Manuscript Em01-E@2 · Em01-E-v2.tex · section · 54f8f8c2

Complete internal and vertex ledger

Evaluated complete degenerate specialization › Complete internal and vertex ledger · lines 1637–1737

Em01-E@2#eq:appK, Em01-E@2#eq:appC, Em01-E@2#eq:appexact

The tensor maps of Appendix app:kernels give

P e_{0} = - e_{0}, A^{μν, 0} = η^{μν}, b^{0} = 2.

Put

D (w) = R (b - a) δ_{D}^{(4)} (a - z) .

The source response is $T^{0, 0, 0} = - D$ . Therefore

K_{9, der}^{0, 0} = \partial_{b^{0}} D, K_{9, mass}^{0, 0} = 2 b^{0} D,

and all other components vanish. The first five internal rows vanish exactly because $f = 0$ . Thus

K_{ret}^{0, 0} = \partial_{b^{0}} D + 2 b^{0} D .

The actual convolution has the exact local value

(C_{δ})_{I}^{μν} (x_{1}, x_{2}) = δ_{I 0} η^{μν} δ^{- 2} p (\frac{κ ( x _{1} )}{δ}) p (\frac{κ ( x _{2} )}{δ}) .

Indeed the $κ$ -marginal of $ρ_{4}$ is $p$ . The product profile gives the two four-variable convolutions. For sufficiently small scales, the cutoff is invisible near $f (U)$ , including for derivatives.

Set

s = κ (y - a), t = κ (b - a),

A_{δ} = δ^{- 3} p (s / δ) p^{'} ((s - t) / δ), E_{δ} = δ^{- 2} p (s / δ) p ((s - t) / δ) .

The kinetic crossed term is $+ ℓ^{ν} A_{δ}$ , the trace-kinetic term is $- 2 ℓ^{ν} A_{δ}$ , and the trace is $4 E_{δ}$ .

The $ℓ$ -derivative annihilates both profile arguments. Here $K$ is independent of $y$ , and $ℓ^{ν} \partial_{ν} g (y) = 1$ . Consequently

t_{δ} = - i K_{ret}^{0, 0} A_{δ} - 2 i y^{0} K_{ret}^{0, 0} E_{δ} .

All six nonzero row–vertex cells are:

Internal row	Vertex type	Exact transposed contribution
$9_{der}$	Kinetic	$- i ⟨ D, \partial_{b^{0}} (A_{δ} Φ_{0})⟩$ .
$9_{der}$	Trace kinetic	$2 i ⟨ D, \partial_{b^{0}} (A_{δ} Φ_{0})⟩$ .
$9_{der}$	Mass	$2 i ⟨ D, y^{0} \partial_{b^{0}} (E_{δ} Φ_{0})⟩$ .
$9_{mass}$	Kinetic	$2 i ⟨ D, b^{0} A_{δ} Φ_{0} ⟩$ .
$9_{mass}$	Trace kinetic	$- 4 i ⟨ D, b^{0} A_{δ} Φ_{0} ⟩$ .
$9_{mass}$	Mass	$- 4 i ⟨ D, b^{0} y^{0} E_{δ} Φ_{0} ⟩$ .

The other fifteen cells vanish by the fixed data $f = 0$ . The second crossed term vanishes by (eq:applines). Neither vanishing is an asymptotic omission.

Expanding the $b^{0}$ -derivatives gives

t_{δ} (Φ) = - i ⟨ D, δ^{- 4} p (s / δ) p^{''} ((s - t) / δ) Φ_{0} + δ^{- 3} p (s / δ) p^{'} ((s - t) / δ) [- \partial_{b^{0}} Φ_{0} + 2 (b^{0} + y^{0}) Φ_{0}] - 2 δ^{- 2} p (s / δ) p ((s - t) / δ) y^{0} [\partial_{b^{0}} Φ_{0} - 2 b^{0} Φ_{0}] ⟩ .

Displayed formulas in this passage

Em01-E@2#eq:appK · line 1650
$K_{9, der}^{0, 0} = \partial_{b^{0}} D, K_{9, mass}^{0, 0} = 2 b^{0} D,$
Em01-E@2#eq:appC · line 1663
$(C_{δ})_{I}^{μν} (x_{1}, x_{2}) = δ_{I 0} η^{μν} δ^{- 2} p (\frac{κ ( x _{1} )}{δ}) p (\frac{κ ( x _{2} )}{δ}) .$
Em01-E@2#eq:appexact · line 1725
$t_{δ} (Φ) = - i ⟨ D, δ^{- 4} p (s / δ) p^{''} ((s - t) / δ) Φ_{0} + δ^{- 3} p (s / δ) p^{'} ((s - t) / δ) [- \partial_{b^{0}} Φ_{0} + 2 (b^{0} + y^{0}) Φ_{0}] - 2 δ^{- 2} p (s / δ) p ((s - t) / δ) y^{0} [\partial_{b^{0}} Φ_{0} - 2 b^{0} Φ_{0}] ⟩ .$
unlabeled · line 1640
$P e_{0} = - e_{0}, A^{μν, 0} = η^{μν}, b^{0} = 2.$
unlabeled · line 1646
$D (w) = R (b - a) δ_{D}^{(4)} (a - z) .$
unlabeled · line 1658
$K_{ret}^{0, 0} = \partial_{b^{0}} D + 2 b^{0} D .$
unlabeled · line 1677
$s = κ (y - a), t = κ (b - a),$
unlabeled · line 1680
$A_{δ} = δ^{- 3} p (s / δ) p^{'} ((s - t) / δ), E_{δ} = δ^{- 2} p (s / δ) p ((s - t) / δ) .$
unlabeled · line 1692
$t_{δ} = - i K_{ret}^{0, 0} A_{δ} - 2 i y^{0} K_{ret}^{0, 0} E_{δ} .$