E. Notations from section 6

In this Appendix we list the concrete form of the various notations made in section 6.

The polynomials denoted by $ \bar{X}_{p}^{\left( i\right) }$ that enter $ \bar{\Delta}^{\mathrm{int}}$ given in (137) read as

$\displaystyle \bar{X}_{0}^{\left( 1\right) }$ $\displaystyle =$ $\displaystyle 6S^{\ast }\eta C+12t^{\ast \mu }\left(
V_{\mu }C+\eta C_{\mu }\ri...
... }^{\ast }C+V_{[\mu
}C_{\nu ]}-\phi _{\mu \nu }\eta \right) F^{\mu \nu } \notag$ (366)
    $\displaystyle -2\left( 2\eta _{\mu \nu \rho }^{\ast }C+2B_{[\mu \nu }^{\ast }C_...
\tilde{D}_{\lambda \sigma }\varepsilon ^{\mu \nu \rho \lambda \sigma },$ (367)

$\displaystyle \bar{X}_{1}^{\left( 1\right) }$ $\displaystyle =$ $\displaystyle \left[ \left( -2C_{\mu \nu \rho }^{\ast
}\eta -2C_{[\mu \nu }^{\ast }V_{\rho ]}-4H_{[\mu }^{\ast }B_{\nu \rho
]}^{\ast }\right) C\right. \notag$ (368)
    $\displaystyle \left. +\left( -2H_{[\mu }^{\ast }V_{\nu }C_{\rho ]}+2H_{[\mu }^{...
\tilde{D}_{\lambda \sigma }\varepsilon ^{\mu \nu \rho \lambda \sigma }
\notag$ (369)
    $\displaystyle -12H_{\mu }^{\ast }t^{\ast \mu }\eta C+6\left( H_{[\mu }^{\ast }V...
...+2H_{\mu }^{\ast }\eta C_{\nu }+C_{\mu \nu }^{\ast }\eta C\right) F^{\mu
\nu },$ (370)

$\displaystyle \bar{X}_{2}^{\left( 1\right) }$ $\displaystyle =$ $\displaystyle \left[ \left( -2H_{[\mu }^{\ast }C_{\nu
\rho ]}^{\ast }\eta -2H_{[\mu }^{\ast }H_{\nu }^{\ast }V_{\rho ]}\right)
C\right. \notag$ (371)
    $\displaystyle \left. +2H_{[\mu }^{\ast }H_{\nu }^{\ast }C_{\rho ]}\eta \right] ...
... \nu \rho \lambda \sigma }+6H_{\mu
}^{\ast }H_{\nu }^{\ast }\eta CF^{\mu \nu },$ (372)

$\displaystyle \bar{X}_{3}^{\left( 1\right) }=4H_{\mu }^{\ast }H_{\nu }^{\ast }H_{\rho }^{\ast }\eta CD^{\mu \nu \rho },$ (373)

$\displaystyle \bar{X}_{0}^{\left( 2\right) }$ $\displaystyle =$ $\displaystyle -12\cdot 5!\left( S^{\ast }\eta +2t^{\ast
\mu }V_{\mu }+2B_{\mu \...
... }\tilde{D}_{\lambda \sigma }\mathcal{G}
^{\mu \nu \rho \lambda \sigma } \notag$ (374)
    $\displaystyle +4!\cdot 4!t^{\ast \mu }\eta \mathcal{\tilde{G}}_{\mu }-4!\cdot 4...
...\mu \nu \rho \lambda
}+6\cdot 4!\left( \phi ^{\ast \mu \nu }\eta \right. \notag$ (375)
    $\displaystyle \left. -K^{\mu \nu \rho }V_{\rho }\right) \tilde{D}_{\mu \nu }-3\...
...e{K}_{\mu \nu }\eta -4V_{[\mu }\mathcal{\tilde{G}}_{\nu
]}\right) F^{\mu \nu },$ (376)

$\displaystyle \bar{X}_{1}^{\left( 2\right) }$ $\displaystyle =$ $\displaystyle -4\cdot 5!\left( C_{\mu \nu \rho }^{\ast
}\eta +C_{[\mu \nu }^{\a...
...t) \tilde{D}_{\lambda \sigma }\mathcal{G}^{\mu \nu \rho
\lambda \sigma } \notag$ (377)
    $\displaystyle -12\cdot 5!\left( C_{\mu \nu }^{\ast }F^{\mu \nu }\eta -2H_{\mu }...
...-12\cdot 4!H_{[\mu }^{\ast }\mathcal{\tilde{G}}_{\nu ]}\eta
F^{\mu \nu } \notag$ (378)
    $\displaystyle -12\cdot 4!\left( C_{\mu \nu }^{\ast }\eta +H_{[\mu }^{\ast }V_{\...
}-6\cdot 4!H_{\mu }^{\ast }K^{\mu \nu \rho }\eta \tilde{D}_{\nu \rho },$ (379)

$\displaystyle \bar{X}_{2}^{\left( 2\right) }$ $\displaystyle =$ $\displaystyle -4\cdot 5!\left( H_{[\mu }^{\ast }C_{\nu
\rho ]}^{\ast }\eta +H_{...
\tilde{D}_{\lambda \sigma }\mathcal{G}^{\mu \nu \rho \lambda \sigma } \notag$ (380)
    $\displaystyle -12\cdot 4!H_{\mu }^{\ast }H_{\nu }^{\ast }\eta \tilde{D}_{\rho \...
...12\cdot 5!H_{\mu }^{\ast }H_{\nu }^{\ast
}\eta F^{\mu \nu }\mathcal{\tilde{G}},$ (381)

$\displaystyle \bar{X}_{3}^{\left( 2\right) }=-4\cdot 5!H_{\mu }^{\ast }H_{\nu }...
...st }\eta \tilde{D}_{\lambda \sigma }\mathcal{G}^{\mu \nu \rho \lambda \sigma },$ (382)

$\displaystyle \bar{X}_{0}^{\left( 3\right) }=-6\cdot 5!S^{\ast }\tilde{\eta}+12...
...mu }+4!B^{\mu \nu }\tilde{D}_{\mu \nu }-36 \tilde{\eta}_{\mu \nu }F^{\mu \nu },$ (383)

$\displaystyle \bar{X}_{1}^{\left( 3\right) }$ $\displaystyle =$ $\displaystyle 2\cdot 5!C_{\mu \nu \rho }^{\ast }\tilde{D}
_{\lambda \sigma }\et...
...\ast }t^{\ast \mu }-C_{\mu \nu }^{\ast }F^{\mu \nu }\right)
\tilde{\eta} \notag$ (384)
    $\displaystyle +6\cdot 4!C_{\mu \nu }^{\ast }\tilde{D}_{\rho \lambda }\eta ^{\mu...
...ta ^{\mu \nu \rho
}+6\cdot 4!H_{[\mu }^{\ast }\tilde{\eta}_{\nu ]}F^{\mu \nu },$ (385)

$\displaystyle \bar{X}_{2}^{\left( 3\right) }=2\cdot 5!H_{[\mu }^{\ast }C_{\nu \...
...{\rho \lambda }\eta ^{\mu \nu \rho \lambda }-5F^{\mu \nu }\tilde{\eta}\right) ,$ (386)

$\displaystyle \bar{X}_{3}^{\left( 3\right) }=2\cdot 5!H_{\mu }^{\ast }H_{\nu }^...
...{\rho }^{\ast }\tilde{D}_{\lambda \sigma }\eta ^{\mu \nu \rho \lambda \sigma }.$ (387)

The functions appearing in (146) and denoted by $ U_{p}^{\left(
i\right) }$ are of the form

$\displaystyle U_{0}^{\left( 1\right) }=-9\left( 2k_{1}\phi ^{\mu \nu }-\tfrac{k...
...\left( 2B_{\mu \nu }^{\ast }C+V_{[\mu }C_{\nu ]}-\phi _{\mu \nu }\eta \right) ,$ (388)

$\displaystyle U_{1}^{\left( 1\right) }=-9\left( k_{1}\phi ^{\mu \nu }-\tfrac{k_...
...}^{\ast }\eta C-H_{\mu }^{\ast }\left( V_{\nu }C+\eta C_{\nu }\right) \right] ,$ (389)

$\displaystyle U_{2}^{\left( 1\right) }=-9\left( k_{1}\phi ^{\mu \nu }-\tfrac{k_{2}}{20} \tilde{K}^{\mu \nu }\right) H_{\mu }^{\ast }H_{\nu }^{\ast }\eta C,$ (390)

$\displaystyle U_{0}^{\left( 2\right) }=108\left( k_{1}\phi ^{\mu \nu }-\tfrac{k...
...lde{G}} +\eta \tilde{K}_{\mu \nu }-8V_{\mu }\mathcal{\tilde{G}}_{\nu }\right) ,$ (391)

$\displaystyle U_{1}^{\left( 2\right) }$ $\displaystyle =$ $\displaystyle 18\varepsilon ^{\alpha \beta \gamma \delta
\varepsilon }\left( C_...
...^{\mu \nu
}\right) \mathcal{G}_{\alpha \beta \gamma \delta \varepsilon } \notag$ (392)
    $\displaystyle -36\varepsilon _{\rho \alpha \beta \gamma \delta }H_{\mu }^{\ast ...
...20}\tilde{K}^{\mu \rho }\right) \eta
\mathcal{G}^{\alpha \beta \gamma \delta },$ (393)

$\displaystyle U_{2}^{\left( 2\right) }=18\varepsilon _{\alpha \beta \gamma \del...
...st }H_{\nu }^{\ast }\eta \mathcal{G}^{\alpha \beta \gamma \delta \varepsilon },$ (394)

$\displaystyle U_{0}^{\left( 3\right) }=9\varepsilon _{\nu \rho \alpha \beta \ga...
...o }-\tfrac{k_{2}}{20}\tilde{K}^{\nu \rho }\right) \eta ^{\alpha \beta \gamma },$ (395)

$\displaystyle U_{1}^{\left( 3\right) }$ $\displaystyle =$ $\displaystyle \tfrac{9}{4}\varepsilon _{\alpha \beta \gamma
\delta \varepsilon ...
...lde{K}^{\mu \nu }\right) \eta ^{\alpha \beta \gamma \delta
\varepsilon } \notag$ (396)
    $\displaystyle -18\varepsilon _{\rho \beta \gamma \delta \varepsilon }H_{\mu }^{...
..._{2}}{20}\tilde{K}^{\mu \rho }\right)
\eta ^{\beta \gamma \delta \varepsilon },$ (397)

$\displaystyle U_{2}^{\left( 3\right) }=9\varepsilon _{\alpha \beta \gamma \delt...
...20}\tilde{K}^{\mu \nu }\right) \eta ^{\alpha \beta \gamma \delta \varepsilon }.$ (398)

Ashkbiz Danehkar