$Z_{0s}^{ab}$ by demanding that our final result is conformally invariant in section \[ssec:NumericsChecks\]. In this section we will only compute the structure of divergences subtracted by $Z_{0s}^{ab}$, by computing the 1-loop self-energies of the lowest total angular momentum mode of the scalar fields in the cutoff scheme.

Throughout this subsection we work in units where $\mu=R_{S^3}=1$, so the reader should not be alarmed when seeing functions which depend on dimensionful quantities. These always consistently vanish between the diagrams. As before we start with dimensional regularization before moving on to the cutoff regularization method.

### Dimensional regularization on $S^3$ {#sssec:CurvedSpaceDimReg}

To apply dimensional regularization on curved space we let our theory live on $S^3\times\mathds{R}^{d+1}$ with $d=-\epsilon$. We extend the Coulomb gauge condition to involve all components of the gauge field other than $A_0$, [^14]

$$\begin{aligned}
\label{eq:ExtendedCoulombGauge}
\partial_iA_i+\partial_aA_a = 0.\end{aligned}$$

Now $A_i$ has a different mode expansion than before since its divergence doesn’t vanish anymore. The expansions of $A_i$ and $A_a$ are,

$$\begin{aligned}
A_a &= \sum_{\alpha,j_{\alpha}\geq 0}\cal{A}^{\alpha}_aS^{\alpha},\\
A_i &= \sum_{\alpha,j_{\alpha}>0}\left(A^{\alpha}\cal{V}_i^{\alpha} +\frac{1}{{j_{\alpha}(j_{\alpha}+2)}}\partial_a\cal{A}_a^{\alpha}\partial_iS^{\alpha}\right).\end{aligned}$$

The quadratic part of the action for the gauge field and scalars are then

$$\begin{aligned}
\mathbf{S}^g_2+\mathbf{S}^s_2 &= \int dtd^dx\mathrm{tr}\left(\frac{1}{2}A^{\bar{\alpha}}(-\partial_t^2-\partial_a^2+{(j_{\alpha}+1)})A^{\alpha}+\frac{1}{2}a^{\bar{\alpha}}(-\partial_a^2+{j_{\alpha}(j_{\alpha}+2)})a^{\alpha}+\right.\nonumber\\
&~~~+\frac{1}{2}\cal{A}_a^{\bar{\alpha}}(-\partial^a_t-\partial_c^2+{j_{\alpha}(j_{\alpha}+2)})(\delta^{ab}-\frac{\partial_a\partial_b}{{j_{\alpha}(j_{\alpha}+2)}})\cal{A}^{\alpha}_b+\\
&~~~\left.+\frac{1}{2}\phi_{\bar{a}}^{\bar{\alpha}}(-\partial_t^2-\partial_a^2+{(j_{\alpha}+1)})\phi^{\alpha}_{\bar{a}}\right).\nonumber\end{aligned}$$

The additional interaction vertices are spelled out in appendix \[ssapp:1LoopCurvedDR\].

