After expanding (32) to the order *k*=3, evaluating the integrals of the form of (33), and collecting the terms that scale as $1/\text{}{\omega}_{m}^{6},$ we obtain for ${L}_{\text{eff}}^{\mathrm{(}\mathrm{4,6}\mathrm{)}}:$

$$\begin{array}{c}{L}_{\text{eff}}^{\mathrm{(}\mathrm{4,6}\mathrm{)}}=\frac{1}{12{\omega}_{m}^{6}}\mathrm{(}\frac{1}{6}\frac{45}{64}\mathrm{[}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}[[{\text{ad}}_{{L}_{0}}^{3}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}\mathrm{}\mathrm{]}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}]]\\ +\frac{1}{2}\frac{1}{32}\mathrm{[}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}[[{\text{ad}}_{{L}_{0}}^{2}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}{\text{\hspace{0.17em}ad}}_{{L}_{0}}{L}_{\text{dr}\mathrm{,0}}\mathrm{}\mathrm{]}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}]]\\ +\frac{1}{2}\frac{7}{16}\mathrm{[}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}[[{\text{ad}}_{{L}_{0}}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}{\text{\hspace{0.17em}ad}}_{{L}_{0}}^{2}{L}_{\text{dr}\mathrm{,0}}\mathrm{}\mathrm{]}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}]]\\ -\frac{1}{6}\frac{9}{8}\mathrm{[}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}[[{L}_{\text{dr}\mathrm{,0}}\mathrm{,}{\text{\hspace{0.17em}ad}}_{{L}_{0}}^{3}{L}_{\text{dr}\mathrm{,0}}\mathrm{}\mathrm{]}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}]]\\ +\frac{1}{2}\frac{27}{64}\mathrm{[}{\text{ad}}_{{L}_{0}}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}[[{\text{ad}}_{{L}_{0}}^{2}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}\mathrm{}\mathrm{]}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}]]\\ -\frac{1}{2}\frac{15}{16}\mathrm{[}{\text{ad}}_{{L}_{0}}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}[[{L}_{\text{dr}\mathrm{,0}}\mathrm{,}{\text{\hspace{0.17em}ad}}_{{L}_{0}}^{2}{L}_{\text{dr}\mathrm{,0}}\mathrm{}\mathrm{]}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}]]\\ +\frac{1}{2}\frac{21}{64}\mathrm{[}{\text{ad}}_{{L}_{0}}^{2}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}[[{\text{ad}}_{{L}_{0}}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}\mathrm{}\mathrm{]}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}]]\\ -\frac{1}{2}\frac{33}{32}\mathrm{[}{\text{ad}}_{{L}_{0}}^{2}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}[[{L}_{\text{dr}\mathrm{,0}}\mathrm{,}{\text{\hspace{0.17em}ad}}_{{L}_{0}}{L}_{\text{dr}\mathrm{,0}}\mathrm{}\mathrm{]}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}]]\\ +\frac{1}{6}\frac{45}{64}[[{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}\mathrm{[}{\text{ad}}_{{L}_{0}}^{3}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}]]\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}\mathrm{]}\\ +\frac{1}{2}\frac{1}{32}[[{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}\mathrm{[}{\text{ad}}_{{L}_{0}}^{2}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}{\text{\hspace{0.17em}ad}}_{{L}_{0}}{L}_{\text{dr}\mathrm{,0}}]]\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}\mathrm{]}\\ +\frac{1}{2}\frac{7}{16}[[{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}\mathrm{[}{\text{ad}}_{{L}_{0}}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}{\text{\hspace{0.17em}ad}}_{{L}_{0}}^{2}{L}_{\text{dr}\mathrm{,0}}]]\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}\mathrm{]}\\ -\frac{1}{6}\frac{9}{8}[[{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}\mathrm{[}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}{\text{\hspace{0.17em}ad}}_{{L}_{0}}^{3}{L}_{\text{dr}\mathrm{,0}}]]\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}\mathrm{]}\\ +\frac{1}{2}\frac{27}{64}[[{\text{ad}}_{{L}_{0}}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}\mathrm{[}{\text{ad}}_{{L}_{0}}^{2}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}]]\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}\mathrm{]}\\ -\frac{1}{2}\frac{15}{16}[[{\text{ad}}_{{L}_{0}}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}\mathrm{[}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}{\text{\hspace{0.17em}ad}}_{{L}_{0}}^{2}{L}_{\text{dr}\mathrm{,0}}]]\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}\mathrm{]}\\ +\frac{1}{2}\frac{21}{64}[[{\text{ad}}_{{L}_{0}}^{2}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}\mathrm{[}{\text{ad}}_{{L}_{0}}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}]]\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}\mathrm{]}\\ -\frac{1}{2}\frac{33}{32}[[{\text{ad}}_{{L}_{0}}^{2}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}\mathrm{[}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}{\text{\hspace{0.17em}ad}}_{{L}_{0}}{L}_{\text{dr}\mathrm{,0}}]]\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}\mathrm{]}\\ -\frac{1}{2}\frac{33}{32}[[{\text{ad}}_{{L}_{0}}^{2}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}\mathrm{}\mathrm{]}\mathrm{,}\text{\hspace{0.17em}}\mathrm{[}{\text{ad}}_{{L}_{0}}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}]]\\ +\frac{1}{2}\frac{1}{32}[[{L}_{\text{dr}\mathrm{,0}}\mathrm{,}{\text{\hspace{0.17em}ad}}_{{L}_{0}}^{2}{L}_{\text{dr}\mathrm{,0}}\mathrm{}\mathrm{]}\mathrm{,}\text{\hspace{0.17em}}\mathrm{[}{\text{ad}}_{{L}_{0}}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}]]\\ -\frac{1}{2}\frac{15}{16}[[{\text{ad}}_{{L}_{0}}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}\mathrm{}\mathrm{]}\mathrm{,}\text{\hspace{0.17em}}\mathrm{[}{\text{ad}}_{{L}_{0}}^{2}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}]]\\ +\frac{1}{2}\frac{7}{16}[[{L}_{\text{dr}\mathrm{,0}}\mathrm{,}{\text{\hspace{0.17em}ad}}_{{L}_{0}}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}\mathrm{[}{\text{ad}}_{{L}_{0}}^{2}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}]]\\ +\frac{1}{2}\frac{21}{64}[[{\text{ad}}_{{L}_{0}}^{2}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}\mathrm{}\mathrm{]}\mathrm{,}\text{\hspace{0.17em}}\mathrm{[}{\text{ad}}_{{L}_{0}}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}]]\\ +\frac{1}{2}\frac{7}{16}[[{L}_{\text{dr}\mathrm{,0}}\mathrm{,}{\text{\hspace{0.17em}ad}}_{{L}_{0}}^{2}{L}_{\text{dr}\mathrm{,0}}\mathrm{}\mathrm{]}\mathrm{,}\text{\hspace{0.17em}}\mathrm{[}{\text{ad}}_{{L}_{0}}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}]]\\ +\frac{1}{2}\frac{27}{64}[[{\text{ad}}_{{L}_{0}}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}\mathrm{}\mathrm{]}\mathrm{,}\text{\hspace{0.17em}}\mathrm{[}{\text{ad}}_{{L}_{0}}^{2}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}]]\\ +\frac{1}{2}\frac{1}{32}[[{L}_{\text{dr}\mathrm{,0}}\mathrm{,}{\text{\hspace{0.17em}ad}}_{{L}_{0}}{L}_{\text{dr}\mathrm{,0}}\mathrm{}\mathrm{]}\mathrm{,}\text{\hspace{0.17em}}\mathrm{[}{\text{ad}}_{{L}_{0}}^{2}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}]]\mathrm{)}\end{array}$$(C1)

To simplify this expression, we first combine the terms that are related by commutation. In addition, we use that

$$\mathrm{[}{\text{ad}}_{{L}_{0}}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}[[{\text{ad}}_{{L}_{0}}^{2}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}\mathrm{}\mathrm{]}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}]]$$(C2)

$$+\mathrm{[}{\text{ad}}_{{L}_{0}}^{2}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}[[{\text{ad}}_{{L}_{0}}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}\mathrm{}\mathrm{]}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}]]$$(C3)

$$=[[{\text{ad}}_{{L}_{0}}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}\mathrm{[}{\text{ad}}_{{L}_{0}}^{2}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}]]\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}\mathrm{]}$$(C4)

$$+[[{\text{ad}}_{{L}_{0}}^{2}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}\mathrm{[}{\text{ad}}_{{L}_{0}}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}]]\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}\mathrm{]}$$(C5)

and

$$\mathrm{[}{\text{ad}}_{{L}_{0}}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}\mathrm{[}{\text{ad}}_{{L}_{0}}^{2}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}]]$$(C6)

$$=\mathrm{[}{\text{ad}}_{{L}_{0}}^{2}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}\mathrm{[}{\text{ad}}_{{L}_{0}}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}{L}_{\text{dr}\mathrm{,0}}]]$$(C7)

$$-\mathrm{[}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}\text{\hspace{0.17em}}\mathrm{[}{\text{ad}}_{{L}_{0}}{L}_{\text{dr}\mathrm{,0}}\mathrm{,}{\text{\hspace{0.17em}ad}}_{{L}_{0}}^{2}{L}_{\text{dr}\mathrm{,0}}]]$$(C8)

With these identities, we simplify the expression to the form given in (34).

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