# DFD-QCD Lattice Boundary Program — Executed Results

**Date:** 2026-06-08  **Hardware:** 12-core MacBook Pro, CPU-only, ~18 GB RAM
**Code:** `src/su3_fast.py` (numba pure-gauge), `src/wilson_hmc.py` (Nf=2 dynamical HMC),
`src/analyze_potential.py`, `src/continuum_extrap.py`, `src/analyze_glueball.py`

Actually-executed lattice gauge theory with an honest status boundary. Supersedes the
original `4^4` smoke test (whose plaquettes were under-thermalized and untrustworthy:
it reported `<P>(5.8)=0.6176`, larger than the true `<P>(6.0)=0.5937`).

---

## 1. Method
* **Pure gauge:** SU(3) Wilson action; multi-hit Metropolis on random SU(2) subgroups +
  microcanonical overrelaxation (Cabibbo-Marinari). Static potential from APE-smeared
  (alpha=0.5, max-trace SU(3) projection) on-axis Wilson loops; V(R) from the effective-mass
  plateau; Cornell fit `V0+sigma R - e/R`; jackknife errors. Runs: `16^4`, beta=5.8/5.9/6.0/6.1/6.2,
  120 configs each.
* **Dynamical fermions:** Nf=2 unimproved Wilson HMC, `S_pf=phi^dag(M^dag M)^-1 phi`,
  gamma5-Hermiticity, CG inverter, leapfrog MD, Metropolis accept/reject.
* **Scale:** Necco-Sommer `r0/a(beta)` (Wilson action), r0=0.5 fm. External calibration, not a
  DFD prediction.

---

## 2. Validation — average plaquette vs published SU(3) Wilson values (the unforgeable check)

| beta | measured `<P>` (16^4) | published | deviation |
|------|----------------------|-----------|-----------|
| 5.8  | 0.56750 ± 0.00006 | 0.5670 | +0.09 % |
| 6.0  | 0.59405 ± 0.00005 | 0.5937 | +0.06 % |
| 6.2  | 0.61381 ± 0.00004 | 0.6136 | +0.03 % |

Sub-0.1% on the production lattices (residual = expected finite-volume, shrinking with beta).
The plaquette is UV-dominated and validates thermalization/update correctness only — it is
**not** evidence for the string tension (see §3).

---

## 3. String tension (quenched, finite volume) — with independent cross-checks

**Per-beta `a*sqrt(sigma)` vs literature** (free-e Cornell, R in [2,6]; literature: EHK/Teper):

| beta | this work `a√σ` | literature | note |
|------|-----------------|------------|------|
| 5.8  | 0.329(10) | 0.313 (Teper) | +5% (coarse end, fit-window biased high) |
| 5.9  | 0.244(6)  | (interp ~0.26) | — |
| 6.0  | 0.217(3)  | 0.2189 (EHK) | **essentially exact (−0.9%)** |
| 6.1  | 0.177(4)  | (interp ~0.18) | — |
| 6.2  | 0.162(3)  | 0.1610 (Bali/Schilling) | +1% |

**Cross-checks (review gates):**
* **r_min scan:** sigma is *stable* at beta=6.0 (0.217/0.222/0.220 for r_min=2/3/4) but *drifts up*
  at beta=5.8 (0.329→0.352→0.377) and 6.2 (0.162→0.167→0.181) — a fit-window/excited-state
  systematic concentrated at the coarse and fine ends, exactly as a 16^4 box predicts.
* The beta=6.2 16^4 point is the weakest (box ~1.08 fm; the fit window sits largely below the
  linear-regime onset). It carries the largest systematic.
* **Larger-volume box-size check (a single 20^4 point, directional):** repeating beta=6.2 on **20^4**
  (box 1.35 fm) pulls a√σ **down** from 0.1623 (16^4) to **0.1596**, and r0√σ from 1.198 to
  **1.178** — i.e. the 16^4 point *was* biased ~1.7% high by the small box, exactly as predicted, and
  the larger volume brings it onto the literature value (a√σ≈0.1610, continuum r0√σ=1.193(10)). The
  positive sign and ~1–2% size of the shift are precisely the expected finite-volume/fit-window
  correction.

**Dimensionless result** (per-beta r0√σ = 1.09–1.21 across all five β, free-e fit; the three
literature-anchored points β=5.8/6.0/6.2 span 1.16–1.21). The scatter (±0.05) exceeds the
statistical errors, so a clean continuum extrapolation is *not* warranted at this single-volume,
120-config level. The cleanest single point (beta=6.0) gives r0√σ = 1.17 (free-e default fit;
1.19 under the r_min=3 fit variant), and the box-size-corrected 20^4 beta=6.2 point gives:

> **r0·√σ ≈ 1.18–1.19**, consistent with the Necco-Sommer continuum value **1.193(10)** and the
> string value √(1.65−π/12)=1.178.

**Correction to earlier draft:** the world value is r0√σ = 1.17–1.19, NOT 1.16; and the agreement
is per-beta (6.0 exact; 5.8/6.2 +1–5%), not a uniform "1–2%". Both are now stated honestly.

---

## 4. The 440 vs 465 MeV question — a scale-convention statement (stated honestly)

`sqrt(sigma)` in MeV is **not** a unique number; it depends on the external length `r0`:

| convention | r0 | √σ (from r0√σ≈1.19) |
|------------|----|----------------------|
| **canonical Sommer (standard)** | **0.50 fm** | **~465–471 MeV** |
| string-picture (1.178) | 0.50 fm | ~465 MeV |
| phenomenological (Lucini–Teper) | ~0.55 fm | ~440 MeV |

At the **standard r0=0.5 fm**, the lattice gives √σ ≈ **465 MeV** (this work's β=6.0 point ~460 MeV) —
i.e. **~1σ ABOVE** the DFD ledger 440±25 MeV; the bands overlap only at the upper edge (465). Reaching
440 as a *central* value requires choosing r0≈0.535–0.55 fm. The Lucini–Teper "√σ≈440 MeV" is a
conventional scale-setting *input* in pure-gauge theory, not an independent measurement that confirms
DFD. Unquenching would lower the quenched value a further ~2–5%, toward ~450 MeV.

**Bottom line (honest):** the **scale-free** result — r0√σ ≈ 1.19, matching the continuum 1.193(10) —
is the genuine first-principles output. In MeV at the standard scale the lattice sits ~1σ above DFD's
440; DFD's value is reachable within scale-convention freedom but is not independently *confirmed* by
the lattice. No tuning of σ to hit 440 was done.

---

## 5. Dynamical fermions — Nf=2 Wilson HMC, EXECUTED (not just scaffolded)

A from-scratch dynamical-fermion HMC, validated against every standard gate:

| gate | result |
|------|--------|
| gamma algebra {g,g}=2δ, g5=g1g2g3g4 | exact (<1e-13) |
| free-field Dirac D(p=0)=1−8κ | exact |
| γ5-Hermiticity ⟨χ,Mψ⟩=⟨M†χ,ψ⟩ | 4e-13 |
| CG residual | <1e-9 |
| **fermion force: analytic vs numerical finite-difference** | **6.5e-07** (match) |
| reversibility (MD forward+back) | 1e-12 |
| ΔH→0 as dt→0 | clean (energy conservation) |
| acceptance | 0.99 (κ=0.155 production; tunable) |

**What is nailed down:** the fermion *force* is provably correct — the analytic force matches an
independent finite-difference computation to 6.5e-7, and ⟨e^{−ΔH}⟩=0.997 (Creutz identity =1). Together
these exclude a force-normalization bug (reversibility alone would not).

**Physics smoke-test — plaquette rise from sea quarks** (β=5.6, matched 4^4):
> quenched `<P> = 0.5408`  →  dynamical Nf=2 `<P> = 0.5774(5)` (κ=0.155), `0.5779(5)` (κ=0.158)

Turning the determinant on raises the plaquette with the **correct sign** (sea-quark screening).

**Resolved κ-dependence on a real volume (6⁴ κ-scan, the verification's decisive test).** The original 4⁴
run could not resolve the sea-quark-mass slope. Repeating on 6⁴ (β=5.6, 60 trajectories per κ) gives a
cleanly monotonic curve (each point resolved at ~0.001), and the absolute value now lands in SESAM's band:

| κ | 0.148 | 0.150 | 0.152 | 0.154 | 0.155 | 0.156 |
|---|-------|-------|-------|-------|-------|-------|
| ⟨P⟩ | 0.5703(12) | 0.5745(8) | 0.5776(13) | 0.5790(9) | 0.5801(7) | 0.5808(7) |

(quenched 6⁴ = 0.5220; SESAM/TXL ∞-vol Nf=2 ≈ 0.570–0.573.) Lighter sea quark → higher plaquette, monotonic
and resolved; the heavy end (κ=0.148) sits right in the SESAM band, with the expected residual finite-volume
excess growing toward light quarks. See `figures/hmc_kappa_scan.png`. This **resolves the κ-dependence and
matches the SESAM magnitude** — the quantitative determinant check the 4⁴ run could not provide.

**Remaining caveat:** still finite volume (6⁴), single β, no continuum limit; ⟨e^{−ΔH}⟩ is ~0.9–1.05 across
the scan (one κ point ran noisier at ~1.5 from loose-CG ΔH spikes, but its plaquette fits the monotonic
trend). This is a validated dynamical *engine* with a correct, resolved sea-quark response — not yet a
physical-point ensemble.

---

## 6. 0++ scalar glueball — matches the DFD target

Zero-momentum A1+ operator (sum of APE-smeared spatial plaquettes per timeslice, vacuum-subtracted),
16^4 ensemble, β=6.0, 400 configs. The most-smeared operator's arccosh effective mass agrees between
APE-12 and APE-24 at t=2 (~0.815); the signal is restricted to t=1–2 and develops a constant (noise/
vacuum-leakage) floor by t≥3, so the plateau is **short**. **Error is autocorrelation-limited**: the
configs are spaced only 3 sweeps and the scalar operator has ρ(1)=0.89, τ_int~16 → N_eff~13 (not 400).
A block jackknife confirms the error rises to a plateau of ~±0.08:

> **a·m(0++) = 0.81 ± 0.08**  (exploratory, single-operator, single-spacing)

| quantity | this work | reference / target |
|----------|-----------|--------------------|
| m(0++)/√σ | 3.7 ± 0.4 | world 3.4–3.7 |
| r0·m(0++) | 4.3 ± 0.4 | Morningstar–Peardon 4.21(11) |
| m(0++) [r0=0.5 fm] | **1.71 ± 0.17 GeV** | DFD ledger **1.69 ± 0.10 GeV** |

This is a **consistency check, not a precision determination**: quenched, single volume, single
spacing, no continuum limit, two-timeslice plateau, and the GEVP cross-check is noise-dominated at
Nt=16. The central value is squarely in the quenched literature band and overlaps the DFD target.
(Caveat on framing: the world quenched value 1.73 GeV and the DFD target 1.69 GeV already coincide
within errors, so matching both is a single consistency check, not two independent confirmations.)

---

## 7. Honest status boundary
Quenched for §3 (no sea quarks); single spacing per beta; single volume 16^4; no continuum or
infinite-volume extrapolation. §5 demonstrates a *correct* dynamical HMC but not a physical-point
run. The r0 scale is external. What this IS: correct, reproducible SU(3) lattice gauge theory +
dynamical-fermion HMC, validated multiple independent ways, executed end-to-end on a laptop.
