Manuscript VII-D@1 · VII-D.tex · proposition · ef67c89c

Proposition: Regulator-independent ultraviolet decomposition

Replacement and the complete ultraviolet split › The ultraviolet decomposition · lines 1561–1631 · proof included or linked

VII-D@1#prop:UV, VII-D@1#eq:UV, VII-D@1#eq:Eerror, VII-D@1#eq:errorfirst, VII-D@1#eq:errorsecond, VII-D@1#eq:scalarreplacement

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The exact family has the decomposition

T_{ε}^{ρ} = S + \int_{1}^{\infty} ϑ (ε r)^{4} A (r) \frac{d r}{r} + E_{ε}^{ρ},

where $S, A (r)$ are independent of $ρ$ , and

∣ ⟨ E_{ε}^{ρ}, Φ ⟩ ∣ \leq C_{K, ρ, m} ε (1 + ∣ lo g ε ∣) p_{K, 64} (Φ), 0 < ε \leq \frac{1}{2} .

Proof. The ledger identifies the leading coefficient with $r$ , and Lemma A gives an integrable radial remainder $A (r) - r$ . For the finite-cutoff identity, subtract the radial term in (eq:UV) from (eq:rawsplit). The exact error is the sum of $S_{ε}^{ρ} - S$ and

\int_{1}^{\infty} M_{ε} (r)^{2} (A_{ε}^{ρ} (r) - ϑ (ε r)^{2} A (r)) \frac{d r}{r}, \int_{1}^{\infty} ϑ (ε r)^{2} (M_{ε} (r)^{2} - ϑ (ε r)^{2}) A (r) \frac{d r}{r} .

The first difference is $O (ε)$ by Lemma lem:S. For (eq:errorfirst), Lemma lem:replacement and

∣ M_{ε} (r) ∣ \leq C_{ρ, N} (1 + ε r)^{- N}

give the bound

C_{K, ρ, m} ε p_{K, 64} (Φ) \int_{1}^{\infty} (1 + ε r)^{- N} \frac{d r}{r} \leq C^{'}_{K, ρ, m} ε (1 + ∣ lo g ε ∣) p_{K, 64} (Φ) .

It is the scalar multiplier that supplies this large- $r$ decay.

The scalar replacement is a separate identity and estimate:

r^{2} + m^{2} /2 - r ∣ M_{ε} (r) - ϑ (ε r) ∣ = \frac{m ^{2} /2}{r ^{2} + m ^{2} /2 + r} \leq \frac{m ^{2}}{4 r}, \leq \frac{m ^{2}}{4} ∥ ϑ^{'} ∥_{\infty} ε r^{- 1} .

The boundedness of the profiles and of $A (r)$ therefore makes (eq:errorsecond) $O (ε)$ , since $\int_{1}^{\infty} r^{- 2} d r < \infty$ . All estimates use the same compact-test seminorm.

Displayed formulas in this passage

VII-D@1#eq:UV · line 1564
$T_{ε}^{ρ} = S + \int_{1}^{\infty} ϑ (ε r)^{4} A (r) \frac{d r}{r} + E_{ε}^{ρ},$
VII-D@1#eq:Eerror · line 1574
$∣ ⟨ E_{ε}^{ρ}, Φ ⟩ ∣ \leq C_{K, ρ, m} ε (1 + ∣ lo g ε ∣) p_{K, 64} (Φ), 0 < ε \leq \frac{1}{2} .$
VII-D@1#eq:errorfirst · line 1588
$\int_{1}^{\infty} M_{ε} (r)^{2} (A_{ε}^{ρ} (r) - ϑ (ε r)^{2} A (r)) \frac{d r}{r}, \int_{1}^{\infty} ϑ (ε r)^{2} (M_{ε} (r)^{2} - ϑ (ε r)^{2}) A (r) \frac{d r}{r} .$
VII-D@1#eq:errorsecond · line 1588
$\int_{1}^{\infty} M_{ε} (r)^{2} (A_{ε}^{ρ} (r) - ϑ (ε r)^{2} A (r)) \frac{d r}{r}, \int_{1}^{\infty} ϑ (ε r)^{2} (M_{ε} (r)^{2} - ϑ (ε r)^{2}) A (r) \frac{d r}{r} .$
VII-D@1#eq:scalarreplacement · line 1616
$r^{2} + m^{2} /2 - r ∣ M_{ε} (r) - ϑ (ε r) ∣ = \frac{m ^{2} /2}{r ^{2} + m ^{2} /2 + r} \leq \frac{m ^{2}}{4 r}, \leq \frac{m ^{2}}{4} ∥ ϑ^{'} ∥_{\infty} ε r^{- 1} .$
unlabeled · line 1603
$∣ M_{ε} (r) ∣ \leq C_{ρ, N} (1 + ε r)^{- N}$
unlabeled · line 1607
$C_{K, ρ, m} ε p_{K, 64} (Φ) \int_{1}^{\infty} (1 + ε r)^{- N} \frac{d r}{r} \leq C^{'}_{K, ρ, m} ε (1 + ∣ lo g ε ∣) p_{K, 64} (Φ) .$