Operational DRED rules (p14r)

Builder/output: dred.py, P14_dred_rules.json; quantum_prescription_defined requires dred_rules_verified, and the release gate recomputes it.

R1, spaces.

The quasi-4-dimensional space Q4S splits as $\mathrm{Q}d\mathrm{S}\oplus \mathrm{Q}(4-d)\mathrm{S}$: $g^{(4)}=\hat{g}+\tilde{g}$, with $\mathrm{tr}{g}^{(4)}=4$, $\mathrm{tr}\hat{g}=d$ and $\mathrm{tr}\tilde{g}=4-d$. The metrics are projectors: $\hat{g}\hat{g}=\hat{g}$, $\tilde{g}\tilde{g}=\tilde{g}$, $\hat{g}\tilde{g}=0$, $g^{(4)}X=X$.

R2, momenta.

The loop momentum $l$ and the external momenta $p$ lie in Q$d$S ($p\cdot \tilde{g}=0$; a stated choice). Field, vertex and polarisation indices are Q4S indices.

R3, lifting.

The exported numerators are polynomials in the lower components $l_r,p_r$ ($r=\mathrm{0..3}$) at fixed external components. They are lifted as $l_r:=l\cdot e_{(r)}=l\cdot {\hat{e}}_{(r)}$ and $p_r=p\cdot e_{(r)}$, with $e_{(r)}$ the $\eta$-orthonormal basis of Q4S. The vertex tensors are lifted with $\eta \to g^{(4)}$, so nothing is truncated.

R4, reduction.

$\int l^{{\mu }_1}\cdots l^{{\mu }_n}F(l^2,l\cdot p)={\sum }_k{C}_k{B}_k$, with $B_k$ the symmetric tensors built from $\hat{g}$ and $p$, and $C_k$ given by the exact Gram (Passarino–Veltman) projection in symbolic $d$. Contracting with the lifted basis gives ${\hat{g}}_{rs}={\eta }_{rs}-{\tilde{G}}_{rs}$, where the evanescent ${\tilde{G}}_{rs}$ satisfy ${\eta }^{rs}{\tilde{G}}_{rs}=4-d$ and ${\tilde{G}}_{rs}{p}^s=0$. The checked closed forms are: the rank-2 tadpole $\hat{g}J/d$; the rank-4 tadpole $J/(d(d+2))$; $B_{00}=(J_{10}{p}^2-J_{02})/((d-1)p^2)$; and $B_{11}=(d{J}_{02}-p^2{J}_{10})/((d-1)p^4)$.

R5, order.

Every index contraction, including the external 4D projection, is done before the Laurent expansion. $d$ stays symbolic through the $\overline{\mathrm{MS}}$ subtraction and the finite-part extraction, so $(d-4)\times$pole terms are retained.

R6, evanescent.

Evanescent remainders do arise in general. The lift of the exported integrands has $\tilde{G}$ parts in every graph, and contraction produces $(4-d)\times$pole finite terms. They are kept. The p14 sentence claiming that none arise is withdrawn.

{Tests (generic placeholders $J=a/ϵ+b+cϵ$, $d=4-2ϵ$; no integral of this theory is evaluated).}

• T1, the ${\eta }_{00}$ part of $\int l_0{l}_{0}F$, has finite part $a/8+b/4$. Substituting $d=4$ before the expansion would give $b/4$; the guarded pipeline raises PrematureD4Substitution.

• T2, ${\eta }^{rs}\times$ the $\tilde{G}$ part, equals $-(4-d)J/d$ with finite part $-a/2$, a $(d-4)\times$pole term.

The mutations $\mathrm{tr}\hat{g}=4$ (rules check fails; T1 becomes $b/4$ and T2 becomes 0) and expand-before-contract (UncontractedObjects) are both rejected. Background literature (nothing is imported from it): W. Siegel, Phys.\ Lett.\ B84 (1979) 193 and B94 (1980) 37; D. Stöckinger, JHEP 0503 (2005) 076, hep-ph/0503129; A. Signer and D. Stöckinger, Nucl.\ Phys.\ B808 (2009) 88, arXiv:0807.4424.