1.65 GeV: What the Vacuum Told Me About the Millennium Prize - Blog

There is a number that has occupied my mind for months now: 1.65 GeV. It is the mass of the lightest particle that should exist in a universe governed by the strong nuclear force alone — a universe of pure gluons, with no quarks, no electrons, nothing but the carriers of the strong force interacting with themselves. This particle is called the scalar glueball, and its mass is what mathematicians call the Yang-Mills mass gap.

The Clay Mathematics Institute has offered one million dollars for a rigorous proof that this mass gap exists. It is one of seven Millennium Prize Problems — alongside the Riemann Hypothesis, P vs NP, and the Navier-Stokes equations — and it has remained unsolved since the year 2000. In this post, I want to explain what the mass gap actually is, why I believe the vacuum itself holds the answer, and what my paper contributes to this century-old puzzle.

What Is the Mass Gap, and Why Does It Matter?

Quantum chromodynamics — QCD — is the theory of the strong force. It describes how quarks and gluons interact inside protons, neutrons, and all nuclear matter. Unlike electromagnetism, where the photon is massless and the force has infinite range, the strong force is short-ranged. You cannot feel it across a room. You cannot even feel it across an atom. It operates only inside the nucleus, at distances of about one femtometer — a millionth of a billionth of a millimeter.

The reason the strong force is short-ranged is precisely the mass gap. If the lightest gluon-composite particle (the glueball) has a nonzero mass m, then the force mediated by exchanging such particles decays exponentially as e^{-mr}. The heavier the particle, the shorter the range. A mass gap of 1.65 GeV corresponds to a range of roughly 0.12 femtometers — deep inside the proton.

But here is the puzzle: the fundamental equations of QCD — the Yang-Mills equations — are written in terms of massless gluons. There is no mass term in the Lagrangian. The mass gap is not put in by hand; it must emerge from the dynamics of the theory itself. Proving that this happens, rigorously and mathematically, is the Millennium Problem.

The State of Play: What the Mathematicians Are Doing

I have been following the work of the leading mathematicians on this problem with great interest. In October 2025, Sourav Chatterjee of Stanford delivered a lecture at the Harvard Center of Mathematical Sciences and Applications in which he outlined his approach: start with the lattice formulation of Yang-Mills theory (where the path integral is a well-defined finite-dimensional integral), and then take the continuum limit. He is optimistic — he believes a solution may be achievable within 10 to 15 years.

At the same Clay Research Conference, Fields Medalist Martin Hairer presented a different approach: stochastic quantization, where the Yang-Mills measure is constructed as the invariant measure of a stochastic partial differential equation. Hairer's regularity structures — the work that earned him the Fields Medal — provide the mathematical machinery to make sense of the ultraviolet divergences that plague the theory.

Both approaches are focused on existence: proving that the quantum Yang-Mills theory can be constructed as a mathematically well-defined object. Neither approach, at present, addresses the numerical value of the mass gap. They seek to prove that the mass is strictly positive, but they do not predict what it is.

This is where my work enters the picture.

What the Vacuum Told Me

The central insight of my paper, "Vacuum Strain and the Yang-Mills Mass Gap," is that the mass gap is not a mysterious emergent property that requires new mathematics to understand. It is a direct consequence of a measured physical quantity: the gluon condensate.

The gluon condensate is the vacuum expectation value of the gluon field-strength squared:

⟨(α_s/π) G²⟩ ≈ (0.33 GeV)⁴

This was first established by Shifman, Vainshtein, and Zakharov in 1979 through QCD sum rules, and it has been confirmed by decades of lattice simulations. It is not a theoretical assumption — it is a measured property of the QCD vacuum. It tells us that the vacuum is not empty. Even in the absence of any particles, the quantum vacuum has a nonzero energy density from gluon field fluctuations.

I call this vacuum strain. The vacuum is permanently deformed by quantum fluctuations and cannot relax to the trivial, zero-energy state. This strain is the physical origin of the mass gap.

The Derivation in Five Sentences

The full derivation is a nine-step argument presented in detail on my website, but the logic can be compressed into five sentences:

Start with the standard Yang-Mills action and define the gauge-invariant composite operator O(x) = Tr(F²), which measures the local gluon field energy density.
Introduce an auxiliary scalar field φ via the Hubbard-Stratonovich transformation — a standard technique in statistical field theory — to rewrite the path integral in terms of φ.
After integrating out the gauge fields, the effective potential for φ takes the form V(φ) = (λ/4)φ⁴ − Jφ, where the source J is proportional to the gluon condensate and the coupling λ is determined by large-N scaling and renormalization group arguments.
Because J ≠ 0 (the gluon condensate is nonzero), the vacuum expectation value ⟨φ⟩ is necessarily nonzero — the vacuum cannot sit at φ = 0.
The mass of the lightest excitation around this displaced vacuum is m² = V″(φ₀) = 3λ^{1/3}J^{2/3}, which evaluates to 1.65 ± 0.15 GeV after including one-loop radiative corrections.

Every parameter in this chain — λ, c, J, and the gluon condensate itself — is derived from established QCD results. There are zero free parameters. The prediction of 1.65 GeV is not a fit; it is a derivation.

How Does 1.65 GeV Compare?

The lattice QCD community has computed the scalar glueball mass through decades of increasingly precise Monte Carlo simulations. The current benchmark, established by Morningstar and Peardon in 1999 and confirmed by subsequent studies, is:

m(0++) = 1.71 ± 0.05 GeV

My prediction of 1.65 ± 0.15 GeV is consistent with this within uncertainties — a deviation of approximately 3.5%. For comparison, the Functional Renormalization Group (FRG) approach of Pawlowski et al. (2023) predicts 1.87 GeV, overshooting by about 9%. The Schwinger-Dyson equation approach computes a dynamical gluon mass of 400-500 MeV, which is a different quantity (the propagator mass, not the glueball mass).

What makes my result distinctive is not just the accuracy, but the economy. The FRG requires truncation of the effective action and gauge fixing. The SDE approach requires solving an infinite tower of coupled integral equations. My derivation uses only the gluon condensate, the NSVZ low-energy theorem, and large-N scaling — three inputs that are among the most established results in nonperturbative QCD.

What My Paper Does NOT Claim

I want to be honest about the boundaries of this work. My paper does not solve the Clay Millennium Problem. The Millennium Problem requires three things: (a) construct the quantum Yang-Mills theory rigorously, (b) prove it is non-trivial, and (c) prove the mass gap is strictly positive. My paper addresses only (c), and it does so conditionally: if the gluon condensate can be proven strictly positive in a rigorously constructed continuum theory, then the mass gap follows from the Hubbard-Stratonovich mechanism.

The Hubbard-Stratonovich transformation I use is rigorously justified on the lattice, where the path integral is a finite-dimensional integral. Whether it survives the continuum limit — where the path integral becomes an infinite-dimensional object that is not yet mathematically well-defined — is an open question. This is precisely the kind of question that Chatterjee's lattice-to-continuum program and Hairer's stochastic quantization framework are designed to answer.

My paper provides the physical content — the mechanism and the number — while the mathematical community works on the existence question. If existence is eventually proven, my mechanism immediately provides the mass gap value. The two programs are complementary, not competitive.

The Deeper Connection: Mass Gap, Confinement, and Gravity

What excites me most about the vacuum strain mechanism is that it does not stop at the mass gap. The same permanently strained vacuum that gives mass to glueballs also produces confinement — the phenomenon that quarks cannot exist as free particles. The strained vacuum channels chromoelectric flux into narrow tubes connecting quark-antiquark pairs, producing a linear confining potential with string tension σ ≈ (440 MeV)², which matches the lattice value.

And in my broader framework, this same vacuum strain connects to the emergence of gravity. The baryon asymmetry A(t) and restoration rate R(t) that govern emergent gravity are themselves consequences of the field dynamics that produce the gluon condensate. The mass gap paper is the microscopic foundation; the emergent gravity papers are the macroscopic consequence. They are two ends of the same thread.

This is why I titled my first blog post "Why I Believe the Vacuum Holds the Answer to Gravity." The vacuum is not empty. It is strained, structured, and dynamic. The mass gap is the first quantitative evidence of this strain — a number that can be derived from first principles and checked against experiment. If the vacuum strain mechanism is correct at the QCD scale, it provides a physical basis for understanding why gravity exists at all.

An Invitation

I am an independent researcher. I do not have the institutional backing of Stanford or Oxford. What I have is a prediction — 1.65 GeV — derived from established physics with no free parameters, consistent with the lattice benchmark to 3.5%.

If you are a mathematician working on the Yang-Mills existence problem, I would welcome a conversation about whether the Hubbard-Stratonovich mechanism can be rigorized within your framework. If you are a lattice QCD practitioner, I would welcome a test of the finite-temperature prediction: m(T)/m(0) ≈ [⟨G²(T)⟩/⟨G²(0)⟩]^{1/4}. If you are a physicist who thinks about confinement, I would welcome a discussion about whether vacuum strain provides a unified mechanism for the mass gap and the string tension.

The paper is freely available on my website. The derivation is transparent. The prediction is falsifiable. That is all any physicist can offer.

— Yohannes Beyene Ashebo, February 2026