nφ)

), (3.16)

where vn(pT ) = 〈cos(nφ)〉, with 〈· · · 〉 denoting an average over particles in a given pT2869

window and over events in a given centrality class, and φ = φ − ψn with the event plane2870

angle ψn = 1n arctan 〈sin(nφ)〉

〈cos(nφ)〉 . Spatial eccentricities, extant at the instant a hydrodynamic2871

flow description becomes applicable, are defined e.g. for the second harmonic as ε2 =2872

〈y 2−x 2〉/〈y 2+x 2〉, where now 〈· · · 〉 is the energy density-weighted average in the transverse2873

x-y plane. These are in turn converted to momentum space anisotropies by hydrodynamic2874

flow. How efficiently this is done is a measure of the transport properties of the strongly2875

coupled QCD matter such as the shear and bulk viscosities. Early flow studies focused2876

on the second flow harmonic coefficient v2 which is very large at RHIC and LHC, and2877

particularly sensitive to the ratio of the shear viscosity to entropy density ratio η/s. In2878

the Fig. 3.29, we show v2 for a Glauber model used in hydrodynamic simulations (left)2879

and the Kharzeev-Levin-Nardi CGC (KLN-CGC) model [234] (right). The eccentricity2880

ε2 in the Glauber model has a weaker dependence on collision centrality relative to the2881

101

pT (GeV) pT (GeV)

00 1 2 0 1 2

5

10

15

20

25

v2

(pe

rce

nt)

RHIC data

Glauber

η/s=0.16

η/s=0.16η/s=0.08

η/s=0.08η/s=10-4

η/s=10-4

0

5

10

15

20

25

v2

(pe

rce

nt)

CGC

η/s=0.24

Figure 3.29: Comparison of data and theoretical predictions using viscous relativistic hydrody-namics for vh2 (pT ) (right) with Glauber-like initial conditions (left) or a simplified implementationof CGC physics (KLN) model (right). Figures adapted from [220].

KLN-CGC model, and therefore requires a lower η/s to fit the data. The value of η/s for2882

the Glauber eccentricity in this model study is equal to 1/4π in natural units [235, 236]2883

conjectured to be a universal bound for strongly-coupled liquids based on applications of2884

the Anti-de Sitter space/Conformal Field Theory (AdS/CFT) correspondence [216, 217].2885

The KLN-CGC value on the other hand gives a number that’s twice as large this prediction.2886

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1 2 3 4 5 6

vn(v

isco

us)/

vn(id

ea

l)

n

RHIC data: 20-30% centrality

vn(η/s=0.08)/vn(ideal)

vn(η/s=0.16)/vn(ideal)

Figure 3.30: Ratio of charged hadron flow vn harmonics from viscous hydrodynamic simulationsand from ideal hydrodynamics [237] (lines only to guide the eye). The ratio is shown for twodifferent values of η/s. Higher harmonics are substantially more affected by shear viscosity thanv2.

Experimental and theoretical developments can help settle what the true value of η/s2887

and in particular hopefully provide essential information on its temperature dependence.2888

Interestingly, the effect of η/s on each of the vn harmonics is different. This is shown2889

strikingly in comparisons of results for the vn moments from event-by-event viscous hydro-2890

dynamic simulations relative to equivalent ideal hydrodynamic simulations in Fig. 3.30. The2891

figure shows the ratio of viscous to ideal moments [237] for n = 2, · · · , 5 for the previously2892

102

discussed values of η/s. The damping of the higher moments vn is quite dramatic with2893

increasing η/s. The results shown are for the Glauber model, and the values for v2,3,4 are in2894

good agreement with available RHIC data [238] for η/s = 0.08. In contrast, the CGC-KLN2895

model, which we saw fits data for the larger η/s = 0.16, does poorly with v3 because of2896

the damping effect noted. The poor agreement of the CGC-KLN model with the higher vn2897

moments can be traced primarily to the absence of event-by-event color charge fluctuations2898

we discussed previously. The IP-Sat model with fluctuations [232, 231] includes these and2899

the results for the moments are closer to the Glauber model that is tuned to fit the data.2900

It is here that we see that the EIC might have significant additional impact on bulk2901

observables in AA collisions. (In addition of course to the absolutely crucial input of estab-2902

lishing the saturation paradigm of entropy production, extracting information on the energy2903

and centrality dependence of Qs and providing information on gluon correlations that may2904

influence thermalization and long range rapidity correlations.) The measurement of dσ/dt2905

in diffractive eA collisions (see Sec. 3.2.2 and Fig. 3.20) allows for a clean determination2906

of the spatial gluon density on average and will help constrain the scale and magnitude of2907

event-by-event fluctuations of color charge densities. Inelastic vector meson production can2908

further constraint the spatial extent of these event-by-event fluctuations [239]. Hadronic2909

multiplicity fluctuations, along with the di-hadron correlation, would also allow one to pin-2910

point the dynamical origin of the energy density fluctuations in heavy ion collisions. Such2911

direct access to spatial information is unique to eA collisions (in contrast to pA collisions)2912

and therefore can only be provided by an EIC.2913

Energy Loss and Hadronization2914

The dramatic suppression of high-pT hadrons discovered at RHIC is an important evidence2915

for production of a dense medium in nuclear collisions. The suppression mechanisms are still2916

under debate, so alternative measurements are required to test models of partonic energy2917

loss and in-medium hadronization. Indeed, inclusive hadron production in AA collisions2918

contains a number of unknowns: (i) the space-time development of the created medium2919

is poorly known; (ii) the energy and (related) virtuality of the produced parton are not2920

observed; (iii) the argument of the fragmentation function, the fractional energy zh of the2921

hadron, is not measured; (iv) the fractional momenta x1, x2 of the colliding partons are not2922

known either, because the recoil jet is often not observed.2923

Electron-ion collisions offer an alternative way to study the mechanisms of energy loss2924

and in-medium hadronization of highly virtual partons. Semi-inclusive deep-inelastic pro-2925

cesses (SIDIS, see Chap. 2) can be used as a testing ground for the high-pT hadron pro-2926

duction seen in the nuclear collisions, as illustrated in Fig. 3.31.2927

The energy (i.e. transverse momentum pT ) of the detected hadrons in AA collisions2928

varies up to 10–20 GeV at RHIC and up to about 100 GeV at LHC. This energy range is only2929

partially covered by available data from the HERMES [240] and CLAS [241] experiments,2930

and is expected to be fully covered by SIDIS measurements at future EIC. The path lengths2931

in the cold nuclear matter and hot medium are also similar, of the order of the nuclear radius.2932

However, SIDIS on nuclear targets allows to test models in much more specific conditions:2933

(i) the nuclear density does not vary with time, its value and spacial distribution are well2934

known; (ii) the initial energy ν (see the Sidebar on page 20) and virtuality Q2 of the2935

hadronizing quark, as well as the fractional energy zh of the detected hadron, are measured.2936

Notice, that the energy of the quark produced in DIS is known only in the region2937

dominated by valence quarks, i.e. at x & 0.1. At smaller x dominated by the sea, the2938

103

h h

qq

A A A

? ?

ee′

cold nuclear

matter

hot nuclear

matter

q

Figure 3.31: Cartoon showing a similarity of the kinematics and geometry in production ofhadrons in a hot matter created in AA collisions and in SIDIS on nuclei.

virtual photon in the nuclear rest frame fluctuates into a qq pair before the interaction,2939

with the quark and antiquark sharing the photon energy ν in the way we can not directly2940

measure, which makes the parton energy and fraction zh not directly observable anymore.2941

Hence one has to conduct measurements at sufficiently large-x and large-Q2 at an EIC.2942

Accurate measurements of different observables, like the magnitude of suppression and2943

broadening at different ν, zh and Q2 with different nuclei should provide a stringent test2944

for the models of energy loss and in-medium hadronization. Furthermore, at an EIC, for2945

the first time one will be able to study open charm and open bottom meson production2946

in eA collisions, as well as the in-medium propagation of the associated heavy quarks:2947

these measurements would allow one to fundamentally test high-energy QCD predictions2948

for energy loss, and confront puzzling measurements of heavy flavor suppression in the QGP2949

at RHIC (see Sec. 3.3.2).2950

h

q

q

q

l l fp

q

γ∗

Figure 3.32: Illustration for the characteristic time scales for in-medium hadronization: theproduction length lp and the length lf of a hadron wave function formation from a colorlessdipole.

A pictorial description of the space-time development of the hadronization process in2951

a medium is presented in Fig. 3.32. A highly virtual parton produced in DIS on a bound2952

nucleon (solid broken line) intensively radiates gluons (corkscrew lines) and dissipates energy2953

though this radiation. Therefore, the production length lp of a (pre)hadron should shrink2954

at zh → 1 as lp ∝ (1 − zh) in order to respect energy conservation (the limit of zh →2955

104

1 corresponds to no energy loss, which is only possible for very short lp). Final state2956

interactions in the medium induce additional gluon radiation compared with hadronization2957

in vacuum, making lp even shorter. Another characteristic time scale, lf , which is usually2958

much longer than lp, is associated with the formation of the wave function of the detected2959

hadron.2960

The magnitude of lp is a frequently debated issue [189]. If lp is short, as is supported2961

by calculations, the attenuation of the colorless dipole becomes the main source of nuclear2962

energy loss and hadron suppression. This model [242] successfully predicted nuclear sup-2963

pression at large zh as reported by HERMES collaboration [243]. However, one may arrive2964

at qualitatively similar effects by assuming that lp is much longer than the nuclear radius2965

[244]. In this case the observed nuclear suppression can be related to the radiative energy2966

loss, akin to that seen in heavy ion collisions. Nevertheless, such a purely partonic energy2967

loss scenario [244] fails to quantitatively explain the observed nuclear attenuation at large2968

zh [240]. The EIC measurements of nuclear effects in SIDIS would clarify the details of par-2969

tonic energy loss and hadronization dynamics by providing stringent tests of the existing2970

models.2971

3.4.3 Connections to Cosmic Ray Physics2972

Ne

utr

ino

-Nu

cle

on

Cro

ss-S

ectio

n (

pb

)

Neutrino Energy (GeV)

Unscreened (LO DGLAP)

Unscreened (NLO DGLAP)

Screened (BFKL/DGLAP)

Color Glass Condensate

103

107

108

109

1010

1011

1012

104

105

Figure 3.33: Predictions of several models with and without parton saturation (CGC) physicsfor the cross sections for neutrino–nucleon scattering at high energies as calculated in [245].

Decisive evidence in favor of parton saturation, which could be uncovered at an EIC,2973

would have a profound impact on the physics of Cosmic Rays. The sources of the observed2974

ultra-high energy cosmic rays must also generate ultra–high energy neutrinos. Deep in-2975

elastic scattering of these neutrinos with nucleons on Earth is very sensitive to the strong2976

interaction dynamics. This is shown in Fig. 3.33 for the cross sections for neutrino–nucleon2977

105

scattering plotted as a function of the incident neutrino energy for several models. As we ar-2978

gued above, the experiments at an EIC would be able to rule out many of the models of high2979

energy strong interactions, resulting in a more precise prediction for the neutrino–nucleon2980

cross section, thus significantly improving the precision of the theoretical predictions for the2981

cosmic ray interactions. The improved precision in our understanding of strong interactions2982

will enhance the ability of the cosmic ray experiments to interpret their measurements ac-2983

curately and will thus allow them to uncover new physics beyond the Standard Model of2984

particle physics.2985

106

Acknowledgment3778

This document is a result of a community wide effort. We particularly thank the fol-3779

lowing colleagues for their contributions to the preparation of this document:3780

J. L. Albacete (IPNO, Universite Paris-Sud 11, CNRS/IN2P3)3781

M. Anselmino (Torino University & INFN)3782

N. Armesto (University of Santiago de Campostella)3783

A. Bacchetta (University of Pavia)3784

T. Burton (Brookhaven National Lab)3785

N.-B. Chang(Shandong University)3786

W.-T. Deng (Frankfurt U., FIAS & Shandong University)3787

A. Dumitru (Baruch College, CUNY)3788

R. Dupre (CEA, Centre de Saclay)3789

S. Fazio (Brookhaven National Lab)3790

V. Guzey (Jefferson Lab)3791

H. Hakobyan(Universidad Tecnica Federico Santa Maria)3792

Y. Hao (Brookhaven National Lab)3793

D. Hasch (INFN, Frascatti)3794

M. Huang(Duke University)3795

C. Hyde (Old Dominion University)3796

B. Kopeliovich (Universidad Tecnica Federico Santa Maria)3797

K. Kumericki (University of Zagreb)3798

M. Lamont (Brookhaven National Lab)3799

T. Lappi (University of Jyvaskyla)3800

J.-H. Lee (Brookhaven National Lab)3801

Y. Lee (Brookhaven National Lab)3802

E. M. Levin (Tel Aviv University & Universidad Tecnica Federico Santa Marıa)3803

F.-L. Lin (Brookhaven National Lab)3804

V. Litvinenko (Brookhaven National Lab)3805

C. Marquet (CERN)3806

A. Metz (Temple University)3807

V. S. Morozov (Jefferson Lab)3808

D. Muller (Ruhr-University Bochum)3809

P. Nadel-Turonski (Jefferson Lab)3810

A. Prokudin (Jefferson Lab)3811

V. Ptitsyn, (Brookhaven National Lab)3812

X. Qian (Caltech)3813

R. Sassot (University de Buenos Aires)3814

G. Schnell (University of Basque Country, Bilbao)3815

P. Schweitzer (University of Connecticut)3816

M. Stratmann (Brookhaven National Lab)3817

M. Sullivan (SLAC)3818

S. Taneja (Stony Brook University & Dalhousie University)3819

T. Toll (Brookhaven National Lab)3820

D. Trbojevic (Brookhaven National Lab)3821

R. Venugopalan (Brookhaven National Lab)3822

138

X. N. Wang (Lawrence Berkeley National Lab & Central China Normal University)3823

B.-W. Xiao (Central China Normal University)3824

Y.-H. Zhang (Jefferson Lab)3825

L. Zheng (Brookhaven National Lab)3826

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