Manuscript Em01-F@3 · Em01-F-v3.tex · section · 225e59af

Coupling, quantum power, and multiplicity

One action and its normalization dictionary › Coupling, quantum power, and multiplicity · lines 890–934

The three counts are distinct:

Quantity	Convention in this paper
Cubic Einstein–Hilbert action coefficient	$λ \int ℓ_{3}$ ; one coupling power.
Marked insertion	$O_{G}^{ret} = λ^{2} B_{f, g} (γ, G, ϕ)$ ; the prefactor is part of its definition.
Matter vertex	$λ V_{g}$ ; one coupling power.
Two selected crossed lines	One graviton line and one scalar line; each contains $ℏ$ .
Selected numerical family	$i$ times the contraction difference, with multiplicity one: overall degree $λ^{3} ℏ^{2}$ .
Response normalization	No $1/ℏ$ , $1/ (i ℏ)$ , or other response prefactor is included.
Loop classification	Not assigned by the coupling count or by this numerical definition.

For different species, the mixed term in $\frac{1}{2} (Γ_{γ} + Γ_{ϕ})^{2}$ is $Γ_{γ} Γ_{ϕ}$ . The two species assignments cancel the factor $1/2$ . The mixed matter derivative already includes the choice of either scalar field in its quadratic stress. No further factor of two is added.

Thus a later definition with an additional $1/ℏ$ response normalization would have a different quantum degree. It would not be the family (eq:apptarget). The distinction between contraction expansion and interacting response normalization is also explicit in the standard functional framework FR (Sections 5–6).