The world inside a proton or a meson is not a neat lineup of separate players. It is a crowded stage where quarks of different flavors throw whispering glances at each other through the gluon field, sometimes crossing invisible boundaries that physicists like to call flavor singlets. In this echo chamber of quantum chromodynamics (QCD), states that look like cleanly separated light-quark mesons and heavy-quarkonia can actually mix. A recent lattice QCD study pulls back the curtain on that mixing in the S-wave channels, using light and charm quarks to explore how flavor-blind glue and quark-antiquark annihilation blend the spectrum. The result is a richer, more interconnected picture of the hadron spectrum and a roadmap for how to hunt glueballs in the messy real world of QCD.
The study is led by Juan Andrés Urrea-Niño of Trinity College Dublin, with collaborators from Bergische Universität Wuppertal, CERN, and colleagues at Trinity. The team uses powerful computer simulations to solve QCD from first principles, tackling a problem that has long frustrated simpler, phenomenological models: how, exactly, can a flavor-singlet light meson mix with a charmonium state when the quark content can annihilate and re-form in different ways? Their answer hinges on two technical advances—the intensified use of distillation profiles to capture quark propagation across the lattice, and a careful construction of a mixed correlation matrix that ties together light-quark mesons and charm quark operators. The result is not just a number, but a new way to see how the spectrum emerges when the quantum fields of light and charm are laid bare side by side.
A shared stage for light and charm quarks
In the language of particle physics, mesons are bound states of a quark and an antiquark. In the simplest picture, you might treat light quarks (up, down, strange) as one family and charm quarks as another, each with its own spectral family of states. But when you allow quark-antiquark annihilation to participate—an inherently quantum process that connects different quark flavors—the lines blur. That blur is the heart of flavor-singlet mixing. It matters especially for flavor-singlet channels, where the same quantum numbers can be accessed by more than one underlying quark content. The experimental landscape already hints at a zoo of near-threshold states (the so-called XY Z excitations) that defy tidy classification. Lattice QCD, which computes QCD on a spacetime lattice, is the only tool that can tackle this without resorting to crude approximations. And that’s where this study makes its mark.
Two technical challenges loom large in such calculations. First, the disconnected diagrams—the contributions to correlation functions that arise when quark loops pop in and out of existence—are notoriously noisy and hard to compute. Second, to resolve states that might be mixtures of light and charm content, you need an operator basis rich enough to sample both sectors. The authors tackle both head on with what they call improved distillation, a method that expands the operator basis by attaching different distillation profiles to the quark propagators. Think of it as giving each operator a different listening post in the lattice’s chorus, so you don’t miss a note that only sounds in a particular flavor’s range.
Crucially, they organize the problem as a mixing correlation matrix that includes both charmonium operators and flavor-singlet light-meson operators, along with the all-important disconnected pieces. By solving a generalized eigenvalue problem (GEVP) for this matrix, they extract the low-lying energy spectrum and ask how the states integrate contributions from each type of operator. The upshot is a direct readout of how much a given energy eigenstate is sampled by light-quark creation operators versus charm-quark operators, and how strongly the two sectors talk to each other through mixing. The work sits amid a growing push to bring gluonic (glueball) operators into the same dialogue, which would let us see whether a given state is genuinely mesonic, gluonic, or some hybrid mix. The study’s authors are setting the stage for that broader conversation.
What the mixing looks like on the lattice
To keep the exploration physically meaningful, the team runs two different lattice ensembles with three light flavors plus one charm flavor (Nf = 3 + 1). One ensemble uses a lighter pion mass of about 420 MeV, and the other uses a heavier light-quark mass with a pion around 800 MeV. Both are designed to let the spectrum breathe: the lighter pion mass approaches the real-world regime where many open-charm decay channels become available, while the heavier mass helps suppress some decays and makes certain effects easier to isolate. Each ensemble is a vast forest of gauge configurations—thousands of samples—to ensure the stochastic noise doesn’t drown the physics. The lattice spacing is small enough to capture the heavy charm scale, and open boundary conditions in time help avoid clutter from topological sampling issues that can plague fine lattices.
In the pseudoscalar channel (the JPC = 0-+ sector), and separately in the vector channel (JPC = 1–), the authors find non-zero off-diagonal correlations between light-meson operators and charmonium operators. In plain terms: the flavor-singlet light meson and the charmonium creation operators are indeed “talking” to each other through the QCD dynamics, even though the two are built from different quark flavors. This explicit mixing is captured by the off-diagonal entries in the mixing correlation matrix, and it remains statistically significant across the two pion-mass ensembles. The team’s GEVP analysis then teases apart the spectrum into states that can be associated, approximately, with light-meson-like and charmonium-like character, while simultaneously respecting that they are not purely one or the other.
One of the clever moves in their analysis is what they call partial pruning. Rather than throwing all operator-types into a single noisy matrix, they separately optimize the light-meson and charmonium blocks using their connected correlators, then weave them back together. This preserves the distinctive flavor content of each block while keeping the condition number of the matrix manageable. It’s a practical trick that keeps the math stable enough to reveal genuine physics rather than numerical mirages. In their language, the result is a correlation matrix that respects both sectors’ identities while still letting the energy eigenstates reveal a shared, mixed personality.
How the mixing reshapes the spectrum
So what happens when you actually solve the GEVP on these mixed matrices? The authors first establish reference solutions from “pure” sectors: a light-meson-only GEVP and a charmonium-only GEVP. They identify the lightest two pseudoscalar states (think of them as the lightest meson ground state and its first excited state in that channel) and the two lowest vector states. Then they solve the full mixing problem and search for the four states closest to those reference levels. Across both channels and both pion masses, the four states that emerge are remarkably consistent with their unmixed counterparts, with a caveat: the spectrum acquires the flavor-singlet mixing texture, and some of the excited-state structure shifts slightly in mass when the mixing is included. This is not a wholesale re-shuffling; it is a gentle adjustment that reveals how each eigenstate is built from both light and charm content rather than from one or the other alone.
What the vectors that accompany the GEVP tell you is striking. If you plot the overlap of each energy eigenstate with each operator, the light-meson-like states overwhelmingly overlap with light-meson operators, and the charmonium-like states with charmonium operators. Yet there are non-zero cross-overlaps. In other words, even when a state is mostly “light” in character, charm content leaks in; and a predominantly “charm” state can borrow a touch of light-quark flavor. This is the quantum world’s version of a duet where the voices remain distinct but harmonize. It matters because it shows that you cannot cleanly separate light-flavor dynamics from charm dynamics in a unified, flavor-singlet channel—at least not if you want a faithful account of the spectrum’s true structure.
The researchers also tried to see how this mixing behaves when they deliberately mix in gluonic operators—Wilson loops that probe glueball-like content. The signal is faint but present, a whisper suggesting that glueball mixing with meson operators could be real in the same flavor-singlet channel. The catch is statistics. Gluonic signals are famously noisy in full QCD with dynamical quarks. The authors acknowledge this and suggest that with more data they could push this line of inquiry further, potentially revealing how glueballs thread through the same energy ladder as quark-antiquark bound states.
Why this matters for glueballs and the big questions in QCD
Charm quarks sit at an energy scale that sits near the threshold where glueball dynamics become relevant. The ηc, the pseudoscalar charmonium, has long been a playground for testing how well lattice QCD handles both quark annihilation effects (disconnected diagrams) and the heavy-quark dynamics that feed into the hyperfine splitting between the ηc and the J/ψ. In this study, including the disconnected contributions and the explicit mixing between light and charm flavor-singlet operators produces a few concrete shifts. For the ground-state ηc, the mass is pulled downward by about 37 MeV when charm-disconnected effects and mixing are included. The first excited states show smaller shifts, but the pattern matches what other lattice and indirect approaches have hinted at. This is not just a numerical curiosity; it’s a window into how nonperturbative QCD reshapes a spectrum that seems, at first glance, to belong to two separate families of particles.
These results also offer a concrete critique of the sometimes-cited OZI rule, which says that flavor-singlet processes that require quark annihilation are suppressed. In the flavor-singlet channels the rule is not absolute. The correlations across flavors imply that the energy eigenstates inherit content from both light and charm sectors. The practical upshot is that any robust lattice study of flavor-singlet mesons, and any attempt to pin down the nature of near-threshold states (some of which could be glueballs or hybrids), must treat mixing as a core feature, not a niche complication.
Equally important is the methodological advance. The improved distillation approach, with multiple profiles and a careful pruning strategy, provides a blueprint for tackling even more ambitious problems: we can incorporate gluonic operators into the same framework, push toward near-physical pion masses, and, crucially, study how these mixings evolve as thresholds open up. The study’s authors are candid about the next steps—more configurations, more statistics, and a fuller operator basis that would include explicit glueball operators on an equal footing. If that path succeeds, lattice QCD could map the full landscape of flavor-singlet states with a clarity that has eluded theorists for decades.
From spectra to stories: how this reshapes our view of the hadron zoo
Beyond the technical triumph, the work reshapes how we tell the story of hadrons. The spectrum is not a neat archipelago of light mesons on one side and heavy quarkonia on the other. It is a continuum of states where flavor, spin, and color weave together, and where nothing is immune to the invisible conversations that quark loops enable. The presence of non-zero off-diagonal mixing in both pseudoscalar and vector channels—across two quite different pion masses—suggests that flavor-singlet states are better thought of as blended profiles rather than pure recipes. That has implications for experimental searches as well. When a signal sits near a charmonium-like scale but carries a hint of light-quark flavor, it may not be a straightforward charmonium or a light meson, but a hybridized object with mixed content. The X(2370) glueball candidate, for instance, sits in a neighborhood where such mixing could be physically meaningful. Lattice studies like this one provide the scaffolding to interpret those resonances with more nuance and fewer shortcuts.
And this is where collaboration matters. The work is a joint achievement of a community spanning Trinity College Dublin, Bergische Universität Wuppertal, and CERN, among others. It embodies how modern theory progresses: never content with a single technique, always cross-checking with multiple operator bases, and always pushing toward a framework that can accommodate the universe’s messy, beautiful mixing. The authors, led by Juan Andrés Urrea-Niño, with colleagues Roman Höllwieser, Francesco Knechtli, Tomasz Korzec, Jacob Finkenrath, and Michael Peardon, demonstrate that to understand the hadron spectrum we must listen to every voice in the chorus and give them all a place on the same stage.
Bottom line: Flavor-singlet mixing is not a fringe effect to be tucked away in a corner of a model. It is a fundamental feature of how QCD assembles the hadron spectrum, and it becomes visible only when we build a rich operator basis and treat the problem with careful, sometimes painstaking numerical artistry. The study’s advances both refine what we know about known states like ηc and ψ(2S) and lay the groundwork for identifying gluonic components that could underlie true glueballs. The universe of hadrons is richer than we once thought, and lattice QCD is teaching us to hear every instrument in the orchestra.
