interaction terms we also need to compute various integrals involving three such spherical functions [^9],

$$\begin{aligned}
\label{eq:3SphericalIntegrals}
B^{\alpha\beta\gamma}  &= \int_{S^3}\cal{S}^{\alpha}\cal{S}^{\beta}\cal{S}^{\gamma} &,
C^{\alpha\beta\gamma}  &= \int_{S^3}\cal{S}^{\alpha}\vec{\cal{V}}^{\beta}\cdot\vec{\nabla}\cal{S}^{\gamma} &,
D^{\alpha\beta\gamma}  &= \int_{S^3}\vec{\cal{V}}^{\alpha}\cdot\vec{\cal{V}}^{\beta}\cal{S}^{\gamma},\nonumber\\
E^{\alpha\beta\gamma}  &= \int_{S^3}\vec{\cal{V}}^{\alpha}\cdot(\vec{\cal{V}}^{\beta}\times\vec{\cal{V}}^{\gamma}) &,
F^{\alpha\beta\gamma}  &= \int_{S^3}\cal{Y}^{{\alpha}\dag}\cal{Y}^{\beta}\cal{S}^{\gamma} &,
G^{\alpha\beta\gamma}  &= \int_{S^3}\cal{Y}^{{\alpha}\dag}\sigma^{\mu}\cal{Y}^{\beta}\vec{\cal{V}}^{\gamma}_{\mu},\nonumber\\
H^{\alpha\beta\gamma}  &= \int_{S^3}\cal{Y}^{\alpha}\varepsilon\cal{Y}^{\beta}\cal{S}^{\gamma} &,
J^{\alpha\beta\gamma}  &= \int_{S^3}\cal{Y}^{{\alpha}\dag}\varepsilon\cal{Y}^{{\beta}\dag}\cal{S}^{\gamma} &,
K^{\alpha\beta\gamma\delta} &= \int_{S^3}\cal{S}^{\alpha}\cal{S}^{\beta}\cal{S}^{\gamma}\cal{S}^{\delta}.\end{aligned}$$

Properties of $S^3$ spherical harmonics, and the solutions of the above integrals can be found in appendix \[app:harmonics\].

The action in terms of the above expansions and integrals is written in (\[eq:QuadraticAction\])-(\[eq:QuarticAction\]). Since the fields $a^{\alpha}$ and $c^{\alpha}$ appear in the action only quadratically, it is convenient to integrate them out first. This was done in the appendix \[sapp:IntegrateOut\] and the resulting action after integrating out is,

$$\begin{aligned}
\mathbf{S}_2 &= \int_0^{\beta} dt \mathrm{tr}\left\{
\frac{1}{2}A^{\bar{\alpha}}(-D_t^2+{(j_{\alpha}+1)}^2)A^{\alpha}+\frac{1}{2}\phi_a^{\bar{\alpha}}(-D_t^2+{(j_{\alpha}+1)}^2)\phi_a^{\alpha}+\right.\nonumber\\
&~\left.+i\psi^{\dag \alpha I}(D_t+{\epsilon_{\alpha}(j_{\alpha}+\frac{1}{2})})\psi^{\alpha}_I\right\},\label{eq:S2}\\
\mathbf{S}_3 &= ig\int_0^{\beta} dt \mathrm{tr}\bigg\{\epsilon_{\alpha}(j_{\alpha}+1)E^{\alpha\beta\gamma}A^{\alpha}A^{\beta}A^{\gamma}-C^{\alpha\beta\gamma}[A^{\beta},\phi_a^{\alpha}]\phi_a^{\gamma}-iG^{\alpha\beta\gamma}\psi^{\dag \alpha I}[A^{\gamma},\psi^{\beta}_I]-\nonumber\\
&~\left.-\frac{i}{2}F^{\bar{\alpha}\beta\gamma}\psi_I^{\alpha}(\rho^{a\dag})^{IJ}[\phi_a^{\gamma},\psi_J^{\beta}]+\frac{i}{2}F^{\alpha\bar{\beta}\gamma}\psi^{\dag I\alpha}\rho^a_{IJ}[\phi_a^{\gamma},\psi^{\dag J\beta}]\right\},\label{eq:S3}\\
\mathbf{S}_4 &= -\frac{1}{2}g^2\int_0^{\beta} dt \mathrm{tr}\left\{
D^{\alpha\gamma\lambda}B^{\bar{\lambda}\beta\delta}[A^{\alpha},\phi_a^{\beta}][A^{\gamma},\phi_a^{\delta}]+\frac{1}{2}B^{\alpha\beta\bar{\lambda}}B^{\lambda\gamma\delta}\mathbf{Q}^{abcd}\phi_a^{\alpha}\phi_b^{\beta}\phi_c^{\gamma}\phi_d^{\delta}-\right.\nonumber\\
&-\frac{D^{\alpha \beta \lambda}D^{\gamma \delta \bar{\lambda}}}{{j_{\lambda}(j_{\lambda}+2)}}[A^\alpha,D_t A^\beta][A^{\gamma},D_t A^{\delta}]-\frac{B^{\alpha \beta \lambda}B^{\gamma \delta \bar{\lambda} }}{{j_{\lambda}(j_{\lambda}+2)}}[\phi^\alpha_a,D_t \phi^\beta_a][\phi^{\gamma}_{b},D_t \phi_{b}^{\delta}]+\nonumber\\
&\left.+\frac{F^{\alpha \beta \lambda}F^{\gamma \delta \bar{\lambda}}}{{j_{\lambda}(j_{\lambda}+2)}}\{\psi^\beta_I,\psi^{\dagger \alpha I}\}\{\psi_J^{\delta}, \psi^{\dagger \gamma J}\}\right\},\label{eq:S4}\end{aligned}$$

where $D_t\equiv \partial_t-i[\alpha,*]$. The other effective vertices which arise from integrating out $a^{\alpha}$ and $c^{\alpha}$ do not contribute to our computation as explained in appendix \[sapp:IntegrateOut\].

Propagators {#ssec:Propagators}
-----------

The propagators of the 