Unintegrated Gluon Densities in QCD

• Unintegrated gluon densities are defined as probability distributions that encode both the longitudinal momentum fraction and transverse momentum of gluons in a proton.
• They underpin kT-factorization and bridge the gap between collinear parton distributions and transverse momentum dependent observables in high-energy QCD.
• Various models, including ABIPSW, GBW, and BFKL-based approaches, offer insights into saturation, evolution, and phenomenological applications in collider experiments.

Unintegrated gluon densities (UGDs) are fundamental objects in QCD high-energy factorization, encoding the probability for finding a gluon in a proton with both specified longitudinal momentum fraction $x$ and transverse momentum $k_T$ (or $\kappa$). UGDs unify the treatment of collinear and transverse degrees of freedom, enabling the description of exclusive and semi-inclusive reactions sensitive to gluon transverse dynamics, and providing the basis for $k_T$-factorization and small-$x$ resummation. The precision, universality, and model dependence of UGDs are central challenges in modern hadronic phenomenology, with significant impact on collider and deep-inelastic scattering observables.

1. Formal Definition and Collinear Limit

In high-energy ($s \to \infty$) QCD, the unintegrated gluon distribution ${\cal F}(x,\kappa^2,\mu^2)$ denotes the probability density for emitting a gluon with momentum fraction $x$ and transverse momentum squared $\kappa^2$, probed at factorization scale $\mu^2$:

$$g(x,\mu^2) = \int_0^{\mu^2} d\kappa^2 \, {\cal F}(x,\kappa^2,\mu^2)$$

This property ensures recovery of the standard collinear gluon PDF upon integration over $\kappa^2$ up to $\mu^2$ (Celiberto, 2019). In practical applications, the shorthand $${\cal F}(x,\kappa^2) \equiv {\cal F}(x,\kappa^2,\mu_F^2)$$ with $\mu_F^2 \sim Q^2$ is often used.

UGDs admit several operator representations. In particular, they can be defined as the Fourier transform of bi-local field-strength correlators with appropriate Wilson lines for gauge invariance:

$$\mathcal{F}(x,\mathbf{k}_T^2) = \frac{1}{x P^+ (2\pi)^3} \int dz^- d^2\mathbf{z}_T \; e^{i x P^+ z^- - i \mathbf{k}_T \cdot \mathbf{z}_T} \; \langle P | F^{+\mu}_a(0) \mathcal{W}_{ab}[0, z] F^{+}_{b,\mu}(z) | P \rangle$$

where $\mathcal{W}_{ab}$ is a gauge link and $F^{+\mu}$ is the gluon field strength tensor (Celiberto, 2019, Bolognino et al., 2022, Mehtar-Tani, 2021, Avsar, 2011).

2. UGD Models and Their Structural Features

Several phenomenologically motivated and theoretically rooted models for UGDs exist. These include:

• ABIPSW (x-independent toy model):

$${\cal F}(x,\kappa^2) = \frac{A}{(2\pi)^2 M^2} \frac{\kappa^2}{\kappa^2 + M^2}$$

Exhibits infrared ($k^2 \to 0$) vanishing $\sim k^2$ and UV saturation to a constant (Celiberto, 2019).

• Gluon-momentum-derivative (“PDF derivative”) model:

$${\cal F}(x,\kappa^2) = \frac{\partial}{\partial\ln\kappa^2} [x g(x,\kappa^2)]$$

By construction, integrates to the collinear gluon density (Celiberto, 2019, Bolognino et al., 2022, Boroun, 2023).

• Ivanov–Nikolaev soft+hard two-component model:

$${\cal F}(x,\kappa^2) = {\cal F}_s^{(B)}(x,\kappa^2)\frac{\kappa_s^2}{\kappa^2 + \kappa_s^2} + {\cal F}_h(x,\kappa^2)\frac{\kappa^2}{\kappa^2 + \kappa_h^2}$$

Soft component dominates at low $k$, hard component matches DGLAP asymptotics (Celiberto, 2019).

• BFKL-based (HSS) model:

$${\cal F}(x,\kappa^2) = \int_{-\infty}^{\infty} \frac{d\nu}{2\pi^2} \nu {\cal C} \frac{\Gamma(\delta - \frac{1}{2} - i\nu)}{\Gamma(\delta)} \left(\frac{1}{x}\right)^{\chi(\frac{1}{2} + i\nu)} \left(\frac{\kappa^2}{Q_0^2}\right)^{\frac{1}{2} + i\nu}$$

Incorporates collinearly-improved NLO BFKL evolution (Celiberto, 2019, Hentschinski, 2011).

• GBW and BGK saturation/dipole-inspired models:

$${\cal F}_{\rm GBW}(x,k_t^2) = \frac{\sigma_0}{4\pi^2 \alpha_s} k_t^2 R_0^2(x) e^{-R_0^2(x) k_t^2}$$

with $R_0^2(x) = (x/x_0)^\lambda$ and parameters fit to HERA data (Celiberto, 2019, Boroun, 2023, Łuszczak et al., 2022, Grinyuk et al., 2012).

• WMR last-step DGLAP+Sudakov:

$${\cal F}(x,\kappa^2;\mu) = T_g(\kappa^2,\mu) \frac{\alpha_s(\kappa^2)}{2\pi} \int_x^1 dz \left[ P_{gg}(z)\frac{x}{z}g(\frac{x}{z},\kappa^2) \Theta\left(\frac{\mu}{\mu + \kappa} - z \right) + ... \right]$$

Features Sudakov suppression at low $k$ and matches collinear PDFs at large $k$ (Celiberto, 2019).

The table below summarizes IR/UV behavior and empirical successes of these models (Bolognino et al., 2022, Celiberto, 2019):

Model	IR Limit	UV Limit	Match to $Q^2$ Data
ABIPSW	$k^2$	const	Good at low $Q^2$
GBW	$k^2$	exp. suppression	Good at high $Q^2$
IN	soft/hard interplay	power-law	Intermediate range
HSS	$\sim k^2$	BFKL tail	Underestimates norm
WMR	Sudakov	DGLAP power-law	Needs tuning

3. Evolution Equations and Matching

The evolution of UGDs at small $x$ is governed by the BFKL equation (with improvement by kinematic constraints and energy-momentum conservation at NLL) (Oliveira et al., 2014, Celiberto et al., 1 Dec 2025):

$$\frac{\partial {\cal F}(x, k_T^2)}{\partial \ln(1/x)} = \int_0^\infty d k_T^{\prime2} K(k_T^2, k_T^{\prime2}) {\cal F}(x, k_T^{\prime2}) - \text{nonlinear(s)}$$

Non-linear extensions, such as the BK (Balitsky-Kovchegov) equation, incorporate gluon recombination and saturation effects, which are crucial for describing dense systems and achieving unitarity (Albacete et al., 2010, Kutak et al., 2013):

$$\frac{\partial N(r, Y)}{\partial Y} = K_{\text{linear}} \otimes N - K_{\text{saturation}} \otimes N^2$$

At large values of the QCD coupling, a diffusive regime emerges, seen also in AdS/CFT approaches to $\mathcal{N}=4$ SYM (Kutak et al., 2013).

The matching to collinear PDFs requires, for instance,

$$g(x, Q^2) = \int_0^{Q^2} d k_T^2 \, {\cal F}(x, k_T^2)$$

and similarly

$${\cal F}(x, k_T^2) \sim \frac{\partial}{\partial \ln k_T^2}[x g(x, k_T^2)]$$

(Celiberto, 2019, Celiberto et al., 1 Dec 2025, Łuszczak et al., 2022).

Combined BFKL+DGLAP evolution schemes further enable smooth interpolation between small-$x$ and high-$Q^2$ regimes (Toton, 2014).

4. Phenomenological Applications and Discriminating Observables

UGDs directly enter the $k_T$-factorization formulas for hard processes:

$$\sigma_{L,T}(Q^2,W) \propto |T_{00,11}(s,Q^2)|^2 \sim \left\{ \int \frac{d^2 \kappa}{\kappa^4} \Phi_{L,T}(Q^2, \kappa) {\cal F}(x, \kappa^2) \right\}^2$$

(Celiberto, 2019).

Exclusive vector-meson leptoproduction ($\rho$ meson) at HERA is highly sensitive to the shape of the UGD. The amplitude ratio $R(Q^2, W) = T_{11}/T_{00}$ is especially discriminating: it is essentially independent of overall normalization but highlights differences in the transverse-momentum structure (Celiberto, 2019, Bolognino et al., 2022). None of the surveyed models fully reproduces HERA data across all $Q^2$, but ABIPSW and GBW models perform best in the intermediate regime (Celiberto, 2019):

$$\chi^2/\text{ndf} \sim 2-4 \quad \text{for ABIPSW/GBW, versus} \gtrsim 5-10 \text{ for others}$$

Additional processes—forward Drell-Yan, hadron–jet, and heavy-flavour production—likewise probe the UGD at different $(x, k_T)$ regions (Celiberto et al., 1 Dec 2025, Boroun, 2023).

5. Saturation, Geometric Scaling, and Strong Coupling Effects

UGDs in saturation models, especially GBW/BGK and rcBK constructions, exhibit geometric scaling: the distributions depend on $k_T^2/Q_s^2(x)$ with $Q_s^2(x) \sim x^{-\lambda}$ (Łuszczak et al., 2022, Boroun, 2023). In strong-coupling scenarios, the saturation scale grows much faster with rapidity than in the weak-coupling regime:

$$Q_s^2(x) \sim x^{-1.06} \quad (\text{strong coupling}) \qquad Q_s^2(x) \sim x^{-0.3} \quad (\text{weak coupling})$$

(Kutak et al., 2013). This implies earlier onset of gluon saturation and denser gluon fields at moderately small $x$.

Anti-shadowing effects in the MD-BFKL equation further enhance UGDs at high rapidity and momentum, modifying scaling properties and bringing predictions into better agreement with global PDF fits (CT18NLO) (Wang et al., 2023).

6. Operator Structure, Gauge Invariance, and Universality

UGDs admit several distinct operator definitions, with their path dependence (Wilson lines) and color structure encoding universality properties and gauge invariance (Avsar, 2011, Dominguez et al., 2011, Mehtar-Tani, 2021). The two widely discussed UGDs at small-$x$ are:

• Weizsäcker–Williams (WW) distribution: number density in light-cone gauge, relevant for processes without initial-state interactions.
• Dipole distribution: Fourier transform of the color dipole; relevant for processes with shockwave/dense-target configurations.

In the large $N_c$ limit, UGDs in complex processes reduce to convolutions of these two building blocks (Dominguez et al., 2011).

Matching purely small-$x$ definitions (BFKL, BK/JIMWLK) to collinear TMDs (CSS formalism) or to higher twist requires careful regulation of rapidity divergences and gauge link paths, especially outside the strict Regge limit (Bolognino et al., 2022, Avsar, 2011, Mehtar-Tani, 2021).

7. Future Directions: Global Analysis and Lattice Computation

Achieving precision and universality for UGDs requires:

• Global fits incorporating exclusive $\rho$ production, inclusive structure functions, and forward observables across HERA, LHC, and EIC kinematics (Celiberto et al., 1 Dec 2025, Bolognino et al., 2022, Celiberto, 2019).
• Implementation of NLO BFKL kernels and collinear resummation for improved evolution (Celiberto et al., 1 Dec 2025).
• Lattice computation of gauge-invariant UGD operators with finite-length Wilson lines, enabling first-principles access to $Q_s(x)$ and nonperturbative features (Mehtar-Tani, 2021).
• Inclusion of spin and polarization effects via appropriate extensions of the TMD basis (Celiberto et al., 1 Dec 2025).

These developments will enable robust and uncertainty-quantified predictions for collider observables sensitive to transverse-momentum-dependent dynamics, closing the longstanding precision gap in the proton’s small-$x$ gluon content.

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Key references:

(Celiberto, 2019, Celiberto et al., 1 Dec 2025, Bolognino et al., 2022, Boroun, 2023, Łuszczak et al., 2022, Mehtar-Tani, 2021, Kutak et al., 2013, Wang et al., 2023, Toton, 2014, Albacete et al., 2010, Dominguez et al., 2011, Hentschinski, 2011, Avsar, 2011, Grinyuk et al., 2012, Boroun, 2023)