Manuscript X-F-ST@1 · X-F-ST.tex · section · e71f03ef

Definitions without duplicate prefactors

F3: three response kernels, support, and admission › Definitions without duplicate prefactors · lines 1469–1519

X-F-ST@1#eq:trees, X-F-ST@1#eq:alltree

Let

Θ = : T_{0} [ϕ] :, k = Δ_{gr}^{R} (\frac{1}{2} g Θ - σ Q^{(1)} [h] r - \frac{1}{2} σ : G_{2} [r, r] :) .

This is an auxiliary derived response, defined after compact smearing by transposition. It is not an independent free field or a full interacting metric.

Define the three coefficient functionals

T_{mat} T_{mix} T_{self} = : A_{1} [Δ_{gr}^{R} (\frac{1}{2} g Θ)] :, = : A_{1} [Δ_{gr}^{R} (- σ Q^{(1)} [h] r)] :, = : A_{1} [Δ_{gr}^{R} (- \frac{1}{2} σ G_{2} [r, r])] : .

All scalar and tensor letters are totally Wick ordered. The degrees in $(ϕ, r)$ are respectively $(3, 0), (1, 1), (1, 2)$ . For the mixed term zero cross blocks make its Wick ordering the ordinary scalar–tensor product.

Use the all-tree expression

Φ_{F} [h, k] = ϕ (f) + κ A_{1} [h + r] + κ^{2} {Q [h + r, h + r] + A_{1} [k] + T_{mat} + T_{mix} + T_{self}} .

By (eq:affine), it is precisely the normal polynomial obtained from the completed tests at

q = η + κ (h + r) + κ^{2} (k + k)

through this formal order. No full scalar Bogoliubov map is applied a second time.

Displayed formulas in this passage

X-F-ST@1#eq:trees · line 1484
$T_{mat} T_{mix} T_{self} = : A_{1} [Δ_{gr}^{R} (\frac{1}{2} g Θ)] :, = : A_{1} [Δ_{gr}^{R} (- σ Q^{(1)} [h] r)] :, = : A_{1} [Δ_{gr}^{R} (- \frac{1}{2} σ G_{2} [r, r])] : .$
X-F-ST@1#eq:alltree · line 1502
$Φ_{F} [h, k] = ϕ (f) + κ A_{1} [h + r] + κ^{2} {Q [h + r, h + r] + A_{1} [k] + T_{mat} + T_{mix} + T_{self}} .$
unlabeled · line 1472
$Θ = : T_{0} [ϕ] :, k = Δ_{gr}^{R} (\frac{1}{2} g Θ - σ Q^{(1)} [h] r - \frac{1}{2} σ : G_{2} [r, r] :) .$
unlabeled · line 1514
$q = η + κ (h + r) + κ^{2} (k + k)$