Manuscript II@2 · II.tex · section · 50d64def

Fixed vector quartic

Comparison example: selected defects and contact quotients › Fixed vector quartic · lines 2613–2692

Use signature $(+ - - -)$ and waves $e^{- ik \cdot x}$ . Consider real antisymmetric $f_{ab c}$ and the fixed action

- \frac{1}{4} \int (F^{a}_{μν} + g f^{a}_{b c} A_{μ}^{b} A_{ν}^{c})^{2} d^{4} x .

Its quartic is fixed, not an arbitrary contact nuisance.

Put

C_{s} = e \sum f_{ab e} f_{c d e}, C_{t} = e \sum f_{a ce} f_{b d e}, C_{u} = e \sum f_{a d e} f_{b ce}, J_{ab c d} = C_{s} - C_{t} + C_{u} .

For transverse physical polarizations,

j_{ij} = (ϵ_{i} \cdot ϵ_{j}) (k_{i} - k_{j}) + 2 (ϵ_{i} \cdot k_{j}) ϵ_{j} - 2 (ϵ_{j} \cdot k_{i}) ϵ_{i} .

Direct differentiation gives

T_{ab c d} = - C_{s} K_{s} - C_{t} K_{t} - C_{u} K_{u},

where $E_{ij} = ϵ_{i} \cdot ϵ_{j}$ and

K_{s} K_{t} K_{u} = \frac{j _{12} \cdot j _{34}}{s _{12}} + E_{13} E_{24} - E_{14} E_{23}, = \frac{j _{13} \cdot j _{24}}{s _{13}} + E_{12} E_{34} - E_{14} E_{23}, = \frac{j _{14} \cdot j _{23}}{s _{14}} + E_{12} E_{34} - E_{13} E_{24} .

The longitudinal identity

j_{1 j}_{ϵ_{1} = k_{1}} = s_{1 j} ϵ_{j} - (k_{1} \cdot ϵ_{j}) (k_{1} + k_{j})

and conservation of the opposite current remove the exchange denominators.

Define

D = E_{34} ϵ_{2} \cdot (k_{3} - k_{4}) + E_{24} ϵ_{3} \cdot (k_{4} - k_{2}) + E_{23} ϵ_{4} \cdot (k_{2} - k_{3}) .

With $ϵ_{1} \mapsto k_{1} / μ$ , the three complete $K$ 's are $D / μ, - D / μ, D / μ$ . For example, the first is obtained from

ϵ_{2} \cdot j_{34} + (k_{1} \cdot ϵ_{3}) E_{24} - (k_{1} \cdot ϵ_{4}) E_{23} = D .

Thus

δ T_{ab c d} = - (D / μ) J_{ab c d} .

At momenta (eq:momenta) scaled by $μ > 0$ , take

ϵ_{1} = ϵ_{2} = (0, 1, 0, 0), ϵ_{3} = ϵ_{4} = (0, 0, 1, 0) .

Then $D = - 2 μ$ , hence $δ T = 2 J$ . For the sparse seed $f_{123} = a, f_{145} = b$ , the species channel $2345$ has $J_{2345} = ab$ , giving $2 ab$ .

This is a defect calculation for that fixed vector quartic. It does not establish survival against an arbitrary vector contact class, and it supplies no spin-two separator.

Displayed formulas in this passage

unlabeled · line 2617
$- \frac{1}{4} \int (F^{a}_{μν} + g f^{a}_{b c} A_{μ}^{b} A_{ν}^{c})^{2} d^{4} x .$
unlabeled · line 2624
$C_{s} = e \sum f_{ab e} f_{c d e}, C_{t} = e \sum f_{a ce} f_{b d e}, C_{u} = e \sum f_{a d e} f_{b ce}, J_{ab c d} = C_{s} - C_{t} + C_{u} .$
unlabeled · line 2631
$j_{ij} = (ϵ_{i} \cdot ϵ_{j}) (k_{i} - k_{j}) + 2 (ϵ_{i} \cdot k_{j}) ϵ_{j} - 2 (ϵ_{j} \cdot k_{i}) ϵ_{i} .$
unlabeled · line 2638
$T_{ab c d} = - C_{s} K_{s} - C_{t} K_{t} - C_{u} K_{u},$
unlabeled · line 2642
$K_{s} K_{t} K_{u} = \frac{j _{12} \cdot j _{34}}{s _{12}} + E_{13} E_{24} - E_{14} E_{23}, = \frac{j _{13} \cdot j _{24}}{s _{13}} + E_{12} E_{34} - E_{14} E_{23}, = \frac{j _{14} \cdot j _{23}}{s _{14}} + E_{12} E_{34} - E_{13} E_{24} .$
unlabeled · line 2653
$j_{1 j}_{ϵ_{1} = k_{1}} = s_{1 j} ϵ_{j} - (k_{1} \cdot ϵ_{j}) (k_{1} + k_{j})$
unlabeled · line 2661
$D = E_{34} ϵ_{2} \cdot (k_{3} - k_{4}) + E_{24} ϵ_{3} \cdot (k_{4} - k_{2}) + E_{23} ϵ_{4} \cdot (k_{2} - k_{3}) .$
unlabeled · line 2670
$ϵ_{2} \cdot j_{34} + (k_{1} \cdot ϵ_{3}) E_{24} - (k_{1} \cdot ϵ_{4}) E_{23} = D .$
unlabeled · line 2676
$δ T_{ab c d} = - (D / μ) J_{ab c d} .$
unlabeled · line 2681
$ϵ_{1} = ϵ_{2} = (0, 1, 0, 0), ϵ_{3} = ϵ_{4} = (0, 0, 1, 0) .$