Manuscript Em02-D@4 · Em02-D-v4.tex · theorem · fcf2fac7

Theorem: 5a–response coincidence cancellation

The 5z coefficient and complete leading assembly › Distributional inner-tip cancellation · lines 1813–1849 · proof included or linked

Em02-D@4#thm:tip-cancellation, Em02-D@4#eq:tip-cancellation

The leading coefficients satisfy

C_{5 a} + C_{resp} = \frac{i κ}{16 π} \int d^{4} a d R (s) d R (t) f (a + s) [Δ_{r} Q ϕ_{2} (a, s, r)]_{r = - t} .

This is a distributional identity including the inner vertex.

Proof. Lemma lem:fundamental gives

\partial_{r^{0}}^{2} R (- r) = Δ_{r} R (- r) + δ^{(4)} (r) .

Pair this identity with the smooth compact $r$ -weight $Q ϕ_{2} (a, s, r)$ , freezing $a, s$ . Then

\int d R (t) [\partial_{r^{0}}^{2} Q ϕ_{2}]_{r = - t} = Q ϕ_{2} (a, s, 0) + \int d R (t) [Δ_{r} Q ϕ_{2}]_{r = - t} .

The first term, multiplied by the prefactor of $C_{5 a}$ , is exactly $- C_{resp}$ . This proves (eq:tip-cancellation).

Displayed formulas in this passage

Em02-D@4#eq:tip-cancellation · line 1816
$C_{5 a} + C_{resp} = \frac{i κ}{16 π} \int d^{4} a d R (s) d R (t) f (a + s) [Δ_{r} Q ϕ_{2} (a, s, r)]_{r = - t} .$
unlabeled · line 1830
$\partial_{r^{0}}^{2} R (- r) = Δ_{r} R (- r) + δ^{(4)} (r) .$
unlabeled · line 1838
$\int d R (t) [\partial_{r^{0}}^{2} Q ϕ_{2}]_{r = - t} = Q ϕ_{2} (a, s, 0) + \int d R (t) [Δ_{r} Q ϕ_{2}]_{r = - t} .$