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

Input statement: Exact regrouping and root

F0: copied analytic inputs › XIV-E: the exact successor package · lines 339–391

XIV-F@1#input:regroup, XIV-F@1#eq:CG, XIV-F@1#eq:Btensor, XIV-F@1#eq:regroup, XIV-F@1#eq:root

With

X_{G} = X [G], X_{h} = X [h], r_{h} = r_{D}^{'} [G] h = P E_{0}^{R} (J_{D}^{'} [G] h),

put

C_{G}^{μ}_{ν} = f \partial_{ν} X_{G}^{μ} - δ^{μ}_{ν} \partial_{ρ} (f X_{G}^{ρ}),

and

B^{α μ} [G; h] = a_{1}^{α β} [h] \partial_{β} X_{G}^{μ} + a_{1}^{α β} [G] \partial_{β} X_{h}^{μ} + a_{2} ˙^{α μ} [G; h] + a_{1}^{α μ} [r_{h}] .

The exact six-branch identity is

M_{ret} [G; h, ψ] = \int C_{G}^{μ}_{ν} X_{h}^{ν} \partial_{μ} ψ d^{4} x + \int f E_{0}^{R} (\partial_{α} (σ_{ε} B^{α μ} [G; h])) \partial_{μ} ψ d^{4} x + \int f p [r_{h}; ψ] d^{4} x .

For $F = E_{m}^{A} f$ , the last term is

M_{9} [G; h, ψ] = \int g (a_{1}^{μν} [r_{h}] \partial_{μ} F \partial_{ν} ψ - m^{2} b_{1} [r_{h}] F ψ) d^{4} x .

“This identity retains all six branches. It involves no characteristic composition and no regularization.”

“Both the derivative and mass terms are retained” XIVE.

Displayed formulas in this passage

XIV-F@1#eq:CG · line 347
$C_{G}^{μ}_{ν} = f \partial_{ν} X_{G}^{μ} - δ^{μ}_{ν} \partial_{ρ} (f X_{G}^{ρ}),$
XIV-F@1#eq:Btensor · line 354
$B^{α μ} [G; h] = a_{1}^{α β} [h] \partial_{β} X_{G}^{μ} + a_{1}^{α β} [G] \partial_{β} X_{h}^{μ} + a_{2} ˙^{α μ} [G; h] + a_{1}^{α μ} [r_{h}] .$
XIV-F@1#eq:regroup · line 365
$M_{ret} [G; h, ψ] = \int C_{G}^{μ}_{ν} X_{h}^{ν} \partial_{μ} ψ d^{4} x + \int f E_{0}^{R} (\partial_{α} (σ_{ε} B^{α μ} [G; h])) \partial_{μ} ψ d^{4} x + \int f p [r_{h}; ψ] d^{4} x .$
XIV-F@1#eq:root · line 378
$M_{9} [G; h, ψ] = \int g (a_{1}^{μν} [r_{h}] \partial_{μ} F \partial_{ν} ψ - m^{2} b_{1} [r_{h}] F ψ) d^{4} x .$
unlabeled · line 342
$X_{G} = X [G], X_{h} = X [h], r_{h} = r_{D}^{'} [G] h = P E_{0}^{R} (J_{D}^{'} [G] h),$