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

Exact branch integrals

Joint Fourier estimates and convergent sectors › Exact branch integrals · lines 696–755

VII-D@1#eq:substitution, VII-D@1#eq:Qbranches, VII-D@1#eq:jointbranches

Set

a_{D} (t, P) = - \frac{t}{4 P} + \frac{i}{8 P ^{2}}, a_{R} (t, P) = - \frac{i}{8 P ^{2}}, ϵ_{D} = - 1, ϵ_{R} = 1.

Throughout the momentum calculation we use the single substitution

p = α - q, p^{'} = β - q, P = ∣ α - q ∣, P^{'} = ∣ β - q ∣.

For $A, B \in {D, R}$ , define

Q_{ε}^{A B} (q; Φ) = \int d t d s d t^{'} d^{3} y \int \frac{d ^{3} α d ^{3} β}{( 2 π ) ^{6}} ϑ (εP) ϑ (ε P^{'}) \times a_{A} (t, P) a_{B} (t^{'}, P^{'}) e^{i [(ϵ_{A} P - ω_{m} (q)) t - (ϵ_{B} P^{'} - ω_{m} (q)) t^{'} + (P - P^{'}) s]} H_{Φ} (t, s, t^{'}, y; α, β) .

For $ε = 0$ , the two profile factors are replaced by one. For each fixed $q$ , the integral is absolutely convergent after taking the joint spatial transform.

The exact positive-cutoff identity is

⟨ T_{ε}^{ρ}, Φ ⟩ = \frac{1}{( 2 π ) ^{3}} \int_{R^{3}} ω_{m} (q) M_{ε} (∣ q ∣)^{2} A, B \in {D, R} \sum Q_{ε}^{A B} (q; Φ) d^{3} q .

The spatial signs follow from

p \cdot (x - y) - p^{'} \cdot (x^{'} - y) + q \cdot (x - x^{'}) = α \cdot (x - y) - β \cdot (x^{'} - y) .

Define $S_{ε}^{ρ}$ by taking all four branches in (eq:jointbranches) on $∣ q ∣ \leq 1$ , and only $D D, D R, R D$ on $∣ q ∣ > 1$ . Define $S$ by the same prescription with all multipliers replaced by one. The high- $∣ q ∣$ integrals defining $S$ are understood after performing the compact time integrations in (eq:Qbranches).

Displayed formulas in this passage

VII-D@1#eq:substitution · line 705
$p = α - q, p^{'} = β - q, P = ∣ α - q ∣, P^{'} = ∣ β - q ∣.$
VII-D@1#eq:Qbranches · line 711
$Q_{ε}^{A B} (q; Φ) = \int d t d s d t^{'} d^{3} y \int \frac{d ^{3} α d ^{3} β}{( 2 π ) ^{6}} ϑ (εP) ϑ (ε P^{'}) \times a_{A} (t, P) a_{B} (t^{'}, P^{'}) e^{i [(ϵ_{A} P - ω_{m} (q)) t - (ϵ_{B} P^{'} - ω_{m} (q)) t^{'} + (P - P^{'}) s]} H_{Φ} (t, s, t^{'}, y; α, β) .$
VII-D@1#eq:jointbranches · line 730
$⟨ T_{ε}^{ρ}, Φ ⟩ = \frac{1}{( 2 π ) ^{3}} \int_{R^{3}} ω_{m} (q) M_{ε} (∣ q ∣)^{2} A, B \in {D, R} \sum Q_{ε}^{A B} (q; Φ) d^{3} q .$
unlabeled · line 699
$a_{D} (t, P) = - \frac{t}{4 P} + \frac{i}{8 P ^{2}}, a_{R} (t, P) = - \frac{i}{8 P ^{2}}, ϵ_{D} = - 1, ϵ_{R} = 1.$
unlabeled · line 739
$p \cdot (x - y) - p^{'} \cdot (x^{'} - y) + q \cdot (x - x^{'}) = α \cdot (x - y) - β \cdot (x^{'} - y) .$

Exact branch integrals

Displayed formulas in this passage

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