Nuclear Unintegrated Gluon Distribution (nUGD)

• The nuclear unintegrated gluon distribution (nUGD) is defined as the momentum-resolved small-x gluon density in a nucleus, characterized by its dependence on both longitudinal fraction and transverse momentum.
• It underpins kₜ-factorization approaches in dipole scattering and saturation physics, serving as the key gluonic input for models in nuclear DIS and heavy-ion collisions.
• nUGD models bridge low-kₜ saturation effects with DGLAP-matched high-kₜ behavior, integrating local nuclear geometry and multiple scattering into observable predictions.

The nuclear unintegrated gluon distribution (nUGD) is the small-$x$ gluon density of a nuclear target resolved simultaneously in longitudinal momentum fraction and transverse momentum, and, in many formulations, also in impact parameter or transverse coordinate. It appears as $\varphi_A(x,k_T^2,b)$, $\phi_A(x,k_T^2,b)$, $\mathcal F_A(x,k_T^2,\boldsymbol b)$, or $xG_A^{ij}(x,\boldsymbol k)$, depending on the formalism, and it is the basic gluonic input for $k_t$-factorization, dipole scattering, saturation physics, nuclear DIS, and the initialization of heavy-ion collisions (Albacete et al., 2010, Moriggi et al., 2020, Mehtar-Tani, 2021).

1. Basic meaning and normalization

In the high-energy literature, an unintegrated gluon distribution keeps explicit the gluon transverse momentum $k_T$ rather than integrating it out into a collinear PDF. A standard relation used in several constructions is

$$xg(x,Q^2)=\int^{Q^2}\frac{dk_t^2}{k_t^2}\,f(x,k_t^2),$$

or, equivalently in approximate form,

$$g(x,Q^2)\sim g(x,Q_0^2)+\int_{Q_0^2}^{Q^2} dk_t^2\, g(x,k_t^2).$$

In the nuclear case the same logic is used, but the target is a nucleus and the distribution may depend on $A$, on the local thickness, or on the impact parameter (Grinyuk et al., 2012, Boroun, 2023).

A useful conceptual distinction is that “nUGD” is not tied to a single universal phenomenological ansatz. In small-$\varphi_A(x,k_T^2,b)$0 physics one encounters dipole-based UGDs, Glauberized dipole constructions, BFKL-inspired momentum-space UGDs, DGLAP-matched dipole UGDs, and gauge-invariant operator definitions with explicit $\varphi_A(x,k_T^2,b)$1-dependence. This suggests that the nUGD is best regarded as a family of closely related target correlators whose common role is to encode nuclear gluon structure differential in transverse momentum, rather than as a single unique function.

At the operator level, a particularly important property is that the unintegrated object can interpolate between the collinear gluon PDF and the small-$\varphi_A(x,k_T^2,b)$2 dipole limit. In the formulation of Hatta, Mueller, and collaborators, the tensor UGD satisfies

$\varphi_A(x,k_T^2,b)$3

while in the strict small-$\varphi_A(x,k_T^2,b)$4 limit it reduces, after the appropriate projection, to the dipole operator built from Wilson lines. This connects nUGDs directly to both nuclear PDFs and saturation correlators (Mehtar-Tani, 2021).

2. Principal constructions of the nUGD

A widely used dipole-based definition, employed in rcBK phenomenology for heavy-ion collisions, starts from the forward dipole amplitude in the fundamental representation, $\varphi_A(x,k_T^2,b)$5, and the adjoint dipole

$\varphi_A(x,k_T^2,b)$6

The local UGD per transverse area is then defined by

$\varphi_A(x,k_T^2,b)$7

In this construction, the nuclear UGD is not introduced as a separate closed formula; instead it is built as a superposition of rcBK-evolved local UGDs with position-dependent initial saturation scales (Albacete et al., 2010).

A second standard route is the Glauber–Mueller dipole construction. One starts from a dipole–proton cross section $\varphi_A(x,k_T^2,b)$8, eikonalizes it through the nuclear thickness function,

$\varphi_A(x,k_T^2,b)$9

and defines the nuclear UGD operationally by a Hankel transform,

$\phi_A(x,k_T^2,b)$0

Here the nuclear modification is entirely encoded in multiple scattering through $\phi_A(x,k_T^2,b)$1, typically obtained from a Woods–Saxon density (Moriggi et al., 2020).

A third line of construction starts from the dipole cross section in models such as BGK. There the proton UGD is obtained from a Fourier–Bessel transform of $\phi_A(x,k_T^2,b)$2, and its large-$\phi_A(x,k_T^2,b)$3 tail is matched to

$\phi_A(x,k_T^2,b)$4

The same procedure can be generalized to nuclei by replacing the proton dipole cross section with a Glauberized nuclear dipole cross section or a BGK-like nuclear model and then matching to a nuclear collinear gluon density $\phi_A(x,k_T^2,b)$5 (Łuszczak et al., 2022).

Framework	Defining object	Nuclearization mechanism
rcBK local-$\phi_A(x,k_T^2,b)$6	$\phi_A(x,k_T^2,b)$7	$\phi_A(x,k_T^2,b)$8
Glauber–Mueller	$\phi_A(x,k_T^2,b)$9, then $\mathcal F_A(x,k_T^2,\boldsymbol b)$0 by Hankel transform	Woods–Saxon $\mathcal F_A(x,k_T^2,\boldsymbol b)$1
Operator definition	$\mathcal F_A(x,k_T^2,\boldsymbol b)$2 from $\mathcal F_A(x,k_T^2,\boldsymbol b)$3-Wilson-line correlator	Replace $\mathcal F_A(x,k_T^2,\boldsymbol b)$4 by $\mathcal F_A(x,k_T^2,\boldsymbol b)$5
BGK/DGLAP-matched	Fourier–Bessel transform of $\mathcal F_A(x,k_T^2,\boldsymbol b)$6	Glauberized $\mathcal F_A(x,k_T^2,\boldsymbol b)$7 plus $\mathcal F_A(x,k_T^2,\boldsymbol b)$8 matching

These constructions are not identical, but they share the same target: a momentum-space representation of nuclear small-$\mathcal F_A(x,k_T^2,\boldsymbol b)$9 gluon structure that can be inserted into factorization formulae.

3. Nuclear geometry, local saturation, and $xG_A^{ij}(x,\boldsymbol k)$0-dependence

A recurrent theme in modern nUGD modeling is that nuclear effects are local and geometric, not merely global. In the rcBK Monte Carlo construction for Pb+Pb collisions, the initial saturation scale at transverse position $xG_A^{ij}(x,\boldsymbol k)$1 is taken as

$xG_A^{ij}(x,\boldsymbol k)$2

where $xG_A^{ij}(x,\boldsymbol k)$3 counts overlapping nucleons. The nucleons are sampled from a Woods–Saxon distribution with a hard-core repulsion of about $xG_A^{ij}(x,\boldsymbol k)$4 fm, projected onto the transverse plane, and the local overlap number is

$xG_A^{ij}(x,\boldsymbol k)$5

The rcBK evolution is then solved for discrete values $xG_A^{ij}(x,\boldsymbol k)$6, $xG_A^{ij}(x,\boldsymbol k)$7, and the corresponding local UGDs are tabulated and interpolated event by event (Albacete et al., 2010).

This construction makes explicit that the nuclear amplitude at a given point is not a simple global $xG_A^{ij}(x,\boldsymbol k)$8 rescaling. In central events many transverse points have large $xG_A^{ij}(x,\boldsymbol k)$9, whereas peripheral events are dominated by $k_t$0 or $k_t$1. The resulting statement that averaging over nuclear geometry does not commute with evolution is important: an nUGD obtained from a single average nuclear saturation scale is not equivalent to an average over locally evolved configurations.

A more global geometric-scaling prescription rescales the saturation scale itself,

$k_t$2

with

$k_t$3

In this case the nuclear scaling variable is $k_t$4, and normalized nuclear momentum-space probability distributions are built directly from geometric-scaling UGDs such as GBW, MPM, or Levin–Tuchin forms (Ramos et al., 12 Jul 2025).

Within Glauber–Mueller approaches, the analogous geometric information is carried by the nuclear thickness $k_t$5. Shadowing weakens at large $k_t$6, because for small dipoles the eikonal factor can be expanded linearly in $k_t$7, while at low and moderate $k_t$8 multiple scattering modifies the nUGD strongly. A common misconception is therefore that the nUGD is entirely specified by an $k_t$9-enhanced saturation scale; the literature shows that the local thickness profile, event geometry, and impact-parameter dependence matter separately (Moriggi et al., 2020).

4. Evolution equations and factorization schemes

In rcBK-based constructions the basic small-$k_T$0 evolution is the Balitsky–Kovchegov equation with running coupling. In the large-$k_T$1 limit,

$k_T$2

supplemented by Balitsky’s running-coupling kernel. The initial condition is a generalized McLerran–Venugopalan form,

$k_T$3

with $k_T$4 or $k_T$5. Larger $k_T$6 yields a steeper falloff of the UGD at $k_T$7 (Albacete et al., 2010).

An alternative analytic route to nonlinear momentum-space evolution is the MD-BFKL equation for the UGD $k_T$8, which contains both a shadowing term and an anti-shadowing term,

$k_T$9

The second term makes explicit that recombination can redistribute gluon momentum and not only suppress it. The detailed analysis was performed for the proton, but the authors argue that the same structure is directly relevant to dense systems, where recombination is amplified (Wang et al., 2023).

Once an nUGD is specified, it enters $$xg(x,Q^2)=\int^{Q^2}\frac{dk_t^2}{k_t^2}\,f(x,k_t^2),$$0-factorization formulae for particle production. In the rcBK heavy-ion model, the inclusive gluon production cross section is written as

$$xg(x,Q^2)=\int^{Q^2}\frac{dk_t^2}{k_t^2}\,f(x,k_t^2),$$1

with $$xg(x,Q^2)=\int^{Q^2}\frac{dk_t^2}{k_t^2}\,f(x,k_t^2),$$2. In the Glauber–Mueller-based heavy-ion calculation one instead uses

$$xg(x,Q^2)=\int^{Q^2}\frac{dk_t^2}{k_t^2}\,f(x,k_t^2),$$3

Both formulae are small-$$xg(x,Q^2)=\int^{Q^2}\frac{dk_t^2}{k_t^2}\,f(x,k_t^2),$$4, semi-hard constructions; neither is a full large-$$xg(x,Q^2)=\int^{Q^2}\frac{dk_t^2}{k_t^2}\,f(x,k_t^2),$$5 or full NLO treatment (Albacete et al., 2010, Moriggi et al., 2020).

5. Phenomenology and observable consequences

In heavy-ion phenomenology, the nUGD is used to predict multiplicity, transverse energy, minijet production, and initial-state nuclear modification. In the rcBK Monte Carlo model for Pb+Pb at $$xg(x,Q^2)=\int^{Q^2}\frac{dk_t^2}{k_t^2}\,f(x,k_t^2),$$6 TeV, the charged multiplicity and transverse energy are obtained from the gluon yield as

$$xg(x,Q^2)=\int^{Q^2}\frac{dk_t^2}{k_t^2}\,f(x,k_t^2),$$7

With $$xg(x,Q^2)=\int^{Q^2}\frac{dk_t^2}{k_t^2}\,f(x,k_t^2),$$8 fixed from high-$$xg(x,Q^2)=\int^{Q^2}\frac{dk_t^2}{k_t^2}\,f(x,k_t^2),$$9 p+p spectra at $$g(x,Q^2)\sim g(x,Q_0^2)+\int_{Q_0^2}^{Q^2} dk_t^2\, g(x,k_t^2).$$0 TeV and $$g(x,Q^2)\sim g(x,Q_0^2)+\int_{Q_0^2}^{Q^2} dk_t^2\, g(x,k_t^2).$$1 fixed from integrated charged multiplicities, the model reproduces the centrality dependence of $$g(x,Q^2)\sim g(x,Q_0^2)+\int_{Q_0^2}^{Q^2} dk_t^2\, g(x,k_t^2).$$2 and $$g(x,Q^2)\sim g(x,Q_0^2)+\int_{Q_0^2}^{Q^2} dk_t^2\, g(x,k_t^2).$$3 in Pb+Pb (Albacete et al., 2010).

In the Glauber–Mueller plus $$g(x,Q^2)\sim g(x,Q_0^2)+\int_{Q_0^2}^{Q^2} dk_t^2\, g(x,k_t^2).$$4-factorization framework, the nUGD provides the initial hard component of hadron production in AA collisions. The shadowing-only nuclear modification factor exhibits a Cronin-like peak around $$g(x,Q^2)\sim g(x,Q_0^2)+\int_{Q_0^2}^{Q^2} dk_t^2\, g(x,k_t^2).$$5 GeV and tends to $$g(x,Q^2)\sim g(x,Q_0^2)+\int_{Q_0^2}^{Q^2} dk_t^2\, g(x,k_t^2).$$6 at higher $$g(x,Q^2)\sim g(x,Q_0^2)+\int_{Q_0^2}^{Q^2} dk_t^2\, g(x,k_t^2).$$7, while a realistic description of ALICE Pb–Pb pion data requires an additional Boltzmann–Gibbs Blast Wave equilibrium component. The resulting picture is that nUGD-based minijet production is necessary but not sufficient: in central heavy-ion collisions, final-state collective dynamics substantially modifies the initial distribution (Moriggi et al., 2020).

The nUGD also enters entropy-based descriptions of initial nuclear matter. When the nuclear UGD is normalized to

$$g(x,Q^2)\sim g(x,Q_0^2)+\int_{Q_0^2}^{Q^2} dk_t^2\, g(x,k_t^2).$$8

it defines a transverse-momentum probability density, and the QCD dynamical entropy becomes a Kullback–Leibler divergence between $$g(x,Q^2)\sim g(x,Q_0^2)+\int_{Q_0^2}^{Q^2} dk_t^2\, g(x,k_t^2).$$9 and $A$0. For geometric-scaling nUGDs, the normalization and the scaling ratio $A$1 make the dynamical entropy almost independent of $A$2, while the entropy density $A$3 retains an explicit $A$4 factor (Ramos et al., 12 Jul 2025).

Exclusive vector meson production provides a complementary probe. The proton analyses of $A$5-meson leptoproduction show that helicity observables such as $A$6 are highly sensitive to the $A$7-shape of the UGD. The same high-energy factorization structure extends to nuclei through

$A$8

This suggests that coherent and incoherent vector-meson production at an EIC would be a sharp discriminator among nUGD models, particularly for their impact-parameter dependence and saturation pattern (Bolognino et al., 2021).

6. Conceptual distinctions, misconceptions, and open problems

The first conceptual issue is non-uniqueness. Small-$A$9 QCD distinguishes the dipole gluon distribution, the Weizsäcker–Williams distribution, and more general gauge-invariant operators. The tensor UGD $\varphi_A(x,k_T^2,b)$00 was proposed precisely to encompass both the collinear gluon PDF and the dipole limit in a single operator definition. This means that “the nUGD” is framework-dependent unless one specifies which operator or which effective definition is being used (Mehtar-Tani, 2021).

A second issue is impact-parameter dependence. In several phenomenological implementations the small-$\varphi_A(x,k_T^2,b)$01 evolution itself is solved without explicit impact parameter, while nuclear geometry is inserted through the initial condition or through $\varphi_A(x,k_T^2,b)$02. This is explicit in the rcBK heavy-ion model, where translational invariance is assumed in the evolution and geometry is encoded through the local $\varphi_A(x,k_T^2,b)$03, and in Glauberized dipole models, where $\varphi_A(x,k_T^2,b)$04-dependence enters eikonally but the underlying proton structure may remain homogeneous (Albacete et al., 2010, Moriggi et al., 2020).

A third issue is matching between saturation physics and collinear evolution. BGK-type constructions and related UGD analyses emphasize that a practical nUGD should connect a dipole-based low-$\varphi_A(x,k_T^2,b)$05 region to a DGLAP-like high-$\varphi_A(x,k_T^2,b)$06 tail through

$\varphi_A(x,k_T^2,b)$07

This is attractive phenomenologically because it links nuclear DIS observables and semi-hard production to the same gluonic input, but it still depends on model choices for $\varphi_A(x,k_T^2,b)$08, the matching scale, and the running-coupling prescription (Łuszczak et al., 2022, Boroun, 2023).

The open theoretical direction is a unified evolution equation for a nuclear operator $\varphi_A(x,k_T^2,b)$09 that reduces to BK/JIMWLK in the strict small-$\varphi_A(x,k_T^2,b)$10 dipole limit and to DGLAP after $\varphi_A(x,k_T^2,b)$11-integration. The operator formulation argues that such an object is target-independent at the level of definition, with the nuclear generalization obtained by replacing $\varphi_A(x,k_T^2,b)$12 by $\varphi_A(x,k_T^2,b)$13. A plausible implication is that future progress on nUGDs will depend less on inventing new standalone parameterizations and more on deriving controlled interpolation schemes between collinear nuclear PDFs, Wilson-line correlators, and impact-parameter dependent saturation dynamics (Mehtar-Tani, 2021).

In that sense, the nUGD is not merely a technical input for $\varphi_A(x,k_T^2,b)$14-factorization. It is the point where nuclear geometry, nonlinear small-$\varphi_A(x,k_T^2,b)$15 evolution, multiple scattering, and momentum-space observables meet.