Manuscript XVI-P@p23 · XVI-P.tex · section · 9b0037a7

4D poles: coordinates and the transversality constraint

Scope of revision p17: the hh evanescent poles and the ch slot › 4D poles: coordinates and the transversality constraint · lines 679–694

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The five p14r $h \cdot h$ directions span the five pair-symmetric 4-derivative 4D structures (rank 5), so span membership is automatic. The sharper statement is a constraint. Longitudinality $p_{m} G^{(mn) (r s)} = 0$ imposes rank-3 linear conditions on the coordinates $z$ , which leaves a 2-dimensional transverse subspace with basis $k_{1} = (\frac{1}{2}, - \frac{1}{2}, - 1, 1, 0)$ and $k_{2} = (1, 0, - 2, 0, 1)$ . The individual pole coordinates are not transverse:

z_{D 01} = i (- \frac{27}{160}, - \frac{71}{240}, \frac{19}{60}, \frac{11}{20}, - \frac{13}{30}), z_{D 03} = i (\frac{7}{96}, \frac{1}{24}, - \frac{1}{8}, - \frac{1}{24}, \frac{1}{12}) .

Their sum is transverse:

z_{D 01 + D 03} = i (- \frac{23}{240}, - \frac{61}{240}, \frac{23}{120}, \frac{61}{120}, - \frac{7}{20}) = i (\frac{61}{120} k_{1} - \frac{7}{20} k_{2}) .

This is the Ward-type statement the ch identity needs at this order. It is incomplete without D11.

Displayed formulas in this passage

unlabeled · line 684
$z_{D 01} = i (- \frac{27}{160}, - \frac{71}{240}, \frac{19}{60}, \frac{11}{20}, - \frac{13}{30}), z_{D 03} = i (\frac{7}{96}, \frac{1}{24}, - \frac{1}{8}, - \frac{1}{24}, \frac{1}{12}) .$
unlabeled · line 689
$z_{D 01 + D 03} = i (- \frac{23}{240}, - \frac{61}{240}, \frac{23}{120}, \frac{61}{120}, - \frac{7}{20}) = i (\frac{61}{120} k_{1} - \frac{7}{20} k_{2}) .$