Manuscript XX-A@1 · XX-A.tex · section · 3a586110

A coefficient template is not an evaluated coefficient

Formal quantum coefficient and the exact Ward-repair contract › A coefficient template is not an evaluated coefficient · lines 1396–1434

XX-A@1#eq:quantum-template

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The complete classical insertion is (eq:observable-jet). Its homogeneous nonlinear part includes all of (eq:n2), not only $N_{2}$ .

A possible renormalized coefficient would have the form

O_{f}^{ren} = J_{R} [O_{0, f} + s F_{1, f} + \frac{s ^{2}}{2} F_{2, f}] + ℏ C_{f} (s) (mod s^{3}, ℏ^{2}),

where $J_{R}$ , its domain, and $C_{f}$ must actually be constructed. Equation (eq:quantum-template) is a specification of the target, not its definition by fiat.

If interactions are included with the amplitude normalization (eq:jet), their action expansion must also be retained. For compact variational arguments around the on-shell background,

s^{- 2} (S [g ˉ + s h, ϕ ˉ + s φ] - S [g ˉ, ϕ ˉ]) = \frac{1}{2} S^{(2)} (z, z) + \frac{s}{6} S^{(3)} (z, z, z) + \frac{s ^{2}}{24} S^{(4)} (z, z, z, z) + O (s^{3}) .

This is a local variational expansion, not a finite subtraction of infinite total background actions.

The cubic and quartic vertices, ghosts, gauge-fixing terms, observable vertices, and their prescribed retarded operations must enter any claimed interacting calculation at matching orders. A single free contraction of $F_{2, f}$ is not the complete interacting one-loop inventory.

Displayed formulas in this passage

XX-A@1#eq:quantum-template · line 1403
$O_{f}^{ren} = J_{R} [O_{0, f} + s F_{1, f} + \frac{s ^{2}}{2} F_{2, f}] + ℏ C_{f} (s) (mod s^{3}, ℏ^{2}),$
unlabeled · line 1419
$s^{- 2} (S [g ˉ + s h, ϕ ˉ + s φ] - S [g ˉ, ϕ ˉ]) = \frac{1}{2} S^{(2)} (z, z) + \frac{s}{6} S^{(3)} (z, z, z) + \frac{s ^{2}}{24} S^{(4)} (z, z, z, z) + O (s^{3}) .$