From QCD Vacuum to Hadron Structure: A New Bridge

Author: Denis Avetisyan

A novel framework connects the nonperturbative dynamics of the QCD vacuum with the observed properties of hadrons, offering a path towards a unified understanding of their internal structure.

This review explores how instanton-based models, light-front quantization, and effective field theories can link the QCD vacuum to parton distribution functions and hadronic observables.

A longstanding challenge in quantum chromodynamics is reconciling the static picture of hadrons-defined by confinement and chiral symmetry breaking-with the dynamic description of their partonic constituents at high energies. This review, ‘The Hadron-Parton Bridge, From the QCD Vacuum to Partons’, proposes a unified framework leveraging the Instanton Liquid Model to connect nonperturbative vacuum dynamics with observable parton distributions via light-front quantization. By constructing hadronic wavefunctions and deriving effective interactions, we establish a pathway to compute partonic observables-including PDFs, GPDs, and gravitational form factors-from first principles. Can this approach provide a comprehensive, multiscale understanding of hadron structure and ultimately illuminate the emergent properties of matter described by QCD?

The Strong Force: A Universe Forged in Interaction

Quantum Chromodynamics, or QCD, stands as the cornerstone of understanding the strong force – one of the four fundamental interactions shaping the universe. This theory describes the behavior of quarks, the fundamental constituents of matter, and gluons, the force carriers mediating the strong interaction between them. Unlike electromagnetism, where charge determines interaction strength, QCD’s strength increases with distance, a phenomenon leading to quark confinement – the inability to observe isolated quarks. This unique characteristic arises from the self-interacting nature of gluons, which themselves carry color charge, resulting in the formation of complex composite particles called hadrons, such as protons and neutrons. Consequently, QCD isn’t merely a description of nuclear physics; it provides a framework for comprehending the very structure of visible matter and the conditions present in the early universe, immediately after the Big Bang, where extreme temperatures and densities allowed quarks and gluons to exist in a deconfined state known as the quark-gluon plasma.

Quantum Chromodynamics (QCD), while remarkably successful in describing the strong force, encounters substantial hurdles when applied to understanding the properties of hadrons – composite particles like protons and neutrons. This difficulty arises from the “nonperturbative” nature of the strong force at the energy scales relevant to hadron structure; unlike electromagnetism, the strong force doesn’t weaken with distance, making traditional approximation techniques ineffective. Consequently, calculating hadron masses and other characteristics directly from QCD’s fundamental equations proves extraordinarily complex, requiring sophisticated computational methods like lattice QCD, which discretizes space-time to make calculations tractable. These methods, while powerful, are computationally intensive and still face limitations in fully capturing the intricacies of hadron behavior, particularly concerning phenomena like confinement – the reason quarks are never observed in isolation – and the origin of hadron masses.

The predictive power of quantum chromodynamics (QCD) diminishes significantly at low energies, a regime where the force between quarks becomes overwhelmingly strong. Conventional perturbative techniques, which rely on approximating interactions as small deviations from free particles, simply fail in these circumstances. This breakdown arises because the strong force’s coupling constant increases dramatically as the energy scale decreases, rendering the perturbative expansion non-convergent. Consequently, physicists have turned to innovative, non-perturbative approaches to probe the QCD vacuum – the seemingly empty space teeming with virtual particles and complex interactions. These methods, including lattice QCD simulations – which discretize spacetime to allow for numerical calculations – and effective field theories, attempt to map the intricate landscape of quark and gluon interactions, ultimately seeking to understand phenomena like hadron masses and the confinement of quarks within particles.

The enduring mysteries of quark confinement and chiral symmetry breaking represent pivotal challenges at the forefront of modern physics. While Quantum Chromodynamics (QCD) successfully describes the strong force, explaining why quarks are never observed in isolation – a phenomenon known as confinement – and understanding the origin of hadron masses through chiral symmetry breaking remain elusive. These aren’t merely academic puzzles; they are fundamental to comprehending the very structure of visible matter. Current research, employing techniques like lattice QCD simulations and effective field theories, strives to map the complex interactions within the QCD vacuum and elucidate the mechanisms responsible for these phenomena, potentially revealing new states of matter and a deeper understanding of the universe’s building blocks. The pursuit of resolving these issues continues to drive innovation in both theoretical and experimental high-energy physics.

The Vacuum as a Fluid: Mapping the Instanton Landscape

The Instanton Liquid Model characterizes the Quantum Chromodynamics (QCD) vacuum not as empty space, but as a highly populated medium of instantons – self-contained, topologically non-trivial solutions to the QCD equations. These instantons, representing tunneling events between different vacuum states, are characterized by a finite size and density. The model proposes these solutions exist with an approximate size of 0.3 femtometers (fm) and a density of approximately one instanton per 3 $fm^3$ . This dense ensemble fundamentally alters the vacuum’s properties, providing a framework for understanding phenomena like chiral symmetry breaking and the generation of dynamical quark mass, which are not predicted by perturbative QCD.

The Instanton Liquid Model generates the quark condensate, $\langle \bar{q}q \rangle$ , through the collective effect of instanton interactions. Chiral symmetry breaking, signaled by a non-zero quark condensate, arises because instantons induce a dynamically generated mass for quarks, even in the limit of massless up and down quarks. The model demonstrates that the density of instantons, approximately 1 / 3 fm^-3, directly contributes to the magnitude of the condensate, providing a quantitative link between the topological structure of the QCD vacuum and the observed spontaneous breaking of chiral symmetry. Calculations within this framework reproduce condensate values consistent with experimental observations and lattice QCD simulations, supporting the model’s validity as a mechanism for chiral symmetry breaking.

The Wilsonian Spirit, a core tenet of effective field theory, asserts that the relevant degrees of freedom at low energies are determined by the physics at short distances. This principle provides strong justification for the Instanton Liquid model, which describes the QCD vacuum through topological solutions – instantons – existing at a characteristic scale of approximately 0.3 fm. By focusing on these short-distance, fundamental configurations, the model successfully explains long-distance phenomena such as chiral symmetry breaking and hadron properties without requiring explicit input from the confinement mechanism itself. This bottom-up approach, rooted in short-distance physics, validates the model’s ability to predict and explain the behavior of the QCD vacuum at larger scales, circumventing the need for direct calculations at energies where non-perturbative effects dominate.

The Instanton Liquid model facilitates the calculation of hadron properties through effective interactions derived from the ensemble of instantons characterizing the QCD vacuum. This approach defines a specific vacuum structure with an average instanton size of 0.3 femtometers (fm) and a density of approximately one instanton per 3 fm³. These parameters, determined by the model, enable the computation of quantities such as hadron masses and decay constants, providing a framework to connect theoretical predictions with experimental observations in quantum chromodynamics. The density and size of these instantons are crucial inputs for calculating the effective interactions governing quark and gluon dynamics within the vacuum.

Probing the Hadron’s Core: Light-Front Dynamics and Parton Structure

Light-Front Quantization (LFQ) is a relativistic quantum mechanical approach that focuses on the dynamics of particles moving at the speed of light, specifically suited for describing high-energy scattering processes. Unlike Instant Form Dynamics which uses equal-time quantization, LFQ employs a light-cone time coordinate, $\theta = t - z$ , resulting in a simplified treatment of Lorentz transformations and facilitating the direct calculation of physically measurable partonic observables. This framework inherently emphasizes the longitudinal momentum fractions of partons within hadrons, making it particularly well-suited for analyzing deep inelastic scattering and other high-energy collisions where the internal structure of hadrons is probed. The use of light-front coordinates also avoids the complexities associated with the negative-norm states that can arise in other relativistic formalisms, providing a more intuitive and computationally tractable approach to understanding hadron structure and interactions.

Gradient Flow Renormalization (GFR) addresses ultraviolet (UV) divergences inherent in Light-Front Quantization (LFQ) calculations by introducing a flow equation that smoothly modifies the fundamental fields. This process effectively regulates high-momentum modes, allowing for the controlled subtraction of divergent contributions. The technique involves evolving fields according to a diffusion-like equation, parameterized by a flow time ‘t’, which acts as an infrared regulator. By performing calculations at finite flow time and then taking the limit as $t \rightarrow 0$ , one obtains finite, physically meaningful results without the need for traditional renormalization schemes involving counterterms. GFR has been shown to preserve the key advantages of LFQ, such as its manifest Lorentz invariance, while providing a robust method for handling divergences in strong interaction calculations.

The evolution of parton distribution functions (PDFs) – which describe the probability of finding a parton within a hadron – is governed by the DGLAP (Dokshitzer-Gribov-Lipatov-Altarelli-Parisi) and ERBL (Eichten-Reya-Larkoski) evolution equations. DGLAP equations, specifically, describe the evolution of PDFs as a function of the resolution scale $Q^2$ at fixed Bjorken-x, accounting for the emission and absorption of gluons. These equations are perturbative and rely on the assumption that $\alpha_s$ (the strong coupling constant) is small. The ERBL equations extend this framework to describe the evolution of generalized parton distributions (GPDs) and are essential for understanding hadron structure at different energy scales. Solving these equations allows for the prediction of PDFs at different momentum fractions and resolution scales, crucial for high-energy scattering processes and collider phenomenology.

Partonic observables, experimentally accessible quantities derived from high-energy collisions, provide a means to probe the internal structure of hadrons in terms of their constituent partons – quarks and gluons. These observables include quantities like deep inelastic scattering cross-sections, jet production rates, and transverse momentum distributions of produced particles. Analysis of these data allows for the determination of Parton Distribution Functions (PDFs), which describe the probability of finding a parton with a specific momentum fraction inside a hadron. Furthermore, measurements sensitive to correlations between partons, such as dihadron correlations and heavy-quark fragmentation functions, reveal information about the dynamics of strong force interactions and the formation of hadronic states. By comparing theoretical predictions, based on models of hadron structure and strong interactions, with experimental data for these observables, we can refine our understanding of the complex internal dynamics of hadrons.

Beyond the Familiar: Exotic Hadrons and the Expanding Landscape of the Strong Force

Hadron spectroscopy functions as a crucial endeavor to chart the energy landscape and inherent characteristics of hadrons – composite particles made of quarks and gluons. By meticulously measuring the masses and decay patterns of these particles, physicists can effectively probe the strong force that binds them. These spectroscopic investigations aren’t merely about cataloging particles; they’re about reverse-engineering the underlying structure. The observed energy levels act as fingerprints, revealing whether a hadron conforms to simple quark models – like a meson composed of a quark-antiquark pair, or a baryon consisting of three quarks – or if it exhibits more complex configurations hinting at the influence of gluonic degrees of freedom and potentially entirely new forms of quark clustering. This detailed mapping provides stringent tests of Quantum Chromodynamics $QCD$ , the fundamental theory describing the strong interaction, and offers invaluable insights into the nature of matter at its most basic level.

Quarkonia, bound states consisting of a heavy quark and its antiquark – such as bottomonium and charmonium – represent uniquely valuable systems for rigorously testing the predictions of Quantum Chromodynamics (QCD). Due to the substantial mass of the heavy quarks, the dynamics within quarkonia are non-relativistic, allowing for perturbative calculations based on QCD to be applied with greater accuracy than in lighter hadron systems. Precise measurements of quarkonia decay rates, energy levels, and electromagnetic transitions provide a stringent check on the theory’s ability to describe the strong force. Discrepancies between experimental observations and theoretical predictions regarding quarkonia can reveal the need for refinements in QCD calculations or point towards the presence of previously unknown physics, such as the influence of next-to-leading order effects or the contribution of exotic decay channels. Consequently, ongoing studies of quarkonia continue to be a cornerstone of efforts to fully understand the fundamental interactions governing matter at its most basic level.

The quest to definitively identify glueballs – hypothetical particles composed entirely of gluons – represents a significant frontier in high-energy physics. Unlike most hadrons, which are built from quarks bound by gluons, glueballs would offer a direct observation of the strong force at work in a purely nonperturbative regime. Theoretical predictions from Quantum Chromodynamics (QCD) suggest the existence of various glueball states with differing masses and decay patterns, but experimentally isolating these particles is incredibly challenging. Because glueballs do not carry color charge, they do not readily decay into easily identifiable final states, often mixing with conventional mesons and obscuring their true nature. Current research focuses on analyzing decay patterns and searching for unambiguous signatures within the vast landscape of hadron resonances, hoping to finally confirm the existence of these elusive particles and provide crucial validation of the nonperturbative aspects of the strong force, which governs the interactions within atomic nuclei.

The landscape of hadron physics has been dramatically reshaped by the confirmed existence of tetraquarks and pentaquarks – particles comprising more than the conventional three quarks found in baryons or quark-antiquark pairs in mesons. These exotic states, initially predicted by theoretical models but only recently observed through experiments like those at the Large Hadron Collider, challenge established understandings of the strong force and how quarks bind together. The observed properties of these multi-quark systems – their masses, decay modes, and internal structures – often deviate from predictions based on simple extensions of meson and baryon phenomenology. Consequently, physicists are compelled to develop increasingly sophisticated theoretical frameworks, incorporating concepts like color-flux tubes, hidden color confinement, and dynamical symmetries, to accurately describe these novel forms of matter and fully explore the implications for quantum chromodynamics $QCD$ . The continued study of tetraquarks and pentaquarks promises to unlock deeper insights into the nonperturbative regime of the strong interaction and potentially reveal new fundamental aspects of nature.

Charting the Future: Unifying Approaches and Expanding Frontiers

A comprehensive understanding of the strong force, governing the interactions of quarks and gluons within hadrons, demands a unified approach bridging the realms of perturbative and nonperturbative quantum chromodynamics (QCD). Perturbative QCD, while highly successful at describing high-energy interactions, falters when confronted with the low-energy phenomena defining hadron masses and structure. Conversely, nonperturbative models, such as the Instanton Liquid, offer insights into these low-energy dynamics by positing the existence of topological fluctuations – instantons – which contribute significantly to the generation of hadron mass. However, these models often lack a direct connection to the well-established calculations of perturbative QCD. Therefore, future progress hinges on developing frameworks that seamlessly integrate both approaches, allowing researchers to leverage the strengths of each and obtain a complete, self-consistent description of the strong interaction across all energy scales. This synergy promises not only a deeper understanding of ordinary hadrons but also the potential to predict the properties of exotic states and unveil the subtle complexities of quantum chromodynamics.

The internal architecture of hadrons-protons, neutrons, and their more exotic counterparts-remains a complex puzzle, but detailed mapping of their energy-momentum tensor (EMT) offers a powerful new avenue for investigation. This tensor, which describes the distribution of energy and momentum within the hadron, acts as a sort of internal pressure map, revealing how quarks and gluons interact and contribute to the particle’s overall structure. By probing the EMT, physicists can move beyond simple pictures of hadrons as mere collections of quarks and gluons, and begin to understand the dynamics governing their confinement and the origin of their mass. Recent theoretical advances, combined with emerging experimental capabilities at facilities like the Electron-Ion Collider, promise to deliver unprecedented insights into the EMT of hadrons, potentially unveiling the intricate interplay between pressure, shear forces, and the subtle mechanisms that define their very existence.

Hadron masses, a fundamental property of these composite particles, are not simply the sum of their constituent quark masses – a discrepancy known as the mass gap. Current research increasingly points to the significant role of topological fluctuations in the quantum vacuum, specifically those seeded by ‘instantons’, as a key mechanism for generating a substantial portion of this mass. These instantons, solutions to the equations of quantum chromodynamics, represent tunneling events that alter the topology of the gluon field, creating a non-perturbative background that effectively dresses the quarks and contributes to their observed mass. Investigations into these fluctuations are complex, requiring advanced computational techniques and theoretical models, but offer a promising pathway towards a complete understanding of how mass emerges from the fundamental interactions governing the strong force.

The continued investigation of exotic hadron states – particles beyond the conventional proton, neutron, and meson families – promises to fundamentally reshape understandings of the strong interaction. Current theoretical models suggest these unusual configurations, such as tetraquarks and pentaquarks, aren’t merely fleeting anomalies but rather manifestations of the strong force’s complex dynamics at play. Crucially, the calculated constituent mass of quarks within these models isn’t a fixed value; it exhibits a demonstrable dependence on the strength of the $'t \text{ Hooft}$ coupling, a parameter governing the intensity of strong force interactions. This suggests that quark mass isn’t an inherent property, but emerges from the interplay of the strong force itself, and that studying exotic hadrons-with their unique internal structures-will offer a novel window into this emergent mass generation and the broader landscape of quantum chromodynamics.

The study seeks essential connections. It distills complex quantum chromodynamics into observable parton distributions. This mirrors a fundamental principle: abstractions age, principles don’t. As Epicurus stated, “It is impossible to live pleasantly without living prudently, honorably, and justly.” This pursuit of foundational understanding-bridging the QCD vacuum to hadron structure-demands a similar prudence. The research navigates complexities, aiming for clarity in describing the hadron’s internal structure. Every complexity needs an alibi; here, that alibi is a coherent framework linking nonperturbative dynamics to partonic observables. The effective field theory approach exemplifies this reduction to essential components.

Further Shores

The presented synthesis, while attempting parsimony, does not eliminate the fundamental tension inherent in connecting the nonperturbative regime-characterized by topological solitons and chiral symmetry breaking-with the perturbative world of parton distribution functions. The instanton liquid model, even when augmented by light-front quantization, remains a classical picture; a fully quantum mechanical description of the QCD vacuum, and its excitation spectrum, remains elusive. Subsequent investigation must prioritize the development of effective field theories capable of consistently interpolating between these regimes, and quantifying the associated uncertainties.

A crucial, and frequently neglected, aspect is the refinement of the connection between the instanton density and experimentally accessible observables. Current calculations often rely on phenomenological input; a self-consistent determination, derived directly from hadron spectroscopy and deep inelastic scattering data, would constitute a significant advance. Unnecessary is violence against attention; the proliferation of free parameters obscures rather than illuminates.

Ultimately, the pursuit of a truly unified understanding necessitates a re-evaluation of the very notion of ‘parton.’ Are these merely mathematical constructs, convenient for perturbative calculations, or do they reflect a deeper ontological reality? The answer, one suspects, lies not in increasing complexity, but in stripping away superfluous layers – in achieving density of meaning, the new minimalism.

Original article: https://arxiv.org/pdf/2601.15085.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/