Manuscript X-A@1 · X-A.tex · lemma · dc7f5c0a

Lemma: Expansion and insertion signs

Relational expansion and insertion signs · lines 1382–1409 · proof included or linked

X-A@1#lem:Phiexp, X-A@1#eq:Phi0, X-A@1#eq:Phi1, X-A@1#eq:Phi2, X-A@1#eq:A1, X-A@1#eq:A2

$Φ_{g, ε} = Φ^{(0)} + κ Φ^{(1)} + κ^{2} Φ^{(2)} (mod κ^{3})$ , with

Φ^{(0)} Φ^{(1)} Φ^{(2)} = ϕ_{0}, = ϕ_{1} - X_{1}^{μ} \partial_{μ} ϕ_{0}, = ϕ_{2} - X_{1}^{μ} \partial_{μ} ϕ_{1} + (X_{1}^{ν} \partial_{ν} X_{1}^{μ} - X_{2}^{μ}) \partial_{μ} ϕ_{0} + \frac{1}{2} X_{1}^{μ} X_{1}^{ν} \partial_{μ} \partial_{ν} ϕ_{0} .

For a $κ$ -independent scalar, $ϕ_{1} = ϕ_{2} = 0$ . For the on-shell evaluation $ϕ = ρ_{g} [q; ϕ_{0}]$ , the coefficients are

A_{1}^{(g, ε)} A_{2}^{(g, ε)} = ρ_{1, g} - X_{1}^{μ} \partial_{μ} ϕ_{0}, = ρ_{2, g} - X_{1}^{μ} \partial_{μ} ρ_{1, g} + (X_{1}^{ν} \partial_{ν} X_{1}^{μ} - X_{2}^{μ}) \partial_{μ} ϕ_{0} + \frac{1}{2} X_{1}^{μ} X_{1}^{ν} \partial_{μ} \partial_{ν} ϕ_{0} .

Proof. Taylor expansion of $ϕ (X_{q}^{- 1} (z))$ with (eq:inverse).

Displayed formulas in this passage

X-A@1#eq:Phi0 · line 1385
$Φ^{(0)} Φ^{(1)} Φ^{(2)} = ϕ_{0}, = ϕ_{1} - X_{1}^{μ} \partial_{μ} ϕ_{0}, = ϕ_{2} - X_{1}^{μ} \partial_{μ} ϕ_{1} + (X_{1}^{ν} \partial_{ν} X_{1}^{μ} - X_{2}^{μ}) \partial_{μ} ϕ_{0} + \frac{1}{2} X_{1}^{μ} X_{1}^{ν} \partial_{μ} \partial_{ν} ϕ_{0} .$
X-A@1#eq:Phi1 · line 1385
$Φ^{(0)} Φ^{(1)} Φ^{(2)} = ϕ_{0}, = ϕ_{1} - X_{1}^{μ} \partial_{μ} ϕ_{0}, = ϕ_{2} - X_{1}^{μ} \partial_{μ} ϕ_{1} + (X_{1}^{ν} \partial_{ν} X_{1}^{μ} - X_{2}^{μ}) \partial_{μ} ϕ_{0} + \frac{1}{2} X_{1}^{μ} X_{1}^{ν} \partial_{μ} \partial_{ν} ϕ_{0} .$
X-A@1#eq:Phi2 · line 1385
$Φ^{(0)} Φ^{(1)} Φ^{(2)} = ϕ_{0}, = ϕ_{1} - X_{1}^{μ} \partial_{μ} ϕ_{0}, = ϕ_{2} - X_{1}^{μ} \partial_{μ} ϕ_{1} + (X_{1}^{ν} \partial_{ν} X_{1}^{μ} - X_{2}^{μ}) \partial_{μ} ϕ_{0} + \frac{1}{2} X_{1}^{μ} X_{1}^{ν} \partial_{μ} \partial_{ν} ϕ_{0} .$
X-A@1#eq:A1 · line 1397
$A_{1}^{(g, ε)} A_{2}^{(g, ε)} = ρ_{1, g} - X_{1}^{μ} \partial_{μ} ϕ_{0}, = ρ_{2, g} - X_{1}^{μ} \partial_{μ} ρ_{1, g} + (X_{1}^{ν} \partial_{ν} X_{1}^{μ} - X_{2}^{μ}) \partial_{μ} ϕ_{0} + \frac{1}{2} X_{1}^{μ} X_{1}^{ν} \partial_{μ} \partial_{ν} ϕ_{0} .$
X-A@1#eq:A2 · line 1397
$A_{1}^{(g, ε)} A_{2}^{(g, ε)} = ρ_{1, g} - X_{1}^{μ} \partial_{μ} ϕ_{0}, = ρ_{2, g} - X_{1}^{μ} \partial_{μ} ρ_{1, g} + (X_{1}^{ν} \partial_{ν} X_{1}^{μ} - X_{2}^{μ}) \partial_{μ} ϕ_{0} + \frac{1}{2} X_{1}^{μ} X_{1}^{ν} \partial_{μ} \partial_{ν} ϕ_{0} .$