Manuscript Em01-D@2 · Em01-D-v2.tex · section · d3b49151

Patch and internal distributions

The complete retained contraction › Patch and internal distributions · lines 705–804

Em01-D@2#eq:Ksum, Em01-D@2#eq:K47, Em01-D@2#eq:K5plus, Em01-D@2#eq:K5minus, Em01-D@2#eq:K6, Em01-D@2#eq:K8, Em01-D@2#eq:K9der, Em01-D@2#eq:K9mass

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Work on

W = A \times B_{0} \times Z \times Y, w = (a, b, z, y),

with separated endpoint regions and

b^{0} > y^{0} > a^{0} > ε, z^{0} > ε, b^{0} > z^{0} .

The source switch is one throughout the retained variables.

Define

R (t, x) = \frac{Θ ( t )}{2 π} δ_{D} (t^{2} - ∣ x ∣^{2}), B (t, x) = \frac{1}{8 π} Θ (t) Θ (t^{2} - ∣ x ∣^{2}) .

Put

L^{μ, I} (s, a) = A^{λ μ, I} \partial_{a^{λ}} R (s - a),

Q^{L, I, J} (v; a, z) = ∣ α ∣ + ∣ β ∣ \leq 2 \sum q_{α β}^{L, I, J} (v) \partial_{v}^{α} δ_{D}^{(4)} (v - z) \partial_{v}^{β} δ_{D}^{(4)} (v - a),

T^{L, I, J} (b; a, z) = P^{L}_{M} \int R (b - v) Q^{M, I, J} (v; a, z) d^{4} v .

The fixed retained kernel is

K_{ret} = K_{47} + K_{5, +} + K_{5, -} + K_{6} + K_{8} + K_{9, der} + K_{9, mass},

where

K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9, der}^{I, J} K_{9, mass}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ_{D}^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)], = - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .

These formulas adopt the retained architecture supplied in Em01-B and Em01-C Em01B,Em01C. The seven rows are bookkeeping rows, not seven new interaction branches.

Displayed formulas in this passage

Em01-D@2#eq:Ksum · line 749
$K_{ret} = K_{47} + K_{5, +} + K_{5, -} + K_{6} + K_{8} + K_{9, der} + K_{9, mass},$
Em01-D@2#eq:K47 · line 757
$K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9, der}^{I, J} K_{9, mass}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ_{D}^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)], = - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$
Em01-D@2#eq:K5plus · line 757
$K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9, der}^{I, J} K_{9, mass}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ_{D}^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)], = - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$
Em01-D@2#eq:K5minus · line 757
$K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9, der}^{I, J} K_{9, mass}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ_{D}^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)], = - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$
Em01-D@2#eq:K6 · line 757
$K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9, der}^{I, J} K_{9, mass}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ_{D}^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)], = - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$
Em01-D@2#eq:K8 · line 757
$K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9, der}^{I, J} K_{9, mass}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ_{D}^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)], = - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$
Em01-D@2#eq:K9der · line 757
$K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9, der}^{I, J} K_{9, mass}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ_{D}^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)], = - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$
Em01-D@2#eq:K9mass · line 757
$K_{47}^{I, J} K_{5, +}^{I, J} K_{5, -}^{I, J} K_{6}^{I, J} K_{8}^{I, J} K_{9, der}^{I, J} K_{9, mass}^{I, J} = - \partial_{b^{μ}} [(f (b) \partial_{b^{ν}} L^{μ, J} (b, z) - δ^{μ}_{ν} \partial_{b^{ρ}} (f (b) L^{ρ, J} (b, z))) L^{ν, I} (b, a)], = \partial_{b^{μ}} [f (b) \partial_{a^{α}} R (b - a) A^{α β, I} \partial_{a^{β}} L^{μ, J} (a, z)], = \partial_{b^{μ}} [f (b) \partial_{z^{α}} R (b - z) A^{α β, J} \partial_{z^{β}} L^{μ, I} (z, a)], = \partial_{b^{μ}} [f (b) D^{α μ, I, J} \partial_{a^{α}} R (b - a) δ_{D}^{(4)} (a - z)], = - \partial_{b^{μ}} [f (b) A^{α μ}_{L} P^{L}_{M} \partial_{b^{α}} \int B (b - v) Q^{M, I, J} (v; a, z) d^{4} v], = - \partial_{b^{ν}} [g (b) \partial_{b^{μ}} F (b) A^{μν}_{L} T^{L, I, J} (b; a, z)], = - m^{2} g (b) F (b) b_{L} T^{L, I, J} (b; a, z) .$
unlabeled · line 708
$W = A \times B_{0} \times Z \times Y, w = (a, b, z, y),$
unlabeled · line 712
$b^{0} > y^{0} > a^{0} > ε, z^{0} > ε, b^{0} > z^{0} .$
unlabeled · line 719
$R (t, x) = \frac{Θ ( t )}{2 π} δ_{D} (t^{2} - ∣ x ∣^{2}), B (t, x) = \frac{1}{8 π} Θ (t) Θ (t^{2} - ∣ x ∣^{2}) .$
unlabeled · line 729
$L^{μ, I} (s, a) = A^{λ μ, I} \partial_{a^{λ}} R (s - a),$
unlabeled · line 734
$Q^{L, I, J} (v; a, z) = ∣ α ∣ + ∣ β ∣ \leq 2 \sum q_{α β}^{L, I, J} (v) \partial_{v}^{α} δ_{D}^{(4)} (v - z) \partial_{v}^{β} δ_{D}^{(4)} (v - a),$
unlabeled · line 742
$T^{L, I, J} (b; a, z) = P^{L}_{M} \int R (b - v) Q^{M, I, J} (v; a, z) d^{4} v .$

Nearby in this section

lemma: Lemma: Exact response reduction and finite order