Manuscript XIV-F@1 · XIV-F.tex · input · bae5a40b

Input statement: Completion and response formulas

F0: copied analytic inputs › XII-E and XII-C: completion and source conventions · lines 608–641

XIV-F@1#input:XIIE

The slot identity gives

Φ^{(2)}_{D} = A_{2}^{sc} [f, g] (γ, 0; ϕ) - C_{3, D}, r_{D} = - u_{D} = P E_{0}^{R} J_{D} .

The first response is

X^{μ} [r] = - E_{0}^{R} \partial_{α} (σ_{ε} a_{1}^{α μ} [r]), p [r; ψ] = - E_{m}^{R} (V_{1, g} [r] ψ),

where

V_{1, g} [r] ψ = \partial_{μ} (g a_{1}^{μν} [r] \partial_{ν} ψ) + m^{2} g b_{1} [r] ψ,

a_{1}^{μν} [r] = \frac{1}{2} η^{μν} tr_{η} r - r^{μν}, b_{1} [r] = \frac{1}{2} tr_{η} r .

The response roots satisfy

A_{1} [f, g] (r; ϕ) = \int α_{f, g} [ϕ] \cdot r d^{4} x, β_{f, g} [ϕ] = - E_{0}^{A} (P α_{f, g} [ϕ]) .

The completion kernels are admitted tree kernels XIIE.

Displayed formulas in this passage

unlabeled · line 611
$Φ^{(2)}_{D} = A_{2}^{sc} [f, g] (γ, 0; ϕ) - C_{3, D}, r_{D} = - u_{D} = P E_{0}^{R} J_{D} .$
unlabeled · line 617
$X^{μ} [r] = - E_{0}^{R} \partial_{α} (σ_{ε} a_{1}^{α μ} [r]), p [r; ψ] = - E_{m}^{R} (V_{1, g} [r] ψ),$
unlabeled · line 623
$V_{1, g} [r] ψ = \partial_{μ} (g a_{1}^{μν} [r] \partial_{ν} ψ) + m^{2} g b_{1} [r] ψ,$
unlabeled · line 628
$a_{1}^{μν} [r] = \frac{1}{2} η^{μν} tr_{η} r - r^{μν}, b_{1} [r] = \frac{1}{2} tr_{η} r .$
unlabeled · line 634
$A_{1} [f, g] (r; ϕ) = \int α_{f, g} [ϕ] \cdot r d^{4} x, β_{f, g} [ϕ] = - E_{0}^{A} (P α_{f, g} [ϕ]) .$