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DOE OSTI2025

Unraveling nonperturbative QCD with transverse momentum hadronic structures

The main purpose of this thesis is to investigate the theoretical foundations of the factorization theorems that involve transverse quark and gluon momentum distributions, and to better understand the interface between foundational questions and phenomenological applications. Many of these applications involve using transverse parton momentum dependent (TMD) correlation functions as tools for probing the complex nonperturbative structures of the hadrons. We first introduce the concepts of collinear and TMD factorization in a toy-model theory and, by leveraging the comparative simplicity of such a model, we test the range of validity and limitations of the factorization approach. At the same time, we introduce the conventional approaches that are adopted for the phenomenological extractions of the TMD distributions, and catalog the advantages and disadvantages. Following the general theoretical framework of factorization, we then develop a different approach to TMD phenomenology that is able to systematically integrate the perturbative and non perturbative information in parametrizations of TMD distributions, while being completely consistent with their operator definitions and the predictions of QCD, including their nontrivial evolution. The name Hadron Structure Oriented approach, or HSO, is coined in order to emphasize the central role of the nonperturbative nature of QCD and its relation to the partonic structure of hadrons. Finally, we provide a first phenomenological implementation of the HSO approach by extracting TMD distributions relying on low-to-moderate energy Drell-Yan data. The quality and robustness of the results are then compared against higher energy processes like high-Q2 Z0 boson production. Our analysis demonstrates the feasibility of this novel approach and points the way toward the improvements it can achieve in future applications.

Rainaldi, Tommaso [Old Dominion Univ., Norfolk, VA↗
DOE OSTI2026

Measurements of Angular Distributions of Drell-Yan Dimuons in $p+p$ and $p+d$ Interactions at 120 GeV/$c$

We present experimental results on the angular distributions of Drell-Yan muons produced by a 120 GeV/$c$ proton beam interacting with liquid hydrogen and deuterium targets. The dimuon angular distributions in both polar ($θ$) and azimuthal ($ϕ$) angles in the Collins-Soper frame are measured within the kinematic range of $4.5 < m_{μμ} < 10\ \mathrm{GeV}/c^2$, $0.19 < p_T < 2.24\ \mathrm{GeV}/c$, and $0 < x_F < 0.95$. Unlike the results of a previous proton-induced Drell-Yan experiment at a higher energy, the data reveal a pronounced $\cos 2ϕ$ modulation in the angular distributions. Comparison with perturbative QCD (pQCD) predictions shows statistically significant deviations, with p-values of 3.5% for the $p+p$ and 1.5% for the $p+d$ Drell-Yan processes. These results suggest the presence of nonperturbative QCD contributions.

Nagai, K. [Christian Brothers U.; Los Alamos; Taiw↗
DOE OSTI2025

𝐵 → 𝜌⁢ℓ⁢$\bar{v}$ Resonance Form Factors from 𝐵→ 𝜋⁢𝜋⁢ℓ⁢$\bar{v}$ in Lattice QCD

The decay 𝐵 → 𝜌⁢ℓ⁢$\bar{v}$ is an attractive process for determining the magnitude of the smallest Cabibbo-Kobayashi-Maskawa matrix element, |𝑉 𝑢⁢𝑏 |, and can provide new insights into the origin of the long-standing exclusive-inclusive discrepancy in determinations of this standard-model parameter. This requires a nonperturbative QCD calculation of the 𝐵 → 𝜌 form factors 𝑉, 𝐴 0 , 𝐴 1 , and 𝐴 12 . The unstable nature of the 𝜌 resonance has prevented precise lattice QCD calculations of these form factors to date. Here, we present the first lattice QCD calculation of the 𝐵 → 𝜌 form factors in which the 𝜌 is treated properly as a resonance in 𝑃-wave 𝜋⁢𝜋 scattering. To this end, we use the Lellouch-Lüscher finite-volume formalism to compute the 𝐵 → 𝜋⁢𝜋 form factors as a function of both momentum transfer and 𝜋⁢𝜋 invariant mass, and then analytically continue to the 𝜌 resonance pole. This calculation is performed with 2 + 1 dynamical quark flavors at a pion mass of approximately 320 MeV, and demonstrates a clear path toward results at the physical point.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2025

Entanglement, trace anomaly, and confinement in QCD

We formulate confinement in quantum chromodynamics (QCD) as an entropic surface phenomenon. Quark and gluon quantum information is localized on a transverse, entangling two-sphere of radius 𝑅 𝐸⁢𝐸 ; at this radius the QCD vacuum—partitioned by a hadron into interior and exterior regions—reaches its maximal entanglement entropy. Lattice-QCD determinations of the scalar (trace) gravitational form factors fix both 𝑅 𝐸⁢𝐸 and the transverse trace-anomaly density 𝜌 ℎ ⁡(𝑅 𝐸⁢𝐸 ), yielding a parameter-free slope 𝑐 ℎ = 8⁢𝜋 2 ⁢𝑅𝑆$^2_{𝐸⁢𝐸}$𝜌 ℎ ⁡(𝑅 𝐸⁢𝐸 ) and a mechanical entropy 𝑆 𝐸⁢𝐸 ⁡(𝑦) = 𝑐 ℎ ⁢𝑦 that grows linearly with rapidity 𝑦. The entropy gradient ∂ 𝑅 𝑆 𝐸⁢𝐸 changes sign at 𝑅 𝐸⁢𝐸 : it pushes colored degrees of freedom outward for 𝑟 <𝑅 𝐸⁢𝐸 and pulls them inward for 𝑟 >𝑅 𝐸⁢𝐸 , thereby localizing them on the codimension-2 entangling two-sphere Σ ⊥ = 𝑆$^2_{𝑅_{𝐸⁢𝐸}}$ (which, in the infinite-momentum frame (IMF), projects onto the transverse plane)—the “information wall.” This provides a high-energy (large-𝑦) entropic confinement diagnostic that complements—rather than replaces—Wilson’s area-law criterion, which probes long-distance dynamics near the rest frame (𝑦 → 0). Imposing unitarity on an entropic ansatz for the amplitude yields 𝜎⁡(𝑠) ∝ 𝑦 𝛿 . World data favor 𝛿 = 2 for elastic 𝑝⁡𝑝⁡($𝑝\bar{⁡𝑝}$) scattering and heavy-quark photoproduction, whereas 𝜙 photoproduction favors a softer 𝛿 = 0.387. All extracted cross sections remain well below the Froissart-Martin bound. These results provide a confinement criterion quantified directly from nonperturbative QCD inputs, unifying the trace anomaly, entanglement entropy, and high-energy scattering within a single quantitative framework.

Color confinement↗
DOE OSTI2024

A potential model study of the nucleon's charge and mass radius

We study the charge and mass distributions within a nucleon and compute the associated squared radii based on a potential model approach. Different constituent quark configurations such as Δ, Y, and quark-diquark are considered and compared, with model parameters calibrated by experimental measurements of the proton and neutron charge radius. The results suggest that while the charge radius is dictated by quark dynamics, the mass radius is strongly influenced by nonperturbative QCD contributions to a nucleon's mass that are not sensitive to the constituent quarks. As a result, the mass radius could become substantially different from the charge radius. The obtained nucleon mass distributions of different configurations are further used for simulations of the initial conditions in heavy ion collisions. Furthermore, the computed eccentricities ε 2 and ε 3 are found to demonstrate a considerable sensitivity to the input nucleon profiles, especially to the mass radius in the peripheral region as well as for systems with fewer participants.

Gallimore, Daniel [Indiana University, Bloomington↗
DOE OSTI2025

𝐷 0 -meson-tagged jet axes difference in proton-proton collisions at $\sqrt{s}$ = 5.02 TeV

Heavy-flavor quarks produced in proton-proton (pp) collisions provide a unique opportunity to investigate the evolution of quark-initiated parton showers from initial hard scatterings to final-state hadrons. By examining jets that contain heavy-flavor hadrons, this study explores the effects of both perturbative and nonperturbative QCD on jet formation and structure. The angular differences between various jet axes, Δ⁢𝑅 axis , offer insight into the radiation patterns and fragmentation of charm quarks. The first measurement of 𝐷 0 -tagged jet axes differences in pp collisions at $\sqrt{s}$ =5.02 TeV by the ALICE experiment at the LHC is presented for jets with transverse momentum 𝑝$^{ch jet}_{T}$ ≥ 10 GeV/𝑐 and 𝐷 0 mesons with 𝑝$^{D^0}_{T}$ ≥ 5 GeV/𝑐. In this 𝐷 0 -meson-tagged jet measurement, three jet axis definitions, each with different sensitivities to soft, wide-angle radiation, are used: the standard axis, soft drop groomed axis, and winner-takes-all axis. Measurements of the radial distributions of 𝐷 0 mesons with respect to the jet axes, Δ⁢𝑅 axis−D 0 , are reported, along with the angle, Δ⁢𝑅 axis , between the three jet axes. The 𝐷 0 meson emerges as the leading particle in these jets, closely aligning with the winner-takes-all axis and diverging from the standard jet axis. The results also examine how varying the sensitivity to soft radiation with grooming influences the orientation of the soft drop jet axis and uncover that charm-jet structure is more likely to survive grooming when the soft drop axis is further from the 𝐷 0 direction, providing further evidence of the dead-cone effect recently measured by ALICE.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2025

Entanglement as a Probe of Hadronization

Recently, it was discovered that the proton structure at high energies exhibits maximal entanglement. This leads to a simple relation between the proton’s parton distributions and the entropy of hadrons produced in high-energy inelastic interactions, which has been experimentally confirmed. In this Letter, we extend this approach to the production of jets. Here, the maximal entanglement predicts a relation between the jet fragmentation function and the entropy of hadrons produced in jet fragmentation. We test this relation using the ATLAS Collaboration data on jet production at the Large Hadron Collider, and find a good agreement between the prediction based on maximal entanglement within the jet and the data. This study represents the first use of a quantum entanglement framework in an experimental study of the hadronization process, offering a new perspective on the transition from perturbative to nonperturbative QCD. Our results open the door to a more comprehensive understanding of the quantum nature of hadronization.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2025

Asymptotic gauge symmetry and UV extension of the nonperturbative coupling in holographic QCD

We extend our recent analytic study of the strong coupling 𝛼 eff in the nonperturbative and near-perturbative regimes [Phys. Rev. Lett. 133, 181901 (2024)] by imposing rigorous renormalization-group constraints from asymptotically free gauge theories at 𝑄 2 → ∞. The asymptotic boundary conditions modify the scaling properties of 𝛼eff at large values of the momentum transfer 𝑄 2 and lead to a scale-dependent confinement strength 𝜅⁡(𝑄 2 ). This requires that both 𝜅⁡(𝑄 2 ) and 𝛼 eff⁡ (𝑄 2 , 𝜅⁡(𝑄 2 )) remain holomorphic in the complex 𝑄 2 plane, except at the physical cuts associated with the heavy-quark thresholds and the singularity flow trajectory studied in our previous Letter. For color SU(3), a precise connection is found between the scaling exponent of 𝜅⁡(𝑄 2 ) in the ultraviolet, the value of the infrared fixed point of the strong coupling, and the number of flavors in agreement with observations. The nonperturbative analytic model gives an accurate description of the strong coupling across all scales, up to the highest available data.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2024

Effects of final state interactions on Landau singularities

In certain kinematic and particle mass configurations, triangle singularities may lead to line-shapes which mimic the effects of resonances. This well-known effect is scrutinized here in the presence of final-state rescattering. The goal is achieved first by utilizing general arguments provided by Landau equations, and second by applying a modern scattering formalism with explicit two- and three-body unitarity.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2026

Measuring spin correlation between quarks during QCD confinement

The vacuum is now understood to have a rich and complex structure, characterized by fluctuating energy fields and a condensate of virtual quark–antiquark pairs. The spontaneous breaking of the approximate chiral symmetry, signalled by the nonvanishing quark condensate $\langle$$q\bar{q}$$\rangle$, is dynamically generated through topologically nontrivial gauge configurations such as instantons. The precise mechanism linking the chiral symmetry breaking to the mass generation associated with quark confinement remains a profound open question in quantum chromodynamics (QCD)—the fundamental theory of strong interaction. High-energy proton–proton collisions could liberate virtual quark–antiquark pairs from the vacuum that subsequently undergo confinement to form hadrons, whose properties could serve as probes into QCD confinement and the quark condensate. Here we report evidence of spin correlations in $Λ\bar{Λ}$ hyperon pairs inherited from spin-correlated strange quark–antiquark virtual pairs. Measurements by the STAR experiment at the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory reveal a relative polarization signal of (18 ± 4)% that links the virtual spin-correlated quark pairs from the QCD vacuum to their final-state hadron counterparts. Crucially, this correlation vanishes when the hyperon pairs are widely separated in angle, consistent with the decoherence of the quantum system. Our findings provide a new experimental model for exploring the dynamics and interplay of quark confinement and entanglement.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2024

QCD Running Coupling in the Nonperturbative and Near-Perturbative Regimes

We use analytic continuation to extend the gauge-gravity duality nonperturbative description of the strong force coupling into the transition, near-perturbative, regime where perturbative effects become important. By excluding the unphysical region in coupling space from the flow of singularities in the complex plane, we derive a specific relation between the scales relevant at large and short distances; this relation is uniquely fixed by requiring maximal analyticity. The unified effective coupling model gives an accurate description of the data in the nonperturbative and the near-perturbative regions.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

QCD Constraints on Isospin-Dense Matter and the Nuclear Equation of State

Understanding the behavior of dense hadronic matter is a central goal in nuclear physics as it governs the nature and dynamics of astrophysical objects such as supernovae and neutron stars. Because of the nonperturbative nature of quantum chromodynamics (QCD), little is known rigorously about hadronic matter in these extreme conditions. Here, lattice QCD calculations are used to compute thermodynamic quantities and the equation of state of QCD over a wide range of isospin chemical potentials with controlled systematic uncertainties. Agreement is seen with chiral perturbation theory when the chemical potential is small. Comparison to perturbative QCD at large chemical potential allows for an estimate of the gap in the superconducting phase, and this quantity is seen to agree with perturbative determinations. Since the partition function for an isospin chemical potential μ I bounds the partition function for a baryon chemical potential μ B = 3 μ I / 2 , these calculations also provide rigorous nonperturbative QCD bounds on the symmetric nuclear matter equation of state over a wide range of baryon densities for the first time. Published by the American Physical Society 2025

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2024

QCD evolution of entanglement entropy

Entanglement entropy has emerged as a novel tool for probing nonperturbative quantum chromodynamics (QCD) phenomena, such as color confinement in protons. While recent studies have demonstrated its significant capability in describing hadron production in deep inelastic scatterings, the QCD evolution of entanglement entropy remains unexplored. Here, in this work, we investigate the differential rapidity-dependent entanglement entropy within the proton and its connection to final-state hadrons, aiming to elucidate its QCD evolution. Our analysis reveals a strong agreement between the rapidity dependence of von Neumann entropy, obtained from QCD evolution equations, and the corresponding experimental data on hadron entropy. These findings provide compelling evidence for the emergence of a maximally entangled state, offering new insights into the nonperturbative structure of protons.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2025

Nonperturbative and perturbative dynamics of a light QCD axion: Dark matter and the strong 𝐶⁢𝑃 problem

Considerable theoretical efforts have gone into expanding the reach of the quantum chromodynamics (QCD) axion beyond its canonical mass–decay-constant relation. The 𝑍 𝒩 QCD axion model reduces the QCD axion mass naturally, by invoking a discrete 𝑍 𝒩 symmetry through which the axion field is coupled to 𝒩 copies of the Standard Model. Before the QCD phase transition at temperature 𝑇 QCD , the 𝑍 𝒩 potential has a minimum at misalignment angle 𝜃 = 𝜋. At 𝑇 QCD , 𝜃 = 𝜋 becomes a maximum; the axion potential becomes exponentially suppressed and develops 𝒩 minima—only one of which actually solves the strong 𝐶⁢𝑃 problem. Before 𝑇 QCD , 𝜃 relaxes toward 𝜋. After 𝑇 QCD , the axion field starts from around the hilltop and may have sufficient kinetic energy to overcome the newly suppressed potential barriers. Such a field evolution leads to nonperturbative effects via the self-interactions near the hilltop, which can cause the exponential growth of fluctuations and backreaction on the coherent motion. This behavior can influence the relic density of the field and the minimum in which it settles. We conduct the first lattice simulations of the 𝑍 𝒩 QCD axion using 𝒞osmoℒattice to accurately calculate dark matter abundances and find nonperturbative dynamics reduce the abundance by up to a factor of two. We furthermore find that the probability of solving the strong 𝐶⁢𝑃 problem tends to diverge considerably from the naïve expectation of 1/𝒩.

Axions↗
DOE OSTI2024

Transverse Momentum Distributions from Lattice QCD without Wilson Lines

The transverse-momentum-dependent distributions (TMDs), which are defined by gauge-invariant 3D parton correlators with staple-shaped lightlike Wilson lines, can be calculated from quark and gluon correlators fixed in the Coulomb gauge on a Euclidean lattice. These quantities can be expressed gauge invariantly as the correlators of Coulomb-gauge-dressed fields, which reduce to the standard TMD correlators under principal-value prescription in the infinite boost limit. In the framework of large-momentum effective theory, a quasi-TMD defined from such correlators in a large-momentum hadron state can be matched to the TMD via a factorization formula, whose exact form is derived using soft collinear effective theory and verified at one-loop order. Compared to the currently used gauge-invariant correlators, this new method can substantially improve statistical precision and simplify renormalization for the time-reversal-even TMDs, which will greatly enhance the predicative power of lattice QCD in the nonperturbative region.

Effective field theory↗
DOE OSTI2025

Role of the chiral anomaly in polarized deeply inelastic scattering. III. Wess-Zumino-Witten contributions and chiral Ward identities for finite quark mass

We extend our prior results on the worldline computation of the axial vector-vector-vector (AVV) triangle anomaly in polarized deeply inelastic scattering (DIS) to the finite mass case by computing in addition the pseudoscalar-vector-vector (PVV) triangle graph. For the well-studied QED case, we show explicitly how the off-forward AVV pole exactly cancels an identical PVV pole. We then demonstrate the dramatic difference in QCD due to the chiral condensate, which qualitatively modifies anomalous Ward identities. As in the massless case, the anomaly pole in QCD is canceled by the dynamics of a primordial isosinglet pseudoscalar η ¯ -meson, whose Wess-Zumino-Witten coupling to the topological charge density shifts the pole to the physical η ′ mass, with the finite quark mass contribution differing by O ( 10 % ) from the Witten-Veneziano formula. We obtain a compact analytic expression for the finite mass corrections to Shore and Veneziano’s result that the proton’s net quark helicity Δ Σ ∝ χ QCD ′ | m = 0 ( 0 ) , the forward slope of the topological susceptibility in the chiral limit, and show they are of the order of a few percentages. Our prior prediction that the polarized DIS structure function g 1 is quenched by sphaleronlike topological transitions at small x is unaffected by quark mass effects. Our results illustrate how worldline computations of anomalous processes, in synergy with lattice computations and nonet chiral perturbation theory, can uncover novel nonperturbative features of QCD at the Electron-Ion Collider. Published by the American Physical Society 2025

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2026

Accessing the gluon momentum fraction of nucleons through the gradient flow

We calculate the gluon momentum fraction of the nucleon using lattice QCD, with a nonperturbative renormalization technique based on the gradient flow. The gluon momentum fraction is determined on a single Wilson-clover ensemble using 𝑁 𝑓 =2 +1 flavors with pion mass 358 MeV and lattice spacing 0.094 fm. We employ the variational method to reduce excited-state contamination and apply the distillation framework to ensure a large operator basis. To reduce systematic uncertainties, we apply Bayesian model averaging to all fit procedures. We apply matching coefficients to the flow-time dependent lattice results to recover the gluon momentum fraction in the $\overline{MS}$-scheme at 2 GeV. Our final result is ⟨𝑥⟩ 𝑔 ⁢(𝜇 =2 GeV) =0.482⁢(35), where we quote only statistical uncertainties.

Lattice QCD↗
DOE OSTI2025

Nonperturbative heavy-flavor transport approach for hot QCD matter

The heavy charm and bottom quarks are unique probes of the transport properties of the quark-gluon plasma (QGP) and its hadronization in high-energy nuclear collisions. A key challenge in this context is to embed the interactions of the heavy quarks in the expanding medium compatible with the strong-coupling nature of the QGP, and thus to unravel the underlying microscopic mechanisms. In the present work we progress toward this goal by combining recent T -matrix interactions for elastic scattering with an effective transport implementation of gluon radiation, and apply these in a Langevin framework in a viscous hydrodynamic evolution. Hadronization of heavy quarks is evaluated using a modern recombination model with 4-momentum conservation, supplemented with fragmentation constrained by data in proton-proton collisions. Deploying this approach to charm-hadron observables in Pb-Pb collisions at the LHC yields fair agreement with experiment while also identifying areas of further systematic improvement of the simulations and its current input.

Heavy-flavor transport↗