Hadronic Predictions

How it works (in plain language)

What is a baryon, in MFT?

In MFT, every particle is a localised configuration of the elastic medium (the "contraction field"). A baryon — a particle made of quarks, like the proton or neutron — is a special kind of localised configuration with a topological winding: the field rotates from one value at the origin to another at infinity, in a way that can't be smoothly undone. This is the "B = 1 hedgehog soliton" — the simplest non-trivial baryon configuration.

The first prediction of this solver is just: does this kind of soliton exist mathematically? Yes — and the BVP solve here demonstrates that explicitly, with the soliton's topological charge B coming out as exactly 1.

What is f_π?

The pion decay constant f_π ≈ 186 MeV is one of the most important measured constants of low-energy nuclear physics. It sets the scale at which the strong nuclear force becomes nonlinear — it's roughly the "stiffness" of the medium that binds quarks into protons and neutrons.

In conventional QCD, f_π is an empirical input — extracted from pion decay measurements, not derived. In MFT, it's derived from a 4-line algebraic identity: it's exactly the curvature of the silver-ratio potential at the nonlinear vacuum, divided by 4. That gives f_π² = δ in MFT units, and converting to MeV gives 185.94 — a 0.03% match to observation with no fitting.

What is the decuplet?

The Δ, Σ*, Ξ*, Ω particles are the four spin-3/2 baryons of the lightest mass family. Their masses are observed to be 1232, 1385, 1533, 1672 MeV. The gaps between them are 153, 148, 139 MeV — nearly equal but not exactly so.

MFT predicts these gaps are exactly equal, from a symmetry argument: the SU(3) flavour symmetry of the Skyrme reduction means the rotational energy cost of each "step" up the decuplet is the same. The observed 4% dispersion is consistent with explicit symmetry breaking from the up/down/strange quark mass differences (small, but real).

What does the BVP show, exactly?

The boundary-value problem solved here is the radial Euler-Lagrange equation for the hedgehog soliton ansatz U(r) = exp(i τ·r̂ f(r)) with f(0) = π and f(∞) = 0. This is the equation that comes out of the MFT Skyrme reduction in the high-density regime. SciPy's solve_bvp finds the radial profile f(r), and from it we compute:

1. The energy integrals E2 and E4 — should be equal (virial balance for any true soliton). Achieved to ~0.03%.
2. The topological charge B — must be exactly 1 for a baryon. Achieved to 6 decimal places.
3. The total energy ε₀ ≈ 145.85 — matches the standard Adkins-Nappi-Witten reference value from the Skyrme model literature.

In short: the BVP confirms the soliton exists with the right structure. The f_π and decuplet predictions then follow from algebraic facts about the potential, not from the BVP solve itself.

What this solver doesn't predict — and why that's not an MFT-specific problem

The classical Skyrme rotor formula combining the BVP integrals with f_π gives a proton mass around 1.6 GeV — too high by a factor of ~1.7 compared to the observed 938 MeV. This is not an MFT problem. The classical Skyrme model itself has this limitation: Adkins, Nappi, and Witten in 1983 (the foundational Skyrme model paper) explicitly noted that "our results are generally within about 30% of experimental values." This 30%-level error has been present in the Skyrme literature for over 40 years, independent of MFT.

The standard fixes in the QCD literature — which apply equally well to MFT — are:

• Vector mesons (ρ, ω): Adding rho and omega meson exchanges reduces the soliton mass and brings it close to 938 MeV. (Adkins-Nappi 1984; Hidden Local Symmetry approach.)
• Quantum corrections (1/N_c): Beyond rigid-body quantization, the Nambu-Jona-Lasinio quantum-fluctuation calculation gives a nucleon mass around 900 MeV. (Multiple groups, 1990s.)
• Holographic QCD methods: A parameter-free reconstruction using holographic principles gives nucleon mass within 5% of 938 MeV.
• Chiral loop corrections: Order few-percent corrections from one-loop pion-nucleon dynamics.

MFT's published P8 paper makes this point explicitly: "the hedgehog-extraction gap reflects the well-known limitations of the classical Skyrme model... The algebraic predictions of MFT (f_π² = δ, m_σ = 2 f_π, the Z = 0 Q-ball spectrum, the lepton and boson masses) are independent of the hedgehog extraction."

In other words: MFT inherits a known limitation from the classical Skyrme reduction it uses, but MFT's own predictions don't depend on it. The f_π = 186 MeV prediction, the decuplet equal spacings, the σ-meson mass m_σ = 2f_π = 372 MeV, the π⁰ identification at 135.3 MeV (0.2% error from the Z=0 Q-ball spectrum), and all the lepton, neutrino, and boson predictions go through different mechanisms entirely and are unaffected by the Skyrme-rotor mass gap.

Closing the gap further would require either implementing the standard QCD-literature vector-meson corrections in MFT context, or developing an MFT-specific derivation of the Skyrme coupling e from the contraction-field structure (P8 conjectures e = 2δ ≈ 4.83; the BVP extraction gives e ≈ 4.5, a 6% gap consistent with classical-Skyrme corrections). Either path is open research; neither is required for MFT's algebraic hadronic predictions to hold.

Source

The hadronic sector is verified by four scripts in the MFT corpus, each computing a different aspect:

• mft_hedgehog_bvp_v2.py — the BVP solver that this page uses (B=1 hedgehog, ε₀ = 145.85, virial balance).
• mft_skyrme_derivation.py — the chiral closure f_π² = δ, hedgehog → nucleon/decuplet predictions.
• mft_hadronic_landscape.py — the complete hadronic landscape: f_π, m_σ, Z=0 Q-balls, GMOR relation, and nuclear force.
• mft_hadronic_v2.py — the π⁰ fine scan from the Z=0 Q-ball spectrum (0.2% match to observed 135 MeV).
• mft_microphysics.py — the cross-sector consistency verification.

Full physics is in P8 of the corpus (Microphysics from the Silver Ratio, Zenodo: 10.5281/zenodo.19343255). See Particles for the broader MFT context.