Manuscript XIII-B@1 · XIII-B.tex · definition · b8a5c0f5

Definition: Selected vertex

B1: the interaction functional and the printed order · lines 509–531

XIII-B@1#eq:vertex

The only interaction entry is

V_{g} [γ, ϕ] = - \frac{1}{2} \int g γ_{μν} T_{0}^{μν} [ϕ] d^{4} x = \frac{1}{2} \int g (a_{1}^{μν} [γ] \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} b_{1} [γ] ϕ^{2}) d^{4} x,

where

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

T_{0}^{μν} [ϕ] = \partial^{μ} ϕ \partial^{ν} ϕ - \frac{1}{2} η^{μν} ((\partial ϕ)^{2} - m^{2} ϕ^{2}) .

Its parameter $λ$ counts this interaction insertion only.

Displayed formulas in this passage

XIII-B@1#eq:vertex · line 511
$V_{g} [γ, ϕ] = - \frac{1}{2} \int g γ_{μν} T_{0}^{μν} [ϕ] d^{4} x = \frac{1}{2} \int g (a_{1}^{μν} [γ] \partial_{μ} ϕ \partial_{ν} ϕ - m^{2} b_{1} [γ] ϕ^{2}) d^{4} x,$
unlabeled · line 520
$a_{1}^{μν} [r] = \frac{1}{2} η^{μν} tr_{η} r - r^{μν}, b_{1} [r] = \frac{1}{2} tr_{η} r,$
unlabeled · line 524
$T_{0}^{μν} [ϕ] = \partial^{μ} ϕ \partial^{ν} ϕ - \frac{1}{2} η^{μν} ((\partial ϕ)^{2} - m^{2} ϕ^{2}) .$