Inside the Proton: Mapping the Hadron Landscape

Author: Denis Avetisyan

New theoretical methods are revealing the complex internal structure of hadrons and providing insights into exotic states beyond the traditional quark model.

The analysis reveals low-lying <span class="katex-eq" data-katex-display="false">D</span> and <span class="katex-eq" data-katex-display="false">D_s</span> mesons across a spectrum of <span class="katex-eq" data-katex-display="false">J^P</span> values, with established light and strange states-indicated by blue circles and orange squares-corresponding to two-pole structures observed within the UChPT analysis, as detailed in Ref. [79]. — The analysis reveals low-lying $D$ and $D_s$ mesons across a spectrum of $J^P$ values, with established light and strange states-indicated by blue circles and orange squares-corresponding to two-pole structures observed within the UChPT analysis, as detailed in Ref. [79].

This review details recent progress in applying lattice QCD to calculate hadron spectra, scattering amplitudes, and explore multi-hadron systems including tetraquark states.

Despite longstanding efforts to map the strong force, fully understanding hadron interactions and the excited states they produce remains a significant challenge. This review, focusing on the field of ‘Hadron spectroscopy and interactions’, details recent progress utilizing lattice QCD to address this, particularly through the analysis of finite-volume spectra and scattering amplitudes. Recent calculations have begun to pinpoint resonance poles and explore the properties of multi-hadron systems, including the increasingly intriguing realm of tetraquark states. How will continued refinement of these techniques illuminate the complex landscape of hadron physics and reveal the underlying dynamics governing their interactions?

The Elegant Foundation of Hadron Spectroscopy

The strong force, governing interactions between quarks and gluons, presents a unique challenge to physicists due to its intensity at short distances. Traditional perturbative methods, successful in quantum electrodynamics, fail when applied to the strong force because the coupling strength becomes too large. Consequently, non-perturbative approaches are essential, and Lattice Quantum Chromodynamics (Lattice QCD) emerges as a powerful tool. This technique fundamentally reimagines spacetime not as a continuous entity, but as a four-dimensional lattice of discrete points. By performing calculations on this discretized spacetime, physicists can sidestep the mathematical difficulties encountered with continuous approaches and solve for the properties of hadrons – composite particles made of quarks and gluons. This discretization, while introducing a subtle approximation, allows for a first-principles calculation of hadron masses and other characteristics, offering a crucial bridge between theoretical predictions and experimental observations in the realm of strong interactions.

Lattice Quantum Chromodynamics (QCD) offers a unique pathway to determine the properties of hadrons – particles like protons and neutrons – directly from the fundamental theory of the strong force. This approach discretizes spacetime into a four-dimensional lattice, allowing for numerical solutions to the equations governing quark and gluon interactions. However, achieving precise results is exceptionally demanding computationally. Simulating these interactions requires immense supercomputing power and sophisticated algorithms to manage the complexity arising from the strong force and the need to extrapolate to the continuum limit where the lattice spacing approaches zero. The resources needed scale rapidly with increasing precision and the inclusion of more realistic quark masses, presenting a continuous challenge to even the most powerful computing facilities and driving ongoing advancements in computational techniques and hardware.

Determining the energy levels and resonance parameters from lattice QCD calculations serves as a vital bridge between theoretical prediction and experimental observation in hadron spectroscopy. These parameters, which define the masses and decay characteristics of hadrons, are not simply inputs to a model, but emerge directly from solving the strong interaction equations on a discretized spacetime lattice. By carefully analyzing the correlations between quark fields, physicists can reconstruct the spectrum of hadrons – baryons and mesons – and compare these results with data from experiments like those at the Large Hadron Collider. Discrepancies between theoretical predictions and experimental measurements then guide refinements to the underlying theory, improving understanding of the strong force and the complex internal structure of matter. Furthermore, extracting resonance parameters-such as widths and lifetimes-provides insights into how hadrons interact and decay, revealing details about the fundamental forces governing the universe at its most basic level.

The analytic structure of the <span class="katex-eq" data-katex-display="false">\mathbb{N}\mathbb{N}\mathbb{N}\mathbb{N}</span> scattering amplitude reveals poles and cuts at fixed angles, and further cuts emerge after partial-wave projection, while standard quantization conditions primarily address the elastic two-particle branch cut. — The analytic structure of the $\mathbb{N}\mathbb{N}\mathbb{N}\mathbb{N}$ scattering amplitude reveals poles and cuts at fixed angles, and further cuts emerge after partial-wave projection, while standard quantization conditions primarily address the elastic two-particle branch cut.

Decoding Hadron Spectra Through Correlation Functions

Two-point correlation functions, denoted as $C_2(t) = \langle 0 | O(t) O(0) | 0 \rangle$ , are fundamental to determining the energy spectrum in lattice Quantum Chromodynamics (QCD) simulations. These functions quantify the probability amplitude for an initial state $|0\rangle$ to evolve into a state created by the operator $O$ at time $t$ . The exponential decay of the correlation function at large time separations allows for the extraction of energy eigenvalues, as the decay rate is directly related to the energy difference between the ground state and the excited states. Specifically, fitting the correlation function to a sum of exponentials yields these energy levels, providing a non-perturbative method for calculating hadron masses and other relevant physical quantities. Accurate determination of these energy levels relies on maintaining sufficient time separation to resolve distinct states and minimizing statistical uncertainties in the correlation function.

Analysis of the correlation matrices generated from two-point correlation functions relies on techniques like the Generalized Eigenvalue Problem (GEVP) and the Lanczos Algorithm to extract energy levels. GEVP involves solving a generalized eigenvalue equation of the form $A v = \lambda B v$ , where A and B are constructed from the correlation matrix elements, and $v$ represents the eigenvector corresponding to eigenvalue λ, which directly relates to the energy of the state. The Lanczos Algorithm, an iterative method for finding the eigenvalues and eigenvectors of a Hermitian matrix, is particularly efficient for large, sparse matrices common in lattice QCD. Both methods effectively diagonalize the correlation matrix, allowing for the identification of the dominant eigenvalues corresponding to the ground and excited states, and subsequently, the extraction of hadronic energy levels.

Signal quality in lattice QCD correlation function analysis is frequently enhanced through the application of operator smearing and strategic source selection. Smearing involves modifying local operators with a series of spatial averages, effectively increasing their overlap with the desired excited states and reducing the influence of discretization errors; this typically improves the condition number of the correlation matrix. Wall sources, constructed to have wavefunctions extending across the simulation volume, offer advantages in extracting specific quantum numbers and minimizing boundary effects, particularly when analyzing multi-hadron states or states with extended spatial distributions. The precise implementation of smearing and source construction depends on the specific observable and desired energy level being targeted, requiring careful optimization to maximize statistical precision and minimize systematic uncertainties.

The difference between the ground-state effective energy of a <span class="katex-eq" data-katex-display="false">D D^*</span> system and the sum of effective masses is calculated using two methods: a symmetric GEVP with <span class="katex-eq" data-katex-display="false">N_{op} = 23</span> (blue circles) and an asymmetric correlation function with a local creation and bilocal annihilation operator (orange diamonds). — The difference between the ground-state effective energy of a $D D^*$ system and the sum of effective masses is calculated using two methods: a symmetric GEVP with $N_{op} = 23$ (blue circles) and an asymmetric correlation function with a local creation and bilocal annihilation operator (orange diamonds).

Navigating Finite Volumes and Scattering Amplitudes

Finite-volume calculations of scattering amplitudes are inherently constrained by the imposed boundary conditions, necessitating the application of quantization conditions to ensure physically meaningful results. These conditions arise because only discrete energy levels are permitted within the finite volume, meaning that the outgoing waves in a scattering process must satisfy these constraints. Specifically, the allowed momenta $\vec{p}$ are determined by the volume $V$ via $\vec{p} = \frac{2\pi \vec{n}}{L}$ , where $L$ represents the linear dimension of the volume and $\vec{n}$ is a vector of integers. Consequently, the scattering amplitude, typically expressed as an infinite sum over intermediate states, becomes a discrete sum, and the energy levels are shifted from their continuum values, introducing finite-size effects that must be accounted for during analysis and extrapolation to infinite volume.

Employing a plane-wave basis within finite-volume calculations provides a framework for quantizing the scattering equations, effectively discretizing the momentum space. This discretization is achieved by imposing periodic boundary conditions on the finite volume $V = L^3$ , which restricts the allowed momenta to discrete values $p = \frac{2\pi n}{L}$ , where $n$ is an integer vector. Consequently, the scattering amplitudes are expressed as a sum over these discrete momentum states, allowing for the calculation of energy levels and the extraction of physically relevant quantities like scattering lengths and effective ranges. The plane-wave basis facilitates the translation between continuous, infinite-volume scattering amplitudes and their discrete, finite-volume counterparts, forming the basis for lattice QCD calculations and other numerical methods used to study particle interactions.

The Left-Hand Cut, a branch cut in the complex energy plane, introduces complexities in finite-volume calculations by contributing to the energy spectrum in a manner inconsistent with simple interpretations. Specifically, the presence of the Left-Hand Cut implies the existence of unphysical poles in the scattering amplitude that, when analytically continued to the physical region, manifest as spurious states in the finite-volume energy spectrum. These states do not correspond to actual scattering particles within the finite volume but arise from the analytic structure of the amplitude and necessitate careful subtraction or regularization procedures to accurately extract the desired physical energies and scattering parameters. The impact of the Left-Hand Cut is particularly pronounced at lower energies and requires precise consideration of the volume size and boundary conditions to avoid misinterpreting the resulting spectrum.

The two-particle partial wave amplitude exhibits physical scattering locations and possible poles on both the complex center-of-mass energy plane, defined by a branch cut originating at threshold, and the complex scattering momentum plane, where <span class="katex-eq" data-katex-display="false">\sqrt{s} = \sqrt{m_{1}^{2}+p^{2}} + \sqrt{m_{2}^{2}+p^{2}}</span>. — The two-particle partial wave amplitude exhibits physical scattering locations and possible poles on both the complex center-of-mass energy plane, defined by a branch cut originating at threshold, and the complex scattering momentum plane, where $\sqrt{s} = \sqrt{m_{1}^{2}+p^{2}} + \sqrt{m_{2}^{2}+p^{2}}$ .

Probing the Frontiers of Exotic Hadron Structures

Before venturing into the realm of undiscovered hadron structures, physicists rely on well-established, yet fleeting, particles to rigorously test the computational tools used in their exploration. Conventional hadrons – those with established decay patterns and properties, such as the $f_0(500)$ , $\rho(770)$ , and $\Delta(1232)$ – act as crucial validation points for lattice Quantum Chromodynamics (Lattice QCD). These calculations, performed on powerful supercomputers, attempt to map the strong force that binds quarks and gluons into protons, neutrons, and other hadrons. By accurately reproducing the known properties and decay dynamics of these benchmark particles, researchers gain confidence in the reliability of their methods before applying them to the more complex and uncertain search for exotic hadron states. This process ensures that any novel structures discovered aren’t merely artifacts of the calculation itself, but genuine predictions of the underlying theory.

The quest to identify exotic hadrons, particularly tetraquarks-particles composed of four quarks-represents a significant challenge to established theories of hadron formation. Conventional understanding posits that hadrons are formed from quark-antiquark pairs or three quarks, but the potential existence of more complex arrangements demands a reevaluation of the strong force dynamics. Discovering and characterizing tetraquarks, and other multi-quark states, requires pushing the limits of both experimental capabilities and theoretical modeling. These investigations aren’t simply about adding more particles to the hadron zoo; they probe the fundamental question of how quarks bind together, potentially revealing new emergent phenomena and refining the Standard Model’s description of matter at its most basic level. Confirmation of tetraquark states would demonstrate that the strong force allows for more diverse and complex configurations than previously imagined, reshaping the landscape of particle physics.

The quest to understand tetraquark states – hadrons composed of four quarks – is increasingly focused on the role of diquarks, tightly bound pairs of quarks that may act as building blocks within these exotic structures. Current theoretical models posit that resonances, such as the $D^*D$ combination, could be key to explaining the observed properties of these tetraquarks. Recent advancements in lattice quantum chromodynamics (QCD) calculations are now enabling researchers to simulate these systems with pion masses as low as 200 MeV, a significant step towards achieving more physically realistic conditions. Lowering the pion mass reduces discrepancies arising from approximations in the calculations, providing a more accurate depiction of the strong force interactions governing the tetraquark’s internal structure and ultimately refining the understanding of how matter can combine in ways beyond the conventional proton and neutron.

Charting New Avenues for Investigation Beyond Standard Methods

Finite-volume spectroscopy, a cornerstone of hadron physics, traditionally relies on analyzing energy levels within confined spaces. However, the HAL QCD method presents a powerful, complementary approach by directly examining hadron-hadron correlation functions. This technique bypasses the need to solve the Schrödinger equation within the finite volume, instead focusing on how hadrons interact and propagate as observed through these correlation functions. By meticulously analyzing these functions, researchers can extract key information about scattering amplitudes and bound state energies, offering a distinct perspective on the strong force dynamics. This is particularly valuable for systems where standard methods encounter difficulties, such as those involving multi-hadron states or resonances, and allows for a more complete understanding of how hadrons bind and interact with one another, extending the reach of theoretical predictions.

A comprehensive understanding of the baryon-baryon interaction serves as a foundational element in constructing accurate models of systems containing multiple hadrons. These interactions, governing how baryons – such as protons and neutrons – bind together, dictate the stability and observable characteristics of complex nuclear matter. Precisely mapping these forces allows physicists to move beyond simplified assumptions and predict the properties of exotic hadrons, like tetraquarks and pentaquarks, as well as dense baryonic matter found in neutron stars. Furthermore, detailed knowledge of baryon interactions is essential for interpreting experimental results from heavy-ion collisions, where copious amounts of multi-hadron states are produced, and for refining calculations in nuclear physics that rely on effective potentials derived from these fundamental interactions.

Refining calculations of multi-hadron systems benefits significantly from the application of Chiral Perturbation Theory (UChPT), a technique that provides crucial insights into the underlying dynamics governing these complex interactions. Recent investigations, leveraging increasingly fine lattice spacing in numerical simulations, have demonstrably reduced discretization effects – systematic errors arising from the discrete nature of lattice calculations – in studies of baryon-baryon interactions. This advancement has enabled more precise predictions regarding the existence and properties of exotic hadronic states, notably suggesting a binding energy of approximately 100 MeV for the $b\bar{b}b\bar{b}$ doubly bottom tetraquark. Such results not only enhance the understanding of strong force interactions but also provide a crucial benchmark for validating theoretical models and interpreting experimental observations in the pursuit of a complete picture of hadronic matter.

The pursuit of understanding hadron spectroscopy, as detailed in this study, resonates with a fundamental principle of clear communication. Jürgen Habermas once stated, “The unexamined life is not worth living.” This sentiment applies equally to the rigorous examination of particle interactions. The paper’s focus on calculating scattering amplitudes and identifying resonance poles – essentially, deciphering the ‘language’ of these interactions – demands a commitment to precise analysis. Just as Habermas champions rational discourse, this research utilizes lattice QCD to achieve a deeper, more coherent picture of the subatomic world, striving for clarity amidst complexity. The exploration of multi-hadron systems, including tetraquarks, highlights a search for underlying structure and meaning within the observed phenomena.

What’s Next?

The pursuit of hadron spectroscopy, as detailed within, increasingly resembles an exercise in controlled deconstruction. Lattice QCD offers the tools to dismantle the strong force’s creations, but assembling a coherent picture demands more than just calculating energies. The field now faces the subtle art of interpretation – discerning genuine physical states from artifacts of finite volume or discretization. The location of a pole, while mathematically precise, remains a statement about the model, not necessarily a definitive observation of nature until rigorously connected to scattering data.

A particular challenge lies in the multi-hadron sector. Tetraquarks, and other exotic configurations, are not simply more hadrons; they represent a qualitative shift in how the strong force binds matter. Identifying these states requires not just finding a signal, but proving it arises from a unified, multi-quark structure, not a loosely bound dimer or a cleverly disguised meson-meson resonance. The elegance of a truly minimal description will likely prove to be the key.

Ultimately, the durability of this approach – and the comprehensibility of the resulting hadron landscape – hinges on a commitment to aesthetic principles. A theory cluttered with adjustable parameters, or reliant on ad-hoc assumptions, is a fragile edifice. Beauty and consistency make a system durable and comprehensible; they are not merely stylistic flourishes, but indicators of deep understanding.

Original article: https://arxiv.org/pdf/2602.23244.pdf

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