Gluonic Hot Spots: Nonuniform Gluon Structures

• Gluonic hot spots are localized domains of enhanced gluon density that vary in definition across high-energy proton phenomenology, hot QCD matter, and nonperturbative studies.
• They are modeled as Gaussian-distributed entities with fluctuating positions, impacting phenomena like hollowness, eccentricities, and coherent/incoherent diffraction patterns.
• Their analysis bridges effective subnucleonic scattering models, transport properties near Tc, and structured gluonic excitations in dense and confined states.

Gluonic hot spots are localized or effectively localized gluonic structures whose precise meaning depends on context. In proton-structure phenomenology they are treated as subnucleonic transverse domains of gluon density or color charge whose positions fluctuate event by event and whose correlations affect scattering, collectivity, and diffraction (Albacete et al., 2016, Albacete et al., 2018, Demirci et al., 2022, Demirci et al., 2023). In hot and dense QCD, the same phrase is used more broadly for local hot/cold spots in initial energy density, regions of enhanced quenching power near $T_c$, structured flux-tube excitations, condensate-defined gluonic domains, instanton pseudoparticles, or anisotropic gluonic condensates (Qin et al., 2011, Bluhm et al., 2012, Bicudo et al., 2018, Lee et al., 2015, Liu et al., 2024, 0705.2399). The common thread is nonuniform gluonic structure, but the underlying degrees of freedom, observables, and dynamical interpretation are not universal.

1. Terminological scope and research settings

The recent literature does not use “gluonic hot spots” as a single sharply defined object. In small-$x$ proton phenomenology, the term usually denotes localized lumps of gluon density, color charge, or color field in the transverse plane, often with Gaussian profiles and fluctuating positions (Albacete et al., 2018, Demirci et al., 2022, Demirci et al., 2023). In proton–proton scattering, these hot spots can be treated as compact, effectively black substructures that enter a Glauber-like multiple-scattering series (Albacete et al., 2016). In proton–nucleus and heavy-ion initial-state modeling, related language is applied to event-by-event local over- and under-densities whose number controls harmonic correlations and eccentricities (Qin et al., 2011, Demirci et al., 2021). By contrast, in transport studies of hot pure-glue matter, “hot spots” are an interpretation of enhanced scattering and quenching efficiency near $T_c$, not literal geometric lumps (Bluhm et al., 2012).

Setting	Meaning of hot spot	Representative consequence
$pp$, $pA$, DIS	Localized gluonic substructure in the proton	Hollowness, eccentricities, coherent/incoherent diffraction
Hot QCD matter	Regions of enhanced transport response	Peak in $\hat q/T^3$ near $T_c$
Confinement and dense matter	Localized field or condensate structure	Flux-tube modulations, instanton lumps, gluonic phases

This heterogeneity matters. Several papers explicitly caution that their results should not be read as evidence for isolated particle-like gluonic clumps. The flux-tube study finds structured deformations rather than separate lumps (Bicudo et al., 2018). The transport study invokes hot spots only as an interpretation of nonuniform response (Bluhm et al., 2012). The dense-matter gluonic-phase work does not use the phrase explicitly, but it does describe dynamically generated gluonic condensates that plausibly function as an analogue of localized enhanced gluonic structure (0705.2399).

2. Correlated proton hot spots and proton–proton scattering

A central realization of gluonic hot spots treats the proton as a three-body gluonic system. In the hollowness model, the proton contains $N_{hs}=3$ hot spots, motivated by a valence-parton picture, with gluonic content localized in transverse domains of radius $R_{hs}\ll R_p$. Each hot spot is modeled as a small black disk, and the hot-spot elastic amplitude is taken as

$$\Theta(s_{ij})=\mathrm{i}\,\exp\left(- s_{ij}^2/2R_{hs}^2\right)(1-\mathrm{i}\rho_{hs})\,.$$

The proton is therefore not a smooth continuous matter distribution, but a set of discrete effective scattering degrees of freedom (Albacete et al., 2016).

The transverse positions $x$0 are distributed through a correlated density

$x$1

with a Gaussian one-body factor $x$2 and a correlation factor containing both the center-of-mass constraint

$x$3

and short-range repulsive correlations between pairs of hot spots. The correlation strength is controlled by $x$4, while $x$5 gives the uncorrelated limit $x$6 (Albacete et al., 2016). The same correlated three-hot-spot distribution is then used in Monte Carlo Glauber modeling of $x$7 initial conditions (Albacete et al., 2016).

For fixed hot-spot positions, the $x$8 elastic amplitude is built from a Glauber-like series over all nine hot-spot pairs,

$x$9

followed by averaging over projectile and target hot-spot distributions. The inelasticity density is

$T_c$0

“Hollowness” denotes the regime in which $T_c$1 is suppressed at $T_c$2 and maximal at nonzero $T_c$3, with the paper using

$T_c$4

as the diagnostic conditions (Albacete et al., 2016).

The proposed mechanism is transverse diffusion or growth of the hot spots with increasing collision energy. As $T_c$5 increases, the geometry of the nine pairwise collisions changes and the multiple-scattering series develops stronger shadowing or destructive-interference effects. The model states that the onset of hollowness at $T_c$6 TeV, absent at lower energies, is naturally explained by this growth mechanism. A crucial result is that without nontrivial correlations, $T_c$7 in all cases, so the model cannot generate the central depletion (Albacete et al., 2016).

The wounded-hot-spot formulation extends this picture to initial-condition modeling. Each hot spot that collides at least once is wounded, and its entropy deposition is taken as a Gaussian profile,

$T_c$8

The multiplicity distribution per wounded hot spot is modeled by a double negative binomial distribution, and the total event entropy $T_c$9 is used as a centrality proxy (Albacete et al., 2016). Representative parameter values show hot-spot growth with energy: for the correlated case $pp$0 fm, $pp$1 fm at $pp$2 GeV, $pp$3 fm at $pp$4 TeV, and $pp$5 fm at $pp$6 TeV; for $pp$7, the corresponding values are $pp$8, $pp$9, and $pA$0 fm (Albacete et al., 2016).

3. Initial-state geometry, collectivity, and fluctuation observables

Once hot spots are treated as the proton’s effective subnucleonic degrees of freedom, their geometry directly controls eccentricities and multi-particle correlations. In a Monte Carlo Glauber study of $pA$1 at $pA$2 TeV, the proton is modeled as $pA$3 Gaussian hot spots with short-range repulsive spatial correlations set by a core distance $pA$4. The key observable is the normalized symmetric cumulant

$pA$5

For the standard three-hot-spot setup, spatial correlations are described as indispensable to reproduce the negative sign of $pA$6 in the highest-centrality bins. The events driving the negative sign are those with large $pA$7 and small $pA$8, and the results collapse when plotted against $pA$9, suggesting that ratio as the controlling variable in the framework (Albacete et al., 2018).

The same work emphasizes that the sign and magnitude of $\hat q/T^3$0 depend on a nontrivial interplay of scales $\hat q/T^3$1, $\hat q/T^3$2, and $\hat q/T^3$3. Increasing the number of hot spots to $\hat q/T^3$4 preserves the qualitative effect of correlations, but the uncorrelated case can become compatible with negative $\hat q/T^3$5 within statistical uncertainty in the most central bin. This is therefore not a one-parameter statement about repulsion alone; it is a coupled statement about proton substructure (Albacete et al., 2018).

Hydrodynamic follow-up shows that these geometric differences can survive into final-state flow. Using entropy profiles from the hot-spot Glauber model as input to a $\hat q/T^3$6D viscous hydrodynamic simulation with $\hat q/T^3$7 fm, $\hat q/T^3$8 MeV, vanishing initial transverse flow, vanishing initial shear-stress tensor, and $\hat q/T^3$9 with a minimum at $T_c$0 MeV, the preliminary result is that $T_c$1 is enhanced in the correlated scenario while $T_c$2 is similar within current statistical uncertainties (Albacete et al., 2018).

Analytic dilute-dense CGC calculations reach a related but more specific conclusion: geometric hot-spot fluctuations dominate the eccentricities, while color-charge fluctuations provide only a negligible correction. In that framework the proton contains $T_c$3 Gaussian hot spots of width $T_c$4 inside a Gaussian envelope of width $T_c$5, and the eccentricities are computed from the two-point function of the Glasma energy density at $T_c$6. The study identifies the size and number of hot spots as the most important parameters characterizing $T_c$7, $T_c$8, and $T_c$9 (Demirci et al., 2021). A closely related analysis of proton hot spots in $N_{hs}=3$0 collisions again concludes that geometric fluctuations of hot-spot positions are the dominant source of eccentricity, whereas color-charge fluctuations give only a small correction; the hot-spot size $N_{hs}=3$1, the IR regulator $N_{hs}=3$2, and the hot-spot geometry are the primary controls (Demirci et al., 2023).

In heavy-ion initial-state fluctuation studies, the language broadens from proton substructure to hot/cold spots in the fireball density profile. Modeling the fluctuation field as

$N_{hs}=3$3

with $N_{hs}=3$4 a sum of localized positive and negative domains, one finds that multi-harmonic correlations

$N_{hs}=3$5

decrease as the number of hot/cold spots increases. In the one-spot limit, $N_{hs}=3$6. A Monte Carlo Glauber application to Pb+Pb at $N_{hs}=3$7 TeV infers about $N_{hs}=3$8 hot spots in most central collisions and only a few in very peripheral collisions (Qin et al., 2011).

The phenomenology is not settled. The MAGMA construction, which uses roughly $N_{hs}=3$9 localized charges per Pb nucleus without explicit nucleons and assumes an $R_{hs}\ll R_p$0 form rather than a local $R_{hs}\ll R_p$1 product, can reproduce the near equality of $R_{hs}\ll R_p$2 and $R_{hs}\ll R_p$3 in ultra-central Pb+Pb at the level of initial eccentricities. However, full SONIC evolution with $R_{hs}\ll R_p$4, $R_{hs}\ll R_p$5, $R_{hs}\ll R_p$6 fm/$R_{hs}\ll R_p$7, and $R_{hs}\ll R_p$8 MeV yields centrality-dependent response coefficients with $R_{hs}\ll R_p$9, and switching to the more standard local $$\Theta(s_{ij})=\mathrm{i}\,\exp\left(- s_{ij}^2/2R_{hs}^2\right)(1-\mathrm{i}\rho_{hs})\,.$$0 interaction removes the apparent success. This is presented as a caution that hot-spot explanations of flow data require end-to-end validation with realistic evolution (Snyder et al., 2020).

4. Diffractive and DIS probes of proton hot spots

Exclusive vector meson production provides a direct fluctuation-sensitive probe of proton hot spots. In a dilute-limit CGC model, the proton is described by $$\Theta(s_{ij})=\mathrm{i}\,\exp\left(- s_{ij}^2/2R_{hs}^2\right)(1-\mathrm{i}\rho_{hs})\,.$$1 localized Gaussian hot spots with fluctuating positions and Gaussian color-charge fluctuations. The hot-spot correlator is localized around hot-spot centers $$\Theta(s_{ij})=\mathrm{i}\,\exp\left(- s_{ij}^2/2R_{hs}^2\right)(1-\mathrm{i}\rho_{hs})\,.$$2, each with profile

$$\Theta(s_{ij})=\mathrm{i}\,\exp\left(- s_{ij}^2/2R_{hs}^2\right)(1-\mathrm{i}\rho_{hs})\,.$$3

while the hot-spot centers are distributed with a Gaussian proton profile

$$\Theta(s_{ij})=\mathrm{i}\,\exp\left(- s_{ij}^2/2R_{hs}^2\right)(1-\mathrm{i}\rho_{hs})\,.$$4

A derived scale is the coherent radius

$$\Theta(s_{ij})=\mathrm{i}\,\exp\left(- s_{ij}^2/2R_{hs}^2\right)(1-\mathrm{i}\rho_{hs})\,.$$5

which sets the effective width of the averaged color-charge distribution (Demirci et al., 2022).

The exclusive $$\Theta(s_{ij})=\mathrm{i}\,\exp\left(- s_{ij}^2/2R_{hs}^2\right)(1-\mathrm{i}\rho_{hs})\,.$$6 amplitude is written in the dipole picture, and the coherent and incoherent cross sections follow from the Good–Walker decomposition,

$$\Theta(s_{ij})=\mathrm{i}\,\exp\left(- s_{ij}^2/2R_{hs}^2\right)(1-\mathrm{i}\rho_{hs})\,.$$7

In this language, coherent diffraction probes the average geometry, while incoherent diffraction probes fluctuations around that mean (Demirci et al., 2022).

The $$\Theta(s_{ij})=\mathrm{i}\,\exp\left(- s_{ij}^2/2R_{hs}^2\right)(1-\mathrm{i}\rho_{hs})\,.$$8-dependence separates fluctuation sources. In the dilute Gaussian-hot-spot model, the coherent cross section is sensitive to both the target size and the structure of the probe. The incoherent cross section is dominated by color fluctuations at small transverse momentum transfer, by proton and hot-spot sizes as well as probe structure at medium $$\Theta(s_{ij})=\mathrm{i}\,\exp\left(- s_{ij}^2/2R_{hs}^2\right)(1-\mathrm{i}\rho_{hs})\,.$$9, and again by color fluctuations at large $x$00. The model reproduces the qualitative $x$01-dependence well, but the relative normalization of coherent and incoherent cross sections is too small compared with HERA data, indicating the need for additional proton fluctuations such as saturation-scale fluctuations, more dynamical gluon-field fluctuations, non-dilute effects, or event-by-event fluctuations in hot-spot number (Demirci et al., 2022).

A later synthesis of proton hot spots and $x$02 eccentricities restates the same hierarchy: geometric fluctuations dominate initial eccentricities, but in exclusive vector meson production the fluctuation source depends on $x$03. Small $x$04 is dominated by color-charge fluctuations, intermediate $x$05 probes proton and hot-spot sizes and the probe structure, and large $x$06 is again dominated by color-charge fluctuations. The paper explicitly warns that for realistic charm the coherent slope is not simply a proton-size observable, because probe structure substantially modifies the interpretation (Demirci et al., 2023).

A refined hot-spot description of exclusive $x$07 production adds three elements missing in earlier dilute-limit studies: event-by-event fluctuations in the number of hot spots $x$08, event-by-event fluctuations in each hot spot’s local saturation scale $x$09, and the first relativistic corrections to the $x$10 light-front wave function. The target average runs over a zero-truncated Poisson distribution for $x$11, Gaussian-distributed hot-spot positions with a center-of-mass constraint, and log-normal fluctuations in $x$12. The Bayesian analysis with HERA data finds that saturation-scale fluctuations are more important than fluctuations in the number of hot spots, while the first relativistic correction is essential for describing the observed power-law behavior in the incoherent cross section at large $x$13 (Mäntysaari et al., 9 Sep 2025).

5. Hot gluonic matter and structured gluonic excitations

In hot pure-glue matter, the hot-spot concept appears in a transport rather than a geometric form. A quasiparticle model study relates the jet quenching parameter $x$14 to the specific shear viscosity $x$15 and finds a pronounced maximum of $x$16 near the deconfinement transition temperature $x$17. For a thermal gas of free massless bosons, the relation reduces to

$x$18

which motivates the ratio

$x$19

At high temperature the model gives $x$20, while near $x$21 it rises toward values comparable to the strong-coupling benchmark $x$22. The paper interprets this as evidence that the medium near $x$23 contains particularly effective scattering centers; in the language of the query, these are “gluonic hot spots” in the sense of enhanced quenching and momentum-broadening power, not resolved geometric substructures (Bluhm et al., 2012).

A different nonperturbative setting is the excited QCD flux tube. Lattice SU(3) calculations on a $x$24 lattice at $x$25, using $x$26 gauge configurations and a $x$27-operator basis, extract the ground state $x$28 and excitations such as $x$29 and $x$30, with $x$31 identified as the lowest excitation. Field densities are measured from plaquette–Wilson-loop correlators and mapped into chromoelectric and chromomagnetic profiles. The ground state is a narrow confining flux tube, whereas the excited states display more complex spatial patterns in both the mediator plane and along the charge axis (Bicudo et al., 2018).

The flux-tube study is careful about interpretation. It does not claim sharply localized isolated hot spots. Instead, the excited profiles are described as structured deformations of the flux tube with excitation-dependent field concentrations and nontrivial spatial modulation. This suggests a restricted but technically meaningful use of the term: not separate clumps, but localized enhancements embedded in an extended confining string (Bicudo et al., 2018).

6. Vacuum geometry, instanton localization, and dense-matter gluonic phases

Several works push the hot-spot idea beyond scattering phenomenology toward nonperturbative vacuum structure. One proposal uses the dimension-2 condensate $x$32 to define a generalized, coordinate-dependent $x$33-vacuum made of curved gauge-fixed slices around hadrons and other strongly interacting objects. On such a slice, ordinary derivatives are replaced by covariant derivatives, and the quadratic gluon Lagrangian acquires a curvature term

$x$34

For a maximally symmetric curved slice with $x$35, the propagator takes the massive-vector form

$x$36

In this framework, localized nonperturbative gluonic structure is encoded by condensate-defined curved domains rather than by discrete scattering centers (Lee et al., 2015).

The QCD instanton vacuum gives a more sharply localized realization. At medium resolution

$x$37

the glue is said to be mostly localized in single instantons or anti-instantons and in two-pseudoparticle clusters. The characteristic parameters are

$x$38

The scalar and pseudoscalar glue operators track, respectively, the total number of pseudoparticles and the instanton–anti-instanton imbalance. A major result is that the traceless gluonic energy-momentum tensor has no leading single-instanton contribution and starts only at next-to-leading order, dominated by instanton–anti-instanton molecules. Within Ji’s mass sum rule, the low-scale decomposition reported is

$x$39

with evolved momentum fractions at $x$40 GeV of about $x$41 and $x$42. Here the hot-spot picture is semiclassical and collective: glue concentrated in localized pseudoparticles rather than constituent gluons (Liu et al., 2024).

Dense neutral two-flavor quark matter provides yet another analogue. In a gauged NJL analysis, two gluonic phases—the gluonic cylindrical phase II and the gluonic color-spin locked phase—are found as dynamically realized and energetically favorable minima over a broad coupling range. The relevant condensates involve background gluon fields such as

$x$43

together with temporal gauge backgrounds associated with color neutrality. For $x$44 MeV and $x$45 MeV, the preferred phases are the normal phase for $x$46 MeV, single-plane-wave LOFF for $x$47 MeV, gluonic cylindrical phase II or GCSL for $x$48 MeV, and 2SC for stronger coupling. These phases are anisotropic and involve nonzero chromoelectric or chromomagnetic field strengths. Although not thermal “hot spots,” they are structured gluonic condensate domains that replace an unstable homogeneous state (0705.2399).

Taken together, these lines of work support a broad but technically differentiated conclusion: gluonic hot spots are best understood as context-dependent realizations of nonuniform gluonic structure. In high-energy proton phenomenology they are explicit effective degrees of freedom with fluctuating geometry; in hot-matter transport they are an interpretation of enhanced interaction strength; and in nonperturbative vacuum, confinement, and dense-matter studies they emerge as localized field concentrations or condensate-supported gluonic domains.